[
  {
    "path": ".gitattributes",
    "content": "# Auto detect text files and perform LF normalization\n* text=auto\n\n# Custom for Visual Studio\n*.cs     diff=csharp\n*.sln    merge=union\n*.csproj merge=union\n*.vbproj merge=union\n*.fsproj merge=union\n*.dbproj merge=union\n\n# Standard to msysgit\n*.doc\t diff=astextplain\n*.DOC\t diff=astextplain\n*.docx diff=astextplain\n*.DOCX diff=astextplain\n*.dot  diff=astextplain\n*.DOT  diff=astextplain\n*.pdf  diff=astextplain\n*.PDF\t diff=astextplain\n*.rtf\t diff=astextplain\n*.RTF\t diff=astextplain\n"
  },
  {
    "path": ".gitignore",
    "content": "#################\n## Eclipse\n#################\n\n*.pydevproject\n.project\n.metadata\nbin/\ntmp/\n*.tmp\n*.bak\n*.swp\n*~.nib\nlocal.properties\n.classpath\n.settings/\n.loadpath\n\n# External tool builders\n.externalToolBuilders/\n\n# Locally stored \"Eclipse launch configurations\"\n*.launch\n\n# CDT-specific\n.cproject\n\n# PDT-specific\n.buildpath\n\n\n#################\n## Visual Studio\n#################\n\n## Ignore Visual Studio temporary files, build results, and\n## files generated by popular Visual Studio add-ons.\n\n# User-specific files\n*.suo\n*.user\n*.sln.docstates\n\n# Build results\n\n[Dd]ebug/\n[Rr]elease/\nx64/\nbuild/\n[Bb]in/\n[Oo]bj/\n\n# MSTest test Results\n[Tt]est[Rr]esult*/\n[Bb]uild[Ll]og.*\n\n*_i.c\n*_p.c\n*.ilk\n*.meta\n*.obj\n*.pch\n*.pdb\n*.pgc\n*.pgd\n*.rsp\n*.sbr\n*.tlb\n*.tli\n*.tlh\n*.tmp\n*.tmp_proj\n*.log\n*.vspscc\n*.vssscc\n.builds\n*.pidb\n*.log\n*.scc\n\n# Visual C++ cache files\nipch/\n*.aps\n*.ncb\n*.opensdf\n*.sdf\n*.cachefile\n\n# Visual Studio profiler\n*.psess\n*.vsp\n*.vspx\n\n# Guidance Automation Toolkit\n*.gpState\n\n# ReSharper is a .NET coding add-in\n_ReSharper*/\n*.[Rr]e[Ss]harper\n\n# TeamCity is a build add-in\n_TeamCity*\n\n# DotCover is a Code Coverage Tool\n*.dotCover\n\n# NCrunch\n*.ncrunch*\n.*crunch*.local.xml\n\n# Installshield output folder\n[Ee]xpress/\n\n# DocProject is a documentation generator add-in\nDocProject/buildhelp/\nDocProject/Help/*.HxT\nDocProject/Help/*.HxC\nDocProject/Help/*.hhc\nDocProject/Help/*.hhk\nDocProject/Help/*.hhp\nDocProject/Help/Html2\nDocProject/Help/html\n\n# Click-Once directory\npublish/\n\n# Publish Web Output\n*.Publish.xml\n*.pubxml\n\n# NuGet Packages Directory\n## TODO: If you have NuGet Package Restore enabled, uncomment the next line\n#packages/\n\n# Windows Azure Build Output\ncsx\n*.build.csdef\n\n# Windows Store app package directory\nAppPackages/\n\n# Others\nsql/\n*.Cache\nClientBin/\n[Ss]tyle[Cc]op.*\n~$*\n*~\n*.dbmdl\n*.[Pp]ublish.xml\n*.pfx\n*.publishsettings\n\n# RIA/Silverlight projects\nGenerated_Code/\n\n# Backup & report files from converting an old project file to a newer\n# Visual Studio version. Backup files are not needed, because we have git ;-)\n_UpgradeReport_Files/\nBackup*/\nUpgradeLog*.XML\nUpgradeLog*.htm\n\n# SQL Server files\nApp_Data/*.mdf\nApp_Data/*.ldf\n\n#############\n## Windows detritus\n#############\n\n# Windows image file caches\nThumbs.db\nehthumbs.db\n\n# Folder config file\nDesktop.ini\n\n# Recycle Bin used on file shares\n$RECYCLE.BIN/\n\n# Mac crap\n.DS_Store\n\n\n#############\n## Python\n#############\n\n*.py[co]\n\n# Packages\n*.egg\n*.egg-info\ndist/\nbuild/\neggs/\nparts/\nvar/\nsdist/\ndevelop-eggs/\n.installed.cfg\n\n# Installer logs\npip-log.txt\n\n# Unit test / coverage reports\n.coverage\n.tox\n\n#Translations\n*.mo\n\n#Mr Developer\n.mr.developer.cfg\n\n\n# PyEx stuff\n\nse_normalized/\nlogfile\nstdout\n*.out\n*.py#\n#stdout#\n#stdout.old#\n#tmp#\n*.dot\ntmp.txt\n\n#############\n## Vagrant\n#############\n.vagrant\n\n#############\n## PyCharm\n#############\n.idea\n"
  },
  {
    "path": "README.md",
    "content": "PyExZ3\n======\n\n### Python Exploration with Z3\n\nThis code is a substantial rewrite of the NICE project's\n(http://code.google.com/p/nice-of/) symbolic execution engine for\nPython, now using the Z3 theorem prover (http://z3.codeplex.com). We have\nremoved the NICE-specific dependences, platform-specific code, and\nmade various improvements, documented below, so it can be used\nby anyone wanting to experiment with dynamic symbolic execution.\n\nThe paper [Deconstructing Dynamic Symbolic Execution](http://research.microsoft.com/apps/pubs/?id=233035)\nexplains the basic ideas behind dynamic symbolic execution and the architecture\nof the PyExZ3 tool (as of git tag v1.0).  Bruni, Disney and Flanagan wrote about \nencoding symbolic execution for Python in Python in the same way in their 2008 paper \n[A Peer Architecture for Lightweight Symbolic Execution](http://hoheinzollern.files.wordpress.com/2008/04/seer1.pdf) - they use proxies rather than multiple inheritance for representing symbolic versions of Python types. \n\nIn the limit, **PyExZ3** attempts to *explore* all the feasible paths in a\nPython function by:\n- executing the function on a concrete input to trace a path through the control flow of the function;\n- symbolic executing the path to determine how its conditions depend on the function's input parameters;\n- generating new values for the parameters to drive the function to yet uncovered paths, using Z3.  \n\nFor small programs without loops or recursion, \n**PyExZ3** may be able to explore all feasible paths.\n\nA novel aspect of the rewrite is to rely solely on Python's operator\noverloading to accomplish all the interception needed for symbolic\nexecution; no AST rewriting or bytecode instrumentation is required,\nThis significantly improves the robustness and portability of **PyExZ3**, \nas well as reducing its size.\n\n### Setup instructions:\n\n- Make sure that you use Python 32-bit (64-bit) if-and-only-if you use the Z3 32-bit (64-bit) binaries. \nTesting so far has been on Python 3.2.3 and 32-bit.\n- Install Python 3.2.3 (https://www.python.org/download/releases/3.2.3/)\n- Install the latest \"unstable\" release of Z3 to directory Z3HOME from http://z3.codeplex.com/releases\n(click on the \"Planned\" link on the right to get the latest binaries for all platforms)\n- Add Z3HOME\\bin to PATH and PYTHONPATH\n- MacOS: setup.sh for Homebrew default locations for Python and Z3; see end for MacOS specific instructions\n- Optional:\n-- install GraphViz utilities (http://graphviz.org/)\n\n### Check that everything works:\n\n- `python run_tests.py test` should pass all tests\n\n- `python pyexz3.py test\\FILE.py` to run a single test from the test directory\n\n### Usage of PyExZ3\n\n- **Basic usage**: give a Python file `FILE.py` as input. By default, `pyexz3` expects `FILE.py` \nto contain a function named `FILE` where symbolic execution will start:\n\n  - `pyexz3 FILE.py`\n\n- **Starting function**: You can override the default starting function with `--start MAIN`,\nwhere `MAIN` is the name of a  function in `FILE`: \n\n  - pyexz3 `--start=MAIN` FILE.py\n\n- **Bounding the number of iterations** of the path exploration is essential when\nanalyzing functions with loops and/or recursion. Specify a bound using the `max-iters` flag:\n\n  - pyexz3 `--max-iters=42` FILE.py\n\n- **Arguments to starting function**: by default, pyexz3 associates a symbolic integer\n(with initial value 0) for each parameter of the starting function. Import from\n`symbolic.args` to get the `@concrete` and `@symbolic` decorators that let you override\nthe defaults on the starting function:\n\n```\nfrom symbolic.args import *\n\n@concrete(a=1,b=2)\n@symbolic(c=3)\ndef startingfun(a,b,c,d):\n    ...\n```\n  \nThe `@concrete` decorator declares that a parameter will not be treated symbolically and\nprovides an initial value for the parameter.\nThe `@symbolic` decorator declares that a parameter will be treated symbolically - the type \nof the associated initial value for the argument will be used to determine the proper symbolic \ntype  (if one exists).   In the above example, parameters `a` and `b` are treated concretely\nand will have initial values `1` and `2` (for all paths explored), and parameter `c` will \nbe treated as a symbolic integer input with the initial value `3` (its value can change after\nfirst path has been explored). Since parameter `d` is not specified, it will be treated as a symbolic \ninteger input with the initial value 0:\n\n- **Output**: `pyexz3` prints the list of generated inputs and corresponding observed \nreturn values to standard out; the lists of generated inputs and the corresponding return values are\nreturned by the exploration engine to `pyexz3` where they can be used for other \npurposes, as described below.\n\n- **Expected result functions** are used for testing of `pyexz3`. If the `FILE.py` contains a function named `expected_result` then after path exploration is complete, the list of return values will be compared against the list \nreturned by `expected_result`. More precisely, the two lists are converted into bags and the bags compared for equality. If a function named `expected_result_set` is present instead, the list are converted into sets and the sets are\ncompared for equality.  List equality is too strong a criteria for testing, since small changes to programs can lead to paths being explored in different orders. \n\n- **Import behavior**: the location of the `FILE.py` is added to the import path so that all imports in `FILE.py` \nrelative to that file will work.\n\n- **Other options**\n  - `--graph=DOTFILE`\n  - `--log=LOGFILE`\n\n### MacOS specific\n\n1. Grab yourself a Brew at http://brew.sh/\n2. Get the newest python or python3 version: `brew install python`\n3. Have the system prefer brew python over system python: `echo export PATH='/usr/local/bin:$PATH' >> ~/.bash_profile`  - \n4. Get z3: `brew install homebrew/science/z3`\n5. Clone this repository: `git clone https://github.com/thomasjball/PyExZ3.git` \n6. Set the PATH: `. PyExZ3/setup.sh`  (do not run the setup script in a subshell `./ PyExZ3/`)\n\n### Vagrant specific\n\n[Vagrant](http://www.vagrantup.com/) is a cross-platform tool to manage\nvirtualized development environments. Vagrant runs on Windows, OS X, and\nLinux and can manage virtual machines running on VirtualBox, VMware,\nDocker, and Hyper-V.\n\n1. [Download Vagrant](http://www.vagrantup.com/downloads.html).\n2. Install [VirtualBox](https://www.virtualbox.org/) or configure an\n[alternative\nprovider](http://docs.vagrantup.com/v2/providers/index.html).\n3. Run `vagrant up` from the PyExZ3 directory. The Vagrantfile in the\nrepository tells Vagrant to download a Debian base image, launch it with\nthe default provider (VirtualBox), and run the script `vagrant.sh` to\nprovision the machine.\n4. Once the provisioning is done you can SSH into the machine using\n`vagrant ssh` and PyExZ3 is ready to run. Please note that the\nprovisioning takes a while as Git is compiled from source as Debian's\nGit is incompatible with [CodePlex](http://www.codeplex.com/) where Z3\nis hosted.\n\n### CVC SMT Solver\n\nBy default PyExZ3 uses the Z3 to solve path predicates. Optionally, the \n[CVC SMT](http://cvc4.cs.nyu.edu/web/) solver can be enabled with the \n`--cvc` argument. While the two solvers offer a similar feature set, the \nintegration of CVC differs from Z3 in a number of ways. Most \npredominately, the CVC integration uses an unbounded rational number \nrepresentation for Python numbers, converting to bit vectors only for \nbitwise operations. The Z3 integration uses bounded bit vectors for all \nnumbers. For programs that use any significant number of bitwise \noperations, the default Z3-based configuration is strongly recommended. \nAdditionally, CVC does not support generating models for non-linear \nrelationships causing a few of the included PyExZ3 test cases to fail \nwith a `LogicException`.\n"
  },
  {
    "path": "TODO.md",
    "content": "TODO List\n=========\n\n- add basic support for SymbolicDictionary\n  - \n- need to capture exceptions thrown by code under test as test results\n- interesting question arises about re-initialization of input arguments\nby ExplorationEngine and re-import of module under test in the face of\nmutable initial objects - we want the re-import to be done before the \nre-initialization, but that's not how it currently works. Easiest thing\nto do is only allow empty dictionary to be specified in @symbolic\n- check input/output behavior separately from sym_exe\n- use consist case on names (Caml or C, choose one)\n\n\n"
  },
  {
    "path": "Vagrantfile",
    "content": "# -*- mode: ruby -*-\n# vi: set ft=ruby :\n\nVAGRANTFILE_API_VERSION = \"2\"\n\nVagrant.configure(VAGRANTFILE_API_VERSION) do |config|\n\n    config.vm.define \"linux\", primary: true do |v|\n        v.vm.provision \"shell\", path: \"vagrant.sh\"        \n        v.vm.box = \"chef/debian-7.4\"\n    end\n\n    config.vm.provider \"virtualbox\" do |v|\n        v.memory = 1024\n    end\n\nend\n"
  },
  {
    "path": "copyright.txt",
    "content": "# Files that mention copyright.txt were derived from the NICE project\n#\n# Copyright (c) 2011, EPFL (Ecole Politechnique Federale de Lausanne)\n# All rights reserved.\n#\n# Created by Marco Canini, Daniele Venzano, Dejan Kostic, Jennifer Rexford\n#\n# Updated by Thonas Ball (2014)\n#\n# Redistribution and use in source and binary forms, with or without\n# modification, are permitted provided that the following conditions are met:\n#\n#   -  Redistributions of source code must retain the above copyright notice,\n#      this list of conditions and the following disclaimer.\n#   -  Redistributions in binary form must reproduce the above copyright notice,\n#      this list of conditions and the following disclaimer in the documentation\n#      and/or other materials provided with the distribution.\n#   -  Neither the names of the contributors, nor their associated universities or\n#      organizations may be used to endorse or promote products derived from this\n#      software without specific prior written permission.\n#\n# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS \"AS IS\" AND ANY\n# EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES\n# OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT\n# SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,\n# SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,\n# PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS\n# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT\n# LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE\n# OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.\n#\n"
  },
  {
    "path": "fail/arrayindex.py",
    "content": "A = [0, 1]\n\n# the index operation A[i] uses the underlying runtime representation\n# of the Python int, so we have no way of capturing this \"conditional\n# through lookup\" operation via inheritance from int, as done with \n# SymbolicInteger\n\n# see test\\arrayindex2.py for the rewriting we would need to do\n# to make the lookup explicit (greatly expanding the search space)\n\ndef arrayindex(i):\n  if A[i]:\n    return A[i]\n  else:\n    return \"OTHER\"\n\ndef expected_result():\n  return [ 1, \"OTHER\" ]\n"
  },
  {
    "path": "fail/dictbool.py",
    "content": "from symbolic.args import *\n\n@symbolic(d={})\ndef dictbool(d):\n    x = d or {}\n    if x == {}:\n        return 0\n    return 1\n\ndef expected_result():\n\treturn [0, 1]\n    \n"
  },
  {
    "path": "fail/divzero.py",
    "content": "# this one fails because we always start with zero for SymbolicIntegers\n# we should have a few seed values to avoid this.\n\ndef divzero(in1,in2):\n  try:\n    if in1 / in2 >= 0:\n        return 1\n    elif in1 / in2 < 0:\n        return 2\n    return 0\n  except:\n    return \"DIVZERO\"\n    \ndef expected_result():\n    return [0,1,2,\"DIVZERO\"]"
  },
  {
    "path": "fail/git.py",
    "content": "from symbolic.args import *\n\n@symbolic(a=0xdeaddeaddeaddead,b=0xbeefbeefbeefbeef)\ndef git(a,b):\n    i=0;\n    passkeyn=[a,b]\n\n    expandedkey=[]\n    while i<6:\n#    while i<2:  # WORKS\n        expandedkey=expandedkey+passkeyn\n        i=i+1\n        v1=passkeyn[0]\n        v2=passkeyn[1]\n        v3=(v1>>0x30)|(((v1>>0x20)&0xffff)<<0x10)|(((v1>>0x10)&0xffff)<<0x20)|((v1&0xffff)<<0x30)\n        v4=(v2>>0x30)|(((v2>>0x20)&0xffff)<<0x10)|(((v2>>0x10)&0xffff)<<0x20)|((v2&0xffff)<<0x30)\n        v5 = ((v4 & 0xFFFFFF8000000000) >> 39) | ((v3 << 25)&0xffffffffffffffff);\n        v6 = ((v3 & 0xFFFFFF8000000000) >> 39) | ((v4 << 25)&0xffffffffffffffff);\n        v1=(v5>>0x30)|(((v5>>0x20)&0xffff)<<0x10)|(((v5>>0x10)&0xffff)<<0x20)|(((v4 & 0xFFFFFF8000000000) >> 39)<<0x30)\n        v2=(v6>>0x30)|(((v6>>32)&0xffff)<<0x10)|(((v6>>0x10)&0xffff)<<0x20)|(((v3 & 0xFFFFFF8000000000) >> 39)<<0x30)\n        passkeyn=[v1,v2]\n    \n    expandedkey=expandedkey+passkeyn\n    \n    print(expandedkey)\n    if expandedkey==[16045725885737590445, 13758425323549998831, 7044313620519854103485, 8215411798635391606653, 8245388070021240879798, 3384596836810669685695, 9287860625795901259255, 4527376222128629444341, 4093654381503457390331, 4647353382867023162077, 8665057901351565392853, 8816957627389395711965, 6783497306152038280055, 9291067819851303074799]:\n#    WORKS\n#    if expandedkey==[16045725885737590445, 13758425323549998831, 7044313620519854103485, 8215411798635391606653, 8245388070021240879798, 3384596836810669685695]:\n        print(\"HERE\")\n        return 1\n    else:\n        print(\"THERE\")\n        return 2\n\nif __name__ == \"__main__\":\n    git(0xdeaddeaddeaddead,0xbeefbeefbeefbeef)\n"
  },
  {
    "path": "fail/pow.py",
    "content": "# this one fails because we treat the operator ** concretely rather than symbolically\n# so that the concrete value 0**2 is substituted in place of x**2.\n# As a result, we never get to calling the theorem prover\n\ndef pow(x):\n  if 4 == x**2:\n    return \"POW\"\n  else:\n    return \"OTHER\"\n\ndef expected_result():\n  return [ \"OTHER\", \"POW\" ]\n"
  },
  {
    "path": "fail/sqrttest.py",
    "content": "from math import sqrt\n\n# sqrt is handled concretely, just as with pow (**)\n\ndef sqrttest(in1):\n    if sqrt(in1) == 0:\n        return 1\n    elif sqrt(in1) > 0:\n        return 2\n    return 0\n\ndef expected_result():\n    return [1,2]"
  },
  {
    "path": "marktoberdorf_paper/DSE/DSE.mdk",
    "content": "Title       : Deconstructing Dynamic Symbolic Execution\nAuthor      : Thomas Ball, Jakub Daniel\nAffiliation : Microsoft Research, Charles University\nEmail       : tball@microsoft.com, jakub.daniel@d3s.mff.cuni.cz\nDoc Class   : IOS-Book-Article.cls\n\nColorizer   : javascript\nColorizer   : cpp\nColorizer   : cpp2\nBib style   : plainnat\nBibliography: dse\nHeading base: 2\n.rulename   : replace=/(.*)/(\\1)/ font-variant=small-caps\n~MathPre,.math-inline,.math-display: replace=/=\\^=/{~\\buildrel\\triangle\\over=~}/g\n~MathPre,.math-inline,.math-display,~Equation: replace=/\\bSt\\b/\\sigma/g\n~Pre,~Code: language=python\n.code2    : language=cpp2\n.language-cpp2     : replace=\"/\\b([TR])(')?(_.)?/\\(T|$\\1\\2\\3$\\)/g\"\n\n~ HtmlOnly\n[TITLE]\n~\n\n~ TexRaw\n\\begin{frontmatter}              % The preamble begins here.\n%\\pretitle{Pretitle}\n\\title{Deconstructing Dynamic Symbolic Execution}\n%\\runningtitle{IOS Press Style Sample}\n%\\subtitle{Subtitle}\n\\author[A]{\\fnms{Thomas} \\snm{Ball}}\nand \n\\author[B]{\\fnms{Jakub} \\snm{Daniel}}\n\n\\runningauthor{Thomas Ball et al.}\n\\address[A]{Microsoft Research}\n\\address[B]{Charles University}\n~\n\n~ Abstract\nDynamic symbolic execution (DSE) is a well-known technique\nfor automatically generating tests to achieve higher levels\nof coverage in a program. Two keys ideas of DSE are\nto: (1) seed symbolic execution by executing a program on an\ninitial input; (2) use concrete values from the program\nexecution in place of symbolic expressions whenever symbolic\nreasoning is hard or not desired. We describe\nDSE for a simple core language and then present\na minimalist implementation of DSE for Python (in Python) \nthat follows this basic recipe. The code is available \nat https://www.github.com/thomasjball/PyExZ3/ (tagged \"v1.0\") \nand has been designed to make it easy to experiment with and\nextend. \n~\n\n~ TexRaw\n\\begin{keyword}\nSymbolic Execution, Automatic Test Generation, White-box Testing, Automated \nTheorem Provers\n\\end{keyword}\n\\end{frontmatter}\n\\thispagestyle{empty}\n\\pagestyle{empty}\n~\n\n\n[PyExZ3]: https://github.com/thomasjball/PyExZ3/ \n[Z3]: http://z3.codeplex.org/\n[MRO]: https://www.python.org/download/releases/2.3/mro/ \n\n# Introduction     { #sec-intro }\n\n~ MathDefs\n\\defcommand{\\mathkw}[1]{\\textbf{#1}}\n~\n\n<!-- should we motivate more? -- this dives in deep very quickly -->\n\nStatic, path-based symbolic execution explores one control-flow path\nat a time through a (sequential) program $P$, using an automated theorem\nprover (ATP) to determine if the current path $p$ is feasible [@King76;@Clarke76]. \nIdeally, symbolic execution of a path $p$ through program\n$P$ yields a logic formula $\\phi_p$ that describes the set of inputs $I$ (possibly empty)\nto program $P$ such that for any $i \\in I$, the execution $P(i)$ follows path $p$. \n\nIf the formula $\\phi_p$ is unsatisfiable then $I$ is empty and so path $p$ is not feasible; \nif the formula is satisfiable then $I$ is not empty and so path $p$ is feasible.\nIn this case, a model of $\\phi_p$ provides a witness $i \\in I$.  Thus, a model-generating ATP\ncan be used in conjunction with\nsymbolic execution to automatically generate tests to cover paths\nin a program. Combined with a search strategy, one gets, in the limit,\nan exhaustive white-box testing procedure, for which there are many\napplications [@CadarGPDE06; @GodefroidLM12; @CadarS13]. \n\nThe formula $\\phi_p$ is called a *path-condition* of the path $p$. \nWe will see that a given path $p$ can induce many different path-conditions.\nA path-condition $\\psi_p$ for path $p$ is *sound* if \nevery input assignment satisfying $\\psi_p$ defines an\nexecution of program $P$ that follows path $p$ [@Godefroid11]. \nBy its definition, the formula  $\\phi_p$ is sound and the best representation of $p$\n(as for all sound path-conditions $\\psi_p$, we have that $\\psi_p \\implies \\phi_p$).\nIn practice, we attempt to compute sound under-approximations of $\\phi_p$ \nsuch as $\\psi_p$. However, we also find it necessary (and useful) to \ncompute unsound path-conditions.\n\nA path-condition can be translated into the input\nlanguage of an ATP, such as [Z3][Z3][@deMouraB08], which provides an answer\nof \"unsatisfiable\", \"satisfiable\" or \"unknown\", due to theoretical or practical\nlimitations in automatically deciding satisfiability of various logics.\nIn the case that the ATP is able to prove \"satisfiable\" we can query it for \nsatisfying model in order to generate test inputs. A path-condition \nfor $p$ can be thought of as function from a\nprogram's primary inputs to a Boolean output representing whether\nor not $p$ is executed under a given input. Thus, we are asking\nthe ATP to invert a function when we ask it to decide\nthe satisfiability/unsatisfiability of a path-condition.\n\nThe static translation of a path $p$ through a program $P$ into \nthe most precise path-condition $\\phi_p$ is not a simple task, as \nprogramming languages and their semantics are very complex.\nCompletely characterizing the set of inputs $I$ that follow\npath $p$ means providing a symbolic interpretation of \nevery operation in the language so that the\nATP can reason about it. For example, consider a method call in Python. \nPython's algorithm for method resolution order (see [MRO])\ndepends on the inheritance hierarchy of the program, a directed, \nacyclic graph that can evolve during program execution. Symbolically\nencoding Python's method resolution order is possible but non-trivial.\nThere are other reasons it is hard or undesirable to symbolically \nexecute various operations, as will be explained in detail later.\n\n## Dynamic symbolic execution\n\n*Dynamic* symbolic execution (DSE) is a form of path-based\nsymbolic execution based on two insights. First, the approach \nstarts by executing program $P$ on\nsome input $i$, seeding the symbolic execution process\nwith a feasible path [@Korel90;@Korel92;@Gupta00]. \nSecond,  DSE\nuses concrete values from the execution $P(i)$ in place of symbolic expressions \nwhenever symbolic reasoning is not possible or desired [@GodefroidKS05;@CadarE05].\nThe major benefit of DSE is to\nsimplify the construction of a symbolic execution tool by\nleveraging concrete execution behavior (given by\nactually running the program).\nAs DSE combines both \nconcrete and symbolic reasoning, it also has been called \"concolic\" \nexecution [@SenACAV06].\n\n~ Figure { #fig-DSE caption=\"Pseudo-code for dynamic symbolic execution\" }\n```\n  i = an input to program P\n  while defined(i):\n     p = path covered by execution P(i)\n     cond = pathCondition(p)\n     s = ATP(Not(cond))\n     i = s.model()\n```\n~\n\nThe pseudo-code of Figure [#fig-DSE] shows the high level process\nof DSE. The variable `i` represents an input\nto program `P`. Execution of program `P` on the input `i`\ntraces  a path `p`, from which\na logical formula `pathCondition(p)` is constructed.\nFinally, the ATP is called with the negation of the path-condition\nto find a new input (that hopefully will cover a new path).  This\npseudo-code elides a number of details that we will deal with later. \n\n~ Figure { #fig-easy-DSE caption=\"Easy example: computing the maximum of four numbers in Python.\"}\n```\ndef max2(s,t):\n\tif (s < t):\n\t\treturn t\n\telse:\n\t\treturn s\n\ndef max4(a,b,c,d):\n\treturn max2(max2(a,b),max2(c,d))\n```\n~\n\nConsider the  Python function `max4` in Figure [#fig-easy-DSE],\nwhich computes the maximum of four numbers via three calls to\nthe function `max2`. Suppose we execute `max4` with values\nof zero for all four arguments.  In this case, the \nexecution path $p$ contains three comparisons (in the order `(a < b)`, \n`(c < d)`, `(a < c)`), all of which evaluate false.\nThus, the path-condition for path $p$ is `(not(a<b) and not(c<d) and not(a < c))`.\nNegating this condition yields `((a<b) or (c<d) or (a<c))`.\nTaking the execution ordering of the three comparisons into account, we\nderive three expressions from the negated path-condition to generate\nnew inputs that will explore execution prefixes of path $p$ of increasing length:\n\n * *length 0*: `(a<b)`\n * *length 1*: `not (a<b) and (c<d)`\n * *length 2*: `not (a<b) and not (c<d) and (a<c)`\n\nThe purpose of taking execution order into account should be clear, as the\ncomparison `(a<c)` only executes in the case where `(not (a<b) and not (c<d))`\nholds. Integer solutions to the above three systems of constraints are:\n\n * `a == 0 and b == 2 and c == 0 and d == 0`\n * `a == 0 and b == 0 and c == 0 and d == 3`\n * `a == 0 and b == 0 and c == 2 and d == 0`\n\nIn the three cases above, we sought solutions that kept as many of\nthe variables as possible equal to the original input (in which \nall variables are equal to 0). Execution of the `max4` function \non the input corresponding to the first solution produces the path-condition\n`((a<b) and not(c<d) and not(b < c))`, from which we can produce more\ninputs.   For this (loop-free function), there are a finite number\nof path-conditions. We leave it as an exercise to the reader to \nenumerate them all. \n\n## Leveraging concrete values in DSE\n\nWe now consider several situations where we can make use of concrete\nvalues in DSE. In the realm of (unbounded-precision) integer arithmetic \n(e.g., bignum integer arithmetic, as in Python 3.0 onwards),\nit is easy  to come up with  tiny programs that will be *very difficult*,\nif not *impossible*, \nfor any symbolic execution tool to deal with, such as the function `fermat3`\nin Figure [#fig-fermat3].  \n\n\n~ Figure { #fig-fermat3 caption=\"Hard example for symbolic execution\"}\n```\ndef fermat3(x,y,z):\n   if (x > 0 and y > 0 and z > 0):\n      if (x*x*x + y*y*y == z*z*z):\n      \treturn \"Fermat and Wiles were wrong!?!\"\n   return 0\n```\n~\n\n\nFermat's Last Theorem, proved\nby Andrew Wiles in the late 20th century, states that no \nthree positive integers $x$, $y$, and $z$ can satisfy the equation\n$x^n + y^n = z^n$ for any integer value of $n$ greater than two.\nThe function `fermat3` encodes this statement for $n=3$.   It\nis not reasonable to have a computer waste time trying to find\na solution that would cause `fermat3` to print the string \n`\"Fermat and Wiles were wrong!?!\"`.  In cases of complex (non-linear)\narithmetic operations,\nsuch as `x*x*x`, we might choose to handle the operation concretely.\n\nThere are a number of ways to deal with the above issue: one is \nto recognize all non-linear terms in a symbolic expression and replace\nthem with their concrete counterparts during execution. For the `fermat3` \nexample, this would mean that during DSE the symbolic expression \n`(x*x*x + y*y*y == z*z*z)` would be reduced to the constant `False`\nby evaluation on the concrete values of variables `x`, `y` and `z`.\n\nBesides difficult operations (such as non-linear arithmetic),\nother examples of code that we might treat\nconcretely instead of symbolically include\nfunctions that are hard to invert, such as cryptographic hash functions,\nor low-level functions that we do not wish to test (such as \noperating system functions).  Consider\nthe code in Figure [#fig-hash], which applies the function\n`unknown` to argument `x` and compares it to argument `y`.\nBy using the name `unknown` we simply mean to say that we \nwish to model this function as a black box, with no knowledge\nof how it operates internally. \n\n~ Figure { #fig-hash caption=\"Another hard example for symbolic execution\"}\n```\ndef dart(x,y):\n  if (unknown(x) == y):\n     return 1\n  return 0\n```\n~\n\nIn such a case, we can use DSE to execute the function `unknown`\non a specific input (say `5013`) and observe its output\n(say `42`). That is, rather than execute `unknown` symbolically\nand invoke an ATP to invert the function's path-condition, we\nsimply treat the call to `unknown` concretely, substituting\nits return value (in this case `42`) for the specialized expression \n`unknown(5013) == y` to get the predicate `(42 == y)`. \n\nAdding the constraint `(x == 5013)` yields the sound but\nrather specific path-condition \n`(x == 5013) and (42 == y)`.\nNote that the path-condition `(42 == y)` is not sound, as it admits\nany value for the variable `x`, which likely includes many values\nfor which `(unknown(x) == y)` is false.\n\n## Overview \n\nThis introduction elides many important \nissues that arise in implementing DSE for a real language, which we will \nfocus on in the remainder of the paper. These include how to:\n\n* Identify the code under test $P$ and the symbolic inputs to $P$;\n* Trace the control flow path $p$ taken by execution $P(i)$;\n* Reinterpret program operations to compute symbolic expressions;\n* Generate a path-condition from $p$ and the symbolic expressions;\n* Generate a new input $i'$ by negating (part of) the path-condition, translating\nthe path-condition to the input language of an ATP, invoking the ATP, and\nlifting a satisfying model (if any) back up to the source level;\n* Guide the search to expose new paths.\n\nThe rest of this paper is organized as follows. Section [#sec-semantics] \ndescribes an instrumented typing discipline where we lift each type (representing \na set of concrete values) to a symbolic type (representing\na set of pairs of concrete and symbolic values).\nSection [#sec-sp2dse] shows how strongest postconditions defines a symbolic\nsemantics for a small programming language and how strongest postconditions\ncan be refined to model DSE.\nSection [#sec-impl] describes an implementation of DSE for the Python language\nin the Python language that follows the instrumented semantics pattern closely\n(full implementation and tests available at [PyExZ3], tagged \"v1.0\").\nSection [#sec-int2z3] describes the symbolic encoding of Python integer \noperations using two decision procedures of Z3: linear arithmetic with\nuninterpreted functions in place of non-linear operations;\nfixed-width bit-vectors with precise encodings of most operations.\nSection [#sec-extensions] offers a number of ideas for projects\nto extend the capabilities of [PyExZ3].\n\n# Instrumented Types { #sec-semantics }\n\nWe are given a universe of classes/types $U$; a type $T \\in U$ carries\nalong a  set of operations that apply to values of type $T$,\nwhere an operation $o \\in T$  takes an argument list of typed \nvalues as input (the first being of type $T$) and produces a single \ntyped value as output. Nullary (static) operations of type $T$ can be \nused to create values of type $T$ (such as constants, objects, etc.)\n\nA program $P$ has typed input variables\n$v_1 : T_1 \\ldots v_k : T_k$ and a body from the language of statements $S$:\n\n~MathPre\nS \\rightarrow   & v := E\n   | & @skip \n   | & S_1 ; S_2 \n   | & @if E @then S_1 @else S_2 @end\n   | & @while E @do S @end\n~ \n\nThe language of expressions ($E$) is defined by the application of operations\nto values, where constants (nullary operations) and \nprogram variables form the leaves \nof the expression tree and non-nullary operators \nform the interior nodes of the tree.\nFor now, we will consider all values to be immutable.\nThat is, the only source of mutation in the language is the \nassignment statement.\n\nTo introduce symbolic execution into the picture,\nwe can imagine that a type $T \\in U$ has\n(one or more) counterparts in a symbolic universe $U'$. A type $T' \\in U'$\nis a subtype of $T \\in U$ with two purposes:\n\n* First, a value of type $T'$ represents a pair of values: \na concrete value $c$ of (super)type $T$ and a symbolic expression $e$. \nA symbolic expression is a tree\nwhose leaves are either nullary operators (i.e., constants) of a type in $U$\nor are Skolem constants representing the (symbolic) inputs ($v_1 \\ldots v_k$)\nto the program $P$, and whose interior nodes represent operations \nfrom types in $U$. We refer to Skolem constants as \"symbolic constants\"\nfrom this point on. Note that symbolic expressions do not contain\nreferences to program variables.\n\n* Second, the type $T'$ redefines some of the operations $o \\in T$,\nnamely those for which we wish to compute symbolic expressions.\nAn operation $o \\in T'$ has the same parameter list as $o \\in T$, allowing it\nto take inputs with types from both $U$ and $U'$. The return type\nof $o \\in T'$ generally is from $U'$ (though it can be from $U$). \nThus, $o \\in T'$ is a proper function subtype of $o \\in T$. \nThe purpose of $o \\in T'$ is to:\n(1) perform operation $o \\in T$ on the concrete \nvalues associated with its inputs; \n(2) build a symbolic expression tree rooted at operation $o$ \nwhose children are the trees associated with the inputs to $o$. \n\nFigure [#fig-subtype] presents pseudo code for the instrumentation\nof a type $T$ via a type $T'$.\nThe class ``Symbolic`` is used to hold an expression tree (``Expr``).\nGiven a class $T \\in U$, a symbolic type $T' \\in U'$ is defined by inheriting \nfrom both $T$ and ``Symbolic``. This ensures that a $T'$ can be used\nwherever a $T$ is expected. \n\n~ Figure { #fig-subtype caption=\"Type instrumentation to carry both concrete values and symbolic expressions.\" }\n``` cpp2\n  class T' : T, Symbolic {\n    T'(c:T, e:Expr) : T(c), Symbolic(e) {}\n\n    override o(this:T, f1:T_1, ... , fk:T_k) : R' {\n      var c := T.o(this, f1, ... ,fk)\n      var e := new Expr(T.o, expr(self), expr(f1), ..., expr(fk))\n      return new R'(c,e) \n    }\n    ...\n  }\n  \n  class R' : R, Symbolic { ... }\n  \n  function expr(v) = v instanceof Symbolic ? v.getExpr() : v\n```\n~\n\nA type such as $T'$  only can be constructed by providing a concrete value $c$ of \ntype $T$ and a symbolic expression $e$ to the constructor for $T'$.\nThis will be done in exactly two places:\n\n* by the creation of symbolic constants associated with the primary\ninputs ($v_1 \\ldots v_k$) to the program;\n\n* by the instrumented operations as shown  in Figure [#fig-subtype]. \n\nAn instrumented operation $o$ on arguments (``this``, `f1`, ..., `fk`)\nfirst invokes its corresponding underlying\noperator $T.o$ on arguments (``this``, `f1`, ..., `fk`) to get concrete value `c`.\nIt then constructs a new expression tree `e`\nrooted at operator $T.o$, whose children are the result of\nmapping the function `expr` over (``this``, `f1`, ..., `fk`). \nThe helper function `expr(v)`\nevaluates to an expression tree in the case that `v` is of ``Symbolic`` type\n(representing a type in $U'$) and evaluates to `v` itself, a concrete value\nof some type in $U$, otherwise.\nFinally, having computed the values `c` and `e`, the instrumented operator \nreturns ``\\($R'$\\)(c,e)``, where $R$ is the return type of operator $T.o$,\nand $R'$ is a subtype of $R$ from universe $U'$.\n\nLooked at another way, the universe $U'$ represents the \"tainting\" of\ntypes from $U$. Tainted values flow from program inputs to the \noperands of operators. If an operator has been redefined\n(as above) then the taint propagates from its inputs to its outputs.\nOn the other hand, if the operator has not been redefined, then it will\nnot propagate the taint. In the context of DSE, \"taint\" means\nthat the instrumented semantics carries along a symbolic expression tree $e$\nalong with a concrete value $c$.\n\nThe choice of types from the universe $U'$ determines how symbolic\nexpressions are constructed. For each $T \\in U$, the \"most symbolic\"\n(least concrete) choice is the $T'$ that redefines every operator of $T$\n(as shown in Figure [#fig-subtype]).\nThe \"least symbolic\" (most concrete) choice is $T' = T$ which\nredefines no operators.  Let $symbolic(T)$ be the\nset of types in $U'$ that are subtypes of $T$. The types\nin $symbolic(T)$ are partially ordered by subset inclusion on the \nset of operators from $T$ they redefine.\n\n<!-- Example of an instrumentation -->\n\n# From Strongest Postconditions to DSE {#sec-sp2dse}\n\nThe previous section showed how symbolic expressions can be computed via a\nset of instrumented types, where the expressions are computed as a side-effect \nof the execution of program operations.\nThis section shows how these symbolic expressions can be used to form a \n*path-condition* (which then can be compiled into a logic formula and \npassed to an automated theorem prover to find new inputs to drive a \nprogram's execution along new paths). We derive a *path-condition*\ndirectly from the _strongest postcondition_ (symbolic) semantics of our\nprogramming language, refining it to model the basic operations\nof an interpreter. \n\n## Strongest Postconditions\n\nThe strongest postcondition transformer $SP$ [@Dijkstra76] is defined over\na predicate $P$ representing a set of \npre-states and a statement $S$ from our language. The transformer $SP(P,S)$\nyields a predicate $Q$ such that for any state $s$ satisfying\npredicate $P$, the execution of statement $S$ from state $s$,\nif it does not go wrong or diverge, yields a state $s'$ satisfying predicate $Q$. \nThe strongest postcondition for the statements in our language\nis defined by the following five rules:\n\n~ MathPre\n1.  & SP(P, x := E) =^= \\exists y . (x = E [ x \\rightarrow y ]) \\wedge P [ x \\rightarrow y ]\n2.  & SP(P, @skip) =^= P\n3.  & SP(P,S1;S2) =^= SP(SP(P,S1), S2)\n4.  & SP(P,@if E @then S_1 @else S_2 @end) =^=\n    &       SP(P\\wedge E,S_1) \\vee  SP(P \\wedge \\neg E,S_2)\n5.  & SP(P,@while E @do S @end) =^=\n    &      SP(P,@if E @then S; @while E @do S @end @else @skip @end)\n~\n\nRule (1) defines the strongest postcondition for \nthe assignment statement. The assignment is modeled\nlogically by the equality $x = E$ where any free occurrence of $x$ in $E$\nis replaced by the existentially quantified variable $y$, which represents\nthe value of $x$ in the pre-state. The same substitution ($[x \\rightarrow y ]$)\nis applied to the pre-state predicate $P$. \n\nRules (2)-(5) define the strongest postcondition for the four control-flow\nstatements. The rules for the **skip** statement and sequencing (;) are\nstraightforward. \nOf particular interest, note that the rule for the **if-then-else** \nstatement splits cases on the expression $E$. \nIt is here that DSE will choose one of the cases for us, as the concrete\nexecution will evaluate $E$ either to be true or false. This gives rise\nto the path-condition (either $P \\wedge E$ or $P \\wedge \\neg E$). The\nrecursive rule for the **while** loop unfolds as many times as the\nexpression $E$ evaluates true, adding to the path-condition.\n\n## From $SP$ to DSE {#sp-refined}\n\nAssume that an execution begins with the assignment of initial values $c_1 \\ldots c_k$ to the program $P$'s inputs\n$V = \\{ v_1 : T_1 \\ldots v_k : T_k \\}$.  To seed symbolic execution, some of the types $T_i$ are\nreplaced by symbolic counterparts $T'_i$, in which case\n$v_i$ is initialized to the value $sc_i = T'_i (c_i,SC(v_i))$ instead of the value $c_i$,\nwhere $SC(v_i)$ is the symbolic constant representing the initial value of variable $v_i$.\nThe symbolic constant $SC(v_i)$ can be thought of as representing any value of\ntype $T_i$, which includes the value $c_i$.  \n\nLet $V_s$ and $V_c$ partition the variables of $V$ into those variables\nthat are treated symbolically ($V_s$) and those that are treated concretely \n($V_c$). The initial state of the program is characterized by the formula\n\n~ Equation\nInit = (\\bigwedge_{v_i \\in V_s} v_i = sc_i) ~\\wedge~ (~\\bigwedge_{v_i \\in V_c} v_i = c_i)\n~\n\nThus, we see that the initial value of every input variable is characterized by \na symbolic constant $sc_i$ or constant $c_i$.  We assume that every non-input\nvariable in the program is initialized before being used.\n\nThe strongest postcondition is formulated to deal with open programs,\nprograms in which some variables are used before being assigned to. \nThis surfaces in Rule (1) for assignment, which uses existential quantification\nto refer to the value of variable $x$ in the pre-state.\n\nBy construction,\nwe have that every variable is defined before being used.  This means\nthat the precondition $P$ can be reformulated as a pair $<St,P_c>$,\nwhere $St$ is a store mapping variables to values and $P_c$ is\nthe path-condition, a list of symbolic expressions (predicates) \ncorresponding\nto the expressions $E$ evaluated in the context of an **if-then-else** \nstatement. Initially, we have that :\n\n~ Equation\nSt = \\{ (v_i,sc_i) | v_i \\in V_s \\} \\cup \\{ (v_i,c_i) | v_i \\in V_c \\}\n~\n\nrepresenting the initial condition $Init$, and $P_c = []$, the empty list.\nWe use $St'$ to refer to the formula that the store $St$\ninduces: \n\n~ Equation\nSt ' =  \\bigwedge_{(v,V) \\in St} (v = V)\n~\n\nThus, the pair $<St,P_c>$ represents the predicate \n$P = St' \\wedge (\\bigwedge_{c \\in P_c} c)$.\nA store $St$ supports two operations: $St[x]$ which denotes the\nvalue that $x$ maps to under $St$; $St[x \\mapsto V]$, which \nproduces a new store in which $x$ maps to value $V$ and is everywhere \nelse the same as $St$. \n\nNow, we can redefine strongest postcondition for assignment to eliminate the\nuse of existential quantification and model the operation of an interpreter,\nby separating out the notion of the store:\n\n~MathPre\n1. & SP(<St, P_c>, x := E) =^= <St[x \\mapsto eval(St,E)], P_c>\\\\\n~\n\nwhere $eval(St,E)$ evaluates expression $E$ under the store $St$\n(where every occurrence of a free variable\n$v$ in $E$ is replaced by the value $St[v]$). \nThis is the standard substitution rule of a standard operational semantics. \n\nWe also redefine the rule for the **if-then-else** statement so\nthat it chooses which branch to take and appends the appropriate\nsymbolic expression (predicate) to the path-condition $P_c$:\n\n~MathPre\n4. & SP(<St, P_c>, @if E @then S_1 @else S_2 @end) =^=  \n   &   @let choice = eval(St,E) @in\n   &   @if choice @then SP(<St, P_c :: expr(choice) >,S_1) \n   &   @else SP(<St, P_c :: \\neg expr(choice) >,S_2)\n~\n\nThe other strongest postcondition rules remain unchanged.\n\n## Summing it up\n\nWe have shown how the symbolic predicate transformer $SP$ can \nbe refined into a symbolic interpreter operating over the\nsymbolic types defined in the previous section.\nIn the case when every input variable is symbolic and every\noperator is redefined, the path-condition is equivalent to the\n_strongest postcondition_ of the execution path $p$. \nThis guarantees that the path-condition for $p$ is *sound*.\nIn the case where a subset of the input variables are symbolic\nand/or not all operators are redefined, the path-condition of $p$ \nis not guaranteed to be sound. We leave it as an exercise to the\nreader to establish sufficient conditions under which the \nuse of concrete values in place of symbolic expressions is \nguaranteed to result in sound path-conditions. \n\nThis section does not address the compilation of a symbolic\nexpression to the (logic) language of an underlying ATP, nor the\nlifting of a satisfying assignment to a formula back to the\nlevel of the source language. This is best done for a particular\nsource language and ATP, as detailed in the next section. \n\n\n# Architecture of PyExZ3 { #sec-impl }\n\nIn this section we present the high-level architecture\nof a simple DSE tool for the Python language, written in Python, called [PyExZ3]. \nFigure [#fig-arch]\nshows the class diagram (dashed edges are \"has-a\" relationships; solid edges\nare \"is-a\" relationships) of the tool. \n\n~ Figure { #fig-arch caption=\"Classes in PyExZ3\" page-align=here }\n![arch]\n~\n\n[arch]: arch.png \"arch\"  { width=100% }\n\n## Loading the code under test\n\nThe `Loader` class takes as input the name of a Python file (e.g., `foo.py`) \nto import. The loader expects to find a function named\n`foo` inside the file `foo.py`, which will serve as the starting point\nfor symbolic execution. The `FunctionInvocation` class\nwraps this starting point. By default, each parameter to `foo` is\na `SymbolicInteger` unless there is decorator `@symbolic` specifying\nthe type to use for a particular argument.\n\nThe loader provides the capability to reload the\nmodule `foo.py` so that the function `foo` can be \nreexecuted within the same process from the same initial\nstate with different inputs (see the class `ExplorationEngine`)\nvia the `FunctionInvocation` class. \n\nFinally, the loader looks for specially named functions `expected_result`\n(`expected_result_set`) in file `foo.py` to use as a test oracle after\nthe path exploration (by `ExplorationEngine`) has completed. These\nfunctions are expected to return a list of values to check\nagainst the list of return values collected from the executions of \nthe `foo` function.\nThe presence of the function `expected_result` (`expected_result_set`) \nyields a comparison of the two lists as bags (sets). We use such weaker\ntests, rather than list equality, because the order in which paths\nare explored by the `ExplorationEngine` can easily change due to small\ndifferences in the input programs. \n\n## Symbolic types\n\nPython supports multiple inheritance and, more importantly,\nallows user-defined classes to inherit \nfrom its built-in types (such as `object` and `int`).\nWe use these two features two implement\nsymbolic versions of Python objects and integers, \nfollowing the instrumented type approach defined in Section [#sec-semantics]. \n\nThe abstract class `SymbolicType` contains the \nsymbolic expression tree and provides basic functions for constructing \nand accessing the tree.  This class does double duty, as it is used\nto represent the (typed) symbolic constants associated with the parameters to the \nfunction, as well as the expression trees (per Section [#sec-semantics]). Recall\nthat the symbolic constants only appear as leaves of expression trees.\nThis means that the expression tree stored in a `SymbolicType` will\nhave instances of a `SymbolicType` as some of its leaves, namely\nthose leaves representing the symbolic constants.\n<!-- Need to say why this is important -->\nThe abstract class provides an `unwrap` method which returns\nthe pair of concrete value and expression tree associated with\nthe `SymbolicType`, as well as a `wrap` method that takes a \npair of concrete value and expression tree and creates a `SymbolicType`\nencapsulating them. \n\nThe class `SymbolicObject` inherits from both `object` and `SymbolicType` and\noverrides the basic comparison operations (`__eq__`, `__neq__`, `__lt__`, `__le__`,\n`__gt__`, and `__ge__`).\nThe class `SymbolicInteger` inherits from both `int` and `SymbolicObject`\nand overrides a number of `int`'s arithmetic methods\n(`__add__`, `__sub__`, `__mul__`, `__mod__`, `__floordiv_`)\nand bitwise methods\n(`__and__`, `__or__`, `__xor__`, `__lshift__`, `__rshift__`).\n\n\n## Tracing control-flow\n\nAs Python interprets a program, it will evaluate expressions, \nsubstituting the value of a variable in its place in an \nexpression, applying operators (methods) to \nparameter values and assigning the return values of methods\nto variables. Value of type `SymbolicInteger` will simply flow\nthrough this interpretation, without necessitating any change \nto the program or the interpreter. This takes care\nof the case of the strongest-postcondition rule for assignment,\nas elaborated in Section [#sp-refined].\n\nThe strong-postcondition rule for a conditional test requires\na little more work. In Python, any object can be tested in\nan `if` or `while` condition or as the operand of a Boolean operation\n(`and`, `or`, `not`)\nThe Python base class `object` provides a method named `__bool__` that\nthe Python runtime calls whenever it needs to perform such a conditional test.\nThis hook provides us what we need to trace the conditional\ncontrol-flow of a Python execution.  We override this method in the class\n`SymbolicObject` in order to inform the `PathToConstraint` object (defined\nlater) of the symbolic expression for the conditional (as captured by\nthe `SymbolicInteger` subclass). \n\nNote that the use of this hook in combination with the\ntainted types will only trace those conditionals\nin a Python execution whose values inherit from `SymbolicObject`; \nby definition, \"untainted\" conditionals do not depend on symbolic \ninputs so there is no value in adding them to the path-condition.\n\n## Recording path-conditions\n\nA `Predicate` records a conditional (more precisely the symbolic expression\nfound in `SymbolicInteger`) and\nwhich way it evaluated in an execution.  A `Constraint`\nhas a `Predicate`, a parent `Constraint` and a set\nof `Constraint` children. `Constraints` form a tree, where\neach path starting from the root of the tree represents\na path-condition. The tree represents all path-conditions that have\nbeen explored so far.\n\nThe class `PathToConstraint` has a reference to the root of \nthe tree of `Constraint`s\nand is responsible for installing a new `Constraint` in the tree\nwhen notified by the overridden `__bool__` method of `SymbolicObject`.\n`PathToConstraint` also tracks whether or not the current execution\nis following an existing path in the tree and grows the\ntree as needed. In fact, it actually\ntracks whether or not the current execution follows a particular\n*expected path* in the tree.\n\nThe expected path is the result\nof the `ExplorationEngine` picking a constraint $c$ in the tree,\nand asking the ATP if the path-condition consisting of the prefix\nof predicates up to but not including $c$ in the tree, \nfollowed by the negation of $c$'s predicate is satisfiable. If the \nATP returns \"satisfiable\" (with a new input $i$), then the assumption\nis that path-condition prefix is sound (that is, the execution of\nthe program on input $i$ will follow the prefix). \n\nHowever, it is possible \nfor the path-condition to be unsound and for \nthe executed path to diverge early \nfrom the expected path, due to the fact that not every operation\nhas a symbolic encoding.  The tool simply reports the divergence\nand continues to process the execution as usual (as a diverging\npath may lead to some other interesting part of the code). \n\n## From symbolic types to Z3\n\nAs we have explained DSE, the symbolic expressions are \nrepresented at the level of the source language. As detailed later\nin Section [#sec-int2z3], we must translate \nfrom the source language to the input language of an\nautomated theorem prover (ATP), in this case [Z3].  This\nseparation of languages is quite useful, as we may have\nthe need to translate a given symbolic expression\nto the ATP's language multiple times, to make use of different\nfeatures of the underlying ATP. \nFurthermore, this separation\nof concerns allows us to easily retarget the DSE tool to a\ndifferent ATP. \n\nThe base class `Z3Expression` represents a Z3 formula. The two \nsubclasses `Z3Integer` and `Z3BitVector` represent different ways \nto model arithmetic reasoning about integers in Z3. We will describe\nthe details of these encodings in Section [#sec-int2z3].\n\nThe class `Z3Wrapper` is responsible for performing the\ntranslation from the source language (Python) to Z3's input language, \ninvoking Z3, and lifting a Z3 answer back to the level of Python. \nThe `findCounterexample` method does all the work, taking as\ninput a list of `Predicate`s (called `assertions`) \nas well as a single `Predicate` (called\nthe `query`). The `assertions` represent a path-condition\nprefix derived from the `Constraint` tree that we wish the next\nexecution to follow, while `query` represents the predicate\nfollowing the prefix in the tree that we will negate.\n\nThe method constructs the formula\n\n~Equation\n(\\bigwedge_{a \\in asserts} a) \\wedge \\neg query\n~\n\nand asks Z3 if it is satisfiable. The method performs\na standard syntactic \"cone of influence\" (CIF)\nreduction on the `asserts` with respect to the \n`query` to shrink the size of the formula. For example,\nif `asserts` is the set of predicates $\\{ (x<a), (a<0), (y>0) \\}$ and\nthe query is $(x=0)$, then the CIF yields the\nset  $\\{ (x<a), (a<0) \\}$, which does not include the predicate $(y>0)$,\nas the variable $y$ is not in the set of variables (transitively)\nrelated to variable $x$.\n\nIf the formula is satisfiable a model is requested\nfrom Z3 and lifted back to Python's type universe.  Note that\nbecause of the CIF reduction, the model may not mention certain\ninput variables, in which case we simply keep their values from\nthe execution from which the `asserts` and `query` were derived. \n\n## Putting it all together\n\nThe class `ExplorationEngine` ties everything together. It kicks off\nan execution of the Python code under test using `FunctionInvocation`.\nAs the Python code executes, building symbolic expressions via `SymbolicType`\nand its subclasses, callbacks to `PathToConstraint` create\na path-condition, represented by `Constraint` and `Predicate`. \nNewly discovered `Constraints` are added to the end of a deque maintained by\n`ExplorationEngine`.\n\nGiven the first seed execution, `ExplorationEngine` starts the work of \nexploring paths in a breadth-first fashion. It removes a `Constraint` $c$\nfrom the front of its deque and, if $c$ has not been already \"processed\", \nuses `Z3Wrapper` to find a new input (as discussed in the previous section)\nwhere $c$ is the query (to be negated) and the path to $c$ in the \n`Constraint` tree forms the assertions. \n\nA `Constraint` $c$ in the tree is considered \"processed\" if an execution \nhas covered $c'$, a sibling of $c$ in the tree that represents\nthe negation of the predicate associated with $c$, or if constraint $c$\nhas been removed from the deque. \n\n# From Python Integers to Z3 Arithmetic { #sec-int2z3 }\n\nIn languages such as C and Java, integers are finite-precision,\ngenerally limited to the size of a machine word (32 or 64 bits, for example).\nFor such languages, satisfiability of finite-precision integer arithmetic\nis decidable and can be reduced to Z3's theory of bit-vectors, where\neach arithmetic operation is encoded by a circuit. This translation permits \nreasoning about non-linear arithmetic problems, such as \n$\\exists x,y,z : x*z + y \\leq (z/y)+5$.\n\nPython (3.0) integers, however, are not finite-precision. They are only\nlimited by the size of machine memory. This means, for example, that\nPython integers don't overflow or underflow. It also means that\nwe can't hope to decide algorithmically whether or not a given\nequation over integer variables has a solution in general. Hilbert's famous\n10th problem and its solution by Matiyasevich tells us that it is\nundecidable whether or not a polynomial\nequation of the form $p(x_1, \\ldots, x_n) = 0$ with integer coefficients\nhas an solution in the integers.\n\nThis means that we will resort to heuristic approaches in our use\nof the [Z3] ATP.  The special case of linear integer arithmetic (LIA)\nis decidable and supported by Z3. In order to deal with non-linear operations,\nwe use uninterpreted functions (UF). Thus, if Z3 returns\n\"unsatisfiable\" we know that there is no solution, but if the\nZ3 \"satisfiable\", we must treat the answer as a \"don't know\". \nThe class `Z3Integer` is used to translate a symbolic expression\ninto the theory LIA+UF and check for unsatisfiability. We leave it as an\nimplementation exercise to check if a symbolic expression can\nbe converted to LIA (without the use of UF) in order to make\nuse of \"satisfiable\" answers from the LIA solver.\n\nIf the translation to `Z3Integer` does not return \"unsatisfiable\", we\nuse Z3's bit-vector decision procedure (via the class\n`Z3BitVector`) to heuristically\ntry to find satisfiable answers, even in the presence of non-linear \narithmetic. We start with bit-vectors of size $N=32$ and *bound* the values\nof the symbolic constants to fit within 8 bits in order to find \nsatisfiable solutions\nwith small values. Also, because Python integers do not overflow/underflow, \nthe bound helps us reserve space in the bit-vector to allow the\nresults of operations to exceed the bound while not overflowing\nthe bit-vector. As long as Z3 returns \"unsatisfiable\" we increase\nthe bound. If the bound reaches $N$, we increase $N$ by 8 bits,\nleaving the bound where it is and continue. \n\nIf Z3 returns\n\"satisfiable\", it may be the case that Z3 found a solution\nthat involved overflow in the bit-vector world of arithmetic\n(modulo $2^N-1$). Therefore,\nthe solution is validated back in the\nPython world by evaluating the formula under that solution\nusing Python semantics. \nIf the formula does not evaluate to the same\nvalue in both worlds, then we increase $N$ by 8 bits (to \ncreate a gap between the bound and $N$) and continue to search\nfor a solution.\n\nThe process terminates when we find a valid satisfying solution \nor $N=64$ and the bound reaches 64 (in which case, we return \"don't know\").\n\n# Extensions {#sec-extensions}\n\nWe have presented the basics of dynamic symbolic execution \n(for Python).\nA more thorough treatment would deal with other data types besides\nintegers, such as Python dictionaries, strings and lists, each\nof which presents their own challenges for symbolic reasoning. \nThere are many other interesting challenges in DSE, such\nas dealing with user-defined classes (rather than built-in types\nas done here) and multi-threaded execution. \n\n# Acknowledgements\n\nMany thanks to the students of the 2014 Marktoberdorf Summer School\non Dependable Software Systems Engineering\nfor their questions and feedback about the first author's lectures on dynamic\nsymbolic execution. The following students of the summer school\nhelpfully provided tests for the [PyExZ3] tool: Daniel Darvas,\nDamien Rusinek, Christian Dehnert and Thomas Pani. Thanks also to Peter\nChapman for his contributions. \n\n\n<!-- \n# From Python Dictionaries to Z3 Maps { #sec-dict2z3 }\n\nPython dictionaries, which map keys to values,\nintroduce mutation into the picture. The methods are:\n\n- Length: \t`__length__(self)` - will be maintained concretely, as cardinality constraints\nare beyond the scope of what we can handle\n- Get: \t`__getitem__(self,key)` - Select\n- Set: \t`__setitem__(self,key,value)` - Update\n- Lookup: \t`__contains__(self,key)` - \n- Delete: \t`__delitem__(self,key)` - Update \n\n\nA dictionary can be mutated via the `__setitem__` and '__delitem__` methods.\nThe keys stored  in Python dictionary must be hashable values. \nAll of Python's immutable\nbuilt-in objects (such as integers, tuples and strings) are hashable. \nUser-defined classes give rise to hashable mutable objects, where\nthe hash value is the object id (an immutable field of the object).\n\nIdeally, symbolic expression trees should contain only\nimmutable values, as they are a pure function of the initial\n(immutable) state of a program.  However, the reality is that\nthe values associated with keys in a dictionary can be mutable.\nFor example, a dictionary itself can be a value in another\ndictionary.    \n\nAs we will see, for the purposes of modelling a dictionary symbolically, \nhowever, the object id of the value in a (key,value) pair is all that\nneeds to be recorded. The object id is immutable, as mentioned before. \nThus, we can represent the state of the dictionary via an append-only log \nof updates where the parameters \nto each update are immutable values (set and delete are the update\noperations).  Each lookup/get/length operation hangs off of a particular\nprefix of the log. \n\nWe model the dictionary as initially\nempty, containing no mapping (rejects all lookups and get, length is zero).\n\n\n\nThe essential axioms\n\n* Initially empty\n* Select\n* Update\n\nInverting select means that we need disequality on values in\nthe dictionary (SymbolicObject)\n\nEquality of dictionaries (extensional arrays)\n\n-->\n\n# References {-}\n[BIB]\n"
  },
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HEADINGS\n%\n% normal heading\n\\def\\ps@headings{%\n      \\let\\@oddfoot\\@empty\\let\\@evenfoot\\@empty\n      \\def\\@evenhead{\\footnotesize\\rlap{\\thepage}\\hfill\\textit{\\leftmark}\\hfill}%\n      \\def\\@oddhead{\\footnotesize\\hfill\\textit{\\rightmark}\\hfill\\llap{\\thepage}}%\n}%\n% empty RH\n\\def\\ps@empty{\\let\\@mkboth\\@gobbletwo\n     \\def\\@oddhead{\\hfill}\\def\\@oddfoot{}\n\\let\\@evenhead\\@oddhead\\let\\@evenfoot\\@oddfoot}\n%\n% RH  with pagenumber at bottom\n\\def\\ps@plain{\\let\\@mkboth\\@gobbletwo\n     \\def\\@oddhead{\\hfill}\\def\\@oddfoot{}\n\\let\\@evenhead\\@oddhead\n  \\def\\@oddfoot{\\hfill\\footnotesize\\thepage\\hfill}\n  \\let\\@evenfoot\\@oddfoot\n}\n% First page RH\n\\def\\ps@copyright{\\let\\@mkboth\\@gobbletwo\n  \\def\\@evenhead{\\parbox[t]{.75\\textwidth}{\\footnotesize\\raggedright\\itshape\\titleheadline}\\hfill\\footnotesize\\thepage}%\n  \\def\\@oddhead {\\parbox[t]{.75\\textwidth}{\\footnotesize\\raggedright\\itshape\\titleheadline}\\hfill\\footnotesize\\thepage}%\n  \\let\\@oddfoot\\relax%\n  \\let\\@evenfoot\\@oddfoot%\n}\n%\n% HEADLINE: Book Title  \n%           Book Editors\n%           IOS Press, 0000\n%\n\\def\\titleheadline{%\n   \\book@title\\\\\n   \\book@editors\\\\\n   \\@publisher, \\the\\@pubyear}\n%\n\\def\\@copyright{\\@issn/\\the@copyear/\\$\\@price\\ \\copyright@sign\\\n\\the\\@pubyear\\@copyrightowner}%\n%\n\n% ************************ FOOTNOTE\n%\n\\newcommand\\@makefntext[1]{%\n    \\parindent1em\\@makefnmark #1}\n\\def\\@makefnmark{\\@textsuperscript{\\normalfont\\@thefnmark}}%\n%\n% ************************ Counters\n\\setcounter{secnumdepth}{3}\n\\newcounter {section}\n\\newcounter {subsection}[section]\n\\newcounter {subsubsection}[subsection]\n\\newcounter {paragraph}[subsubsection]\n\\newcounter {subparagraph}[paragraph]\n\\renewcommand \\thesection {\\@arabic\\c@section}\n\\renewcommand\\thesubsection   {\\thesection.\\@arabic\\c@subsection}\n\\renewcommand\\thesubsubsection{\\thesubsection .\\@arabic\\c@subsubsection}\n\\renewcommand\\theparagraph    {\\thesubsubsection.\\@arabic\\c@paragraph}\n\\renewcommand\\thesubparagraph {\\theparagraph.\\@arabic\\c@subparagraph}\n%\n% ******************** Sectioning commands\n\\def\\no@harm{\\let\\thanks=\\@gobble \\let\\\\=\\@empty}\n%**************** Section commands\n\\def\\nohyphen{\\pretolerance=10000 \\tolerance=10000\n\\hyphenpenalty=10000 \\exhyphenpenalty=10000}\n\\newcommand\\section{\\@startsection {section}{1}{\\z@}%\n                                   {-\\bigskipamount}%\n                                   {\\medskipamount}%\n                                   {\\normalsize\\bfseries\\nohyphen\\raggedright}}\n\\newcommand\\subsection{\\@startsection {subsection}{2}{\\z@}%\n                                   {-\\medskipamount}%\n                                   {\\medskipamount}%\n                                   {\\normalsize\\itshape\\nohyphen\\raggedright}}\n\\newcommand\\subsubsection{\\@startsection{subsubsection}{3}{\\z@}%\n                                     {-\\medskipamount}%\n                                     {\\smallskipamount}%\n                                     {\\normalsize\\itshape\\nohyphen\\raggedright}}\n\\newcommand\\paragraph{\\@startsection{paragraph}{4}{\\z@}%\n                                    {\\smallskipamount}%\n                                    {-1em}%\n                                    {\\normalsize\\itshape}}\n\\newcommand\\subparagraph{\\@startsection{subparagraph}{5}{\\z@}%\n                                       {0.1pt}%\n                                       {-1em}%\n                                       {\\normalsize\\itshape}}\n% Format for the counter:\n\\def\\@seccntformat#1{\\csname the#1\\endcsname.\\enspace}\n%\n\\def\\appendix{\\par\n   \\setcounter{section}{0}%\n   \\setcounter{subsection}{0}%\n   \\gdef\\thesection{\\Alph{section}}}\n%\n\\def\\acknowledgements{\\section*{\\acknowledgementsname}%\n  \\typeout{\\acknowledgementsname}}\n%\n\\def\\notes{\\section*{Notes}\\footnotesize}\n\\def\\endnotes{\\par \\vskip 6pt plus12pt minus2pt\\relax}\n%****************** LISTS\n\\if@twocolumn\n  \\setlength\\leftmargini  {2em}\n\\else\n  \\setlength\\leftmargini  {2.5em}\n\\fi\n\\leftmargin  \\leftmargini\n\\setlength\\leftmarginii  {2.2em}\n\\setlength\\leftmarginiii {1.87em}\n\\setlength\\leftmarginiv  {1.7em}\n\\if@twocolumn\n  \\setlength\\leftmarginv  {.5em}\n  \\setlength\\leftmarginvi {.5em}\n\\else\n  \\setlength\\leftmarginv  {1em}\n  \\setlength\\leftmarginvi {1em}\n\\fi\n\\setlength  \\labelsep  {.4em}\n\\setlength  \\labelwidth{\\leftmargini}\n\\addtolength\\labelwidth{-\\labelsep}\n%\n\\renewcommand\\theenumi{\\@arabic\\c@enumi}\n\\renewcommand\\theenumii{\\@alph\\c@enumii}\n\\renewcommand\\theenumiii{\\@roman\\c@enumiii}\n\\renewcommand\\theenumiv{\\@Alph\\c@enumiv}\n\\newcommand\\labelenumi{\\theenumi.}\n\\newcommand\\labelenumii{(\\theenumii)}\n\\newcommand\\labelenumiii{\\theenumiii.}\n\\newcommand\\labelenumiv{\\theenumiv.}\n\\renewcommand\\p@enumii{\\theenumi}\n\\renewcommand\\p@enumiii{\\theenumi(\\theenumii)}\n\\renewcommand\\p@enumiv{\\p@enumiii\\theenumiii}\n%\n\\def\\setenumlabel#1{\\gdef\\max@enumlabel{#1}}\n\\setenumlabel{1.}\n%\n\\def\\enumerate{\\@ifnextchar[{\\enumerate@}{\\enumerate@[\\max@enumlabel]}}\n%\n\\def\\enumerate@[#1]{\\ifnum \\@enumdepth >4 \\@toodeep\\else\n \\advance\\@enumdepth \\@ne\n \\edef\\@enumctr{enum\\romannumeral\\the\\@enumdepth}%\n \\list {\\csname label\\@enumctr\\endcsname}%\n {\\usecounter{\\@enumctr}\\def\\makelabel##1{{\\hfill\\rm ##1}}\n\\settowidth{\\labelwidth}{#1}\n\\advance\\labelwidth by\\parindent \\labelsep=0.5em\n \\leftmargin\\z@ \\rightmargin\\z@ \\itemindent=\\labelwidth\n        \\advance\\itemindent\\labelsep\n        \\leftmargin=\\the\\itemindent\\itemindent=\\z@\n        \\partopsep\\z@ \\topsep\\smallskipamount \\parsep\\z@ \\itemsep\\z@ %\\@rightskip\\z@ plus 1fil\n \\listparindent\\z@}\\fi\\setenumlabel{1.}}\n\n%%%%%%%%%%%%%%%%%%%%%% ITEMIZE\n\\newcommand\\labelitemi{\\normalfont\\bfseries \\textbullet}\n\\newcommand\\labelitemii{\\textasteriskcentered}\n\\newcommand\\labelitemiii{\\textasteriskcentered}\n\\newcommand\\labelitemiv{\\textperiodcentered}\n\n\\let\\@itemize@indent\\parindent\n%\n\\def\\itemize{\\@ifnextchar[{\\itemize@}{\\itemize@[]}}\n\\def\\itemize@[#1]{\\ifnum \\@itemdepth >4 \\@toodeep\\else\n  \\advance\\@itemdepth \\@ne\n  \\edef\\@itemitem{labelitem\\romannumeral\\the\\@itemdepth}%\n  \\if.#1. \\else\\def\\@@tempa{#1}\\edef\\@itemitem{@@tempa}\\fi\\list\n{\\csname\\@itemitem\\endcsname}{\\settowidth{\\labelwidth}\n                {\\csname\\@itemitem\\endcsname}\n                \\def\\makelabel##1{##1}\\labelsep=0.5em%ST\n\t\t\\itemindent=\\labelwidth \\advance\\itemindent\\labelsep\n                \\advance\\itemindent\\@itemize@indent\n\t\t\\leftmargin\\the\\itemindent \\itemindent=\\z@\n                \\partopsep\\z@ \\topsep\\smallskipamount \\parsep\\z@ %\\@rightskip\\z@ plus 1fil\n\t\t\\itemsep\\z@ \\listparindent\\z@} \\fi}\n%\n\\newenvironment{description}\n               {\\list{}{\\labelwidth\\z@ \\itemindent-\\leftmargin\n                        \\let\\makelabel\\descriptionlabel}}\n               {\\endlist}\n\\newcommand*\\descriptionlabel[1]{\\hspace\\labelsep\n                                \\normalfont\\bfseries #1}\n\\newenvironment{verse}\n               {\\let\\\\\\@centercr\n                \\list{}{\\itemsep      \\z@\n                        \\itemindent   -1.5em%\n                        \\listparindent\\itemindent\n                        \\rightmargin  \\leftmargin\n                        \\advance\\leftmargin 1.5em}%\n                \\item\\relax}\n               {\\endlist}\n\n\\newenvironment{quotation}\n               {\\list{}{\\small\\listparindent2mm%\n                        \\itemindent\\z@   %\n                        \\rightmargin\\z@   \\leftmargin\\parindent%\n                        \\partopsep\\z@ \\topsep\\smallskipamount \\parsep\\z@%\n                        }%\n                \\item[\\Q@strut]\\relax}\n               {\\endlist}\n\\def\\Q@strut{\\leavevmode\\hbox{\\vrule height9pt depth1pt width0pt}}\n\n\\newenvironment{quote}\n               {\\list{}{\\listparindent\\z@%\n                        \\itemindent    \\listparindent%\n                        \\rightmargin\\z@   \\leftmargin 1.5em%\n                        \\partopsep\\z@ \\topsep6pt \\parsep\\z@%\n                        }%\n                \\item[\\Q@strut]\\relax}\n               {\\endlist}\n%\n%************************** TABULAR\n\\let\\savehline\\hline\n   \\def\\thline{\\noalign{\\vskip3pt}\\savehline\\noalign{\\vskip3pt}}%\n   \\def\\fhline{\\noalign{\\vskip1pt}\\savehline\\noalign{\\vskip7pt}}%\n   \\def\\bhline{\\noalign{\\vskip3pt}\\noalign{\\global\\arrayrulewidth=1\\p@}\\savehline\\noalign{\\global\\arrayrulewidth=.5\\p@}\\noalign{\\vskip3pt}}%\n   \\def\\lhline{\\noalign{\\vskip3pt}\\noalign{\\global\\arrayrulewidth=.3\\p@}\\savehline\\noalign{\\global\\arrayrulewidth=.5\\p@}\\noalign{\\vskip3pt}}\n%\n%************************** MATH SETTINGS\n\\setlength\\mathindent{2em}\n\\setlength\\arraycolsep{1.2\\p@}\n\\setlength\\tabcolsep{6\\p@}\n\\setlength\\arrayrulewidth{.4\\p@}\n\\setlength\\doublerulesep{2\\p@}\n\\setlength\\tabbingsep{\\labelsep}\n\\setlength\\jot{6\\p@}\n\\skip\\@mpfootins = \\skip\\footins\n\\setlength\\fboxsep{3\\p@}\n\\setlength\\fboxrule{.4\\p@}\n\\if@seceqn\n\\@addtoreset {equation}{section}\n\\renewcommand\\theequation{\\thesection.\\@arabic\\c@equation}\n\\else\n\\renewcommand\\theequation{\\@arabic\\c@equation}\n\\fi\n%******* TABLES, FIGURES, ALGORITHM\n\\newcounter{figure}\n\\if@secfloat\n \\@addtoreset{figure}{section}\n \\renewcommand \\thefigure {\\thesection.\\@arabic\\c@figure}\n\\else\n \\renewcommand \\thefigure {\\@arabic\\c@figure}\n\\fi\n\\def\\fps@figure{tbp}\n\\def\\ftype@figure{1}\n\\def\\ext@figure{lof}\n\\def\\fnum@figure{\\figurename~\\thefigure.}\n\\newenvironment{figure}\n               {\\let\\@makecaption\\@makefigurecaption\\let\\@floatboxreset\\@figureboxreset\\@float{figure}}\n               {\\end@float}\n\\newenvironment{figure*}\n               {\\let\\@makecaption\\@makefigurecaption\\let\\@floatboxreset\\@figureboxreset\\@dblfloat{figure}}\n               {\\end@dblfloat}\n\n\\def\\@figureboxreset{%\n        \\reset@font%\n        \\centering%\n        \\@setnobreak%\n        \\@setminipage%\n}\n\n\n\\long\\def\\@makefigurecaption#1#2{\\footnotesize%\n \\vskip\\abovecaptionskip\n\\setbox\\@tempboxa\\hbox{\\textbf{#1}\\enspace #2}%\n  \\ifdim \\wd\\@tempboxa >\\hsize\n    \\unhbox\\@tempboxa\\par\n  \\else\n    \\hbox to\\hsize{\\hfil\\box\\@tempboxa\\hfil}%\n  \\fi}\n%\n% TABLE\n\\newcounter{table}\n\\if@secfloat\n  \\@addtoreset{table}{section}\n\\renewcommand \\thetable{\\thesection.\\@arabic\\c@table}\n\\else\n  \\renewcommand \\thetable{\\@arabic\\c@table}\n\\fi\n\\def\\fps@table{tbp}\n\\def\\ftype@table{2}\n\\def\\ext@table{lot}\n\\def\\fnum@table{\\tablename~\\thetable.}\n%\n\\newenvironment{table}\n               {\\let\\@makecaption\\@maketablecaption%\n               \\let\\@floatboxreset\\@tableboxreset\\@float{table}}\n               {\\end@float}\n\\newenvironment{table*}\n               {\\let\\@makecaption\\@maketablecaption%\n               \\let\\@floatboxreset\\@tableboxreset\\@dblfloat{table}}\n               {\\end@dblfloat}\n%\n\\def\\@tableboxreset{%\n        \\reset@font%\n        \\centering\\footnotesize%\n        \\def\\arraystretch{1.2}\n        \\@setnobreak%\n        \\@setminipage%\n}\n\n\\newlength\\abovecaptionskip\n\\newlength\\belowcaptionskip\n\\setlength\\abovecaptionskip{8\\p@}\n\\setlength\\belowcaptionskip{3\\p@}\n%\n\\newdimen\\tablewidth \\tablewidth\\textwidth\n\\newdimen\\saved@tablewidth \\saved@tablewidth\\textwidth\n%\n\\long\\def\\@maketablecaption#1#2{%\n \\begingroup%\n    \\footnotesize%\n    \\global\\setbox\\@tempboxa\\hbox{\\textbf{#1}\\enspace #2}%\n \\endgroup%\n \\centering%\n \\ifdim \\wd\\@tempboxa>\\tablewidth %\n    \\parbox[t]{\\tablewidth}{\\footnotesize\\textbf{#1}\\enspace #2\\vphantom{Ay}\\par}%\n \\else\n    \\hbox to\\hsize{\\hfill\\box\\@tempboxa\\vphantom{Ay}\\hfill}%\n \\fi%\n \\global\\saved@tablewidth\\tablewidth%\n \\global\\tablewidth\\hsize\\vskip\\belowcaptionskip}\n%\n%\n%%****************** Algorithm\n\\newcounter{algorithm}\n\\if@secfloat\n  \\@addtoreset{algorithm}{section}\n\\renewcommand \\thealgorithm{\\thesection.\\@arabic\\c@algorithm}\n\\else\n  \\renewcommand \\thealgorithm{\\@arabic\\c@algorithm}\n\\fi\n\\def\\fps@algorithm{tbp}\n\\def\\ftype@algorithm{4}\n\\def\\ext@algorithm{loa}\n\\def\\fnum@algorithm{\\algorithmname~\\thealgorithm.}\n%\n\\newenvironment{algorithm}\n               {\\let\\@makecaption\\@makealgorithmcaption%\n               \\let\\@floatboxreset\\@algorithmboxreset\\@float{algorithm}}\n               {\\end@float}\n\\newenvironment{algorithm*}\n               {\\let\\@makecaption\\@makealgorithmcaption%\n               \\let\\@floatboxreset\\@algorithmboxreset\\@dblfloat{algorithm}}\n               {\\end@dblfloat}\n\n\\def\\@algorithmboxreset{%\n        \\reset@font%\n        \\centering\n        \\@setnobreak%\n        \\@setminipage%\n}\n\\long\\def\\@makealgorithmcaption#1#2{\\vskip 2ex \\small\n  \\hbox to \\hsize{\\parbox[t]{\\hsize}{{\\bf #1} #2}}}\n%\n%%%% Program Code:\n\\def\\programcode{%\n\\let\\@makealgorithmcaption\\@makefigurecaption\n\\def\\algorithmname{Program Code}}\n\n\n%********************* COMPATIBILITY WITH OLD LATEX:\n\\DeclareOldFontCommand{\\rm}{\\normalfont\\rmfamily}{\\mathrm}\n\\DeclareOldFontCommand{\\sf}{\\normalfont\\sffamily}{\\mathsf}\n\\DeclareOldFontCommand{\\tt}{\\normalfont\\ttfamily}{\\mathtt}\n\\DeclareOldFontCommand{\\bf}{\\normalfont\\bfseries}{\\mathbf}\n\\DeclareOldFontCommand{\\it}{\\normalfont\\itshape}{\\mathit}\n\\let\\sl\\it\n\\DeclareOldFontCommand{\\sc}{\\normalfont\\scshape}{\\@nomath\\sc}\n\\DeclareRobustCommand*\\cal{\\@fontswitch\\relax\\mathcal}\n\\DeclareRobustCommand*\\mit{\\@fontswitch\\relax\\mathnormal}\n%\n% *********** MATH\n%\n\\if@secthm\n \\@addtoreset{thm}{section}\n \\def\\thethm{\\thesection.\\arabic{thm}}\n\\else\n \\def\\thethm{\\arabic{thm}}\n\\fi\n%\n%***************************** BIBLIOGRAPHY\n\n\\newenvironment{thebibliography}[1]\n     {\\section*{\\refname}\\footnotesize\\rmfamily\\upshape%\n      \\list{\\@biblabel{\\@arabic\\c@enumiv}}%\n           {\\settowidth\\labelwidth{\\@biblabel{#1}}%\n            \\leftmargin\\labelwidth\n            \\setlength\\labelsep{8\\p@}\n            \\advance\\leftmargin\\labelsep\n            \\usecounter{enumiv}%\n            \\let\\p@enumiv\\@empty\n            \\renewcommand\\theenumiv{\\@arabic\\c@enumiv}}%\n      \\sloppy\n      \\clubpenalty4000\n      \\@clubpenalty \\clubpenalty\n      \\widowpenalty4000%\n      \\sfcode`\\.\\@m}\n     {\\def\\@noitemerr\n       {\\@latex@warning{Empty `thebibliography' environment}}%\n      \\endlist}\n%\n\\newcommand\\newblock{\\hskip .11em\\@plus.33em\\@minus.07em}\n%\n\\def\\@citex[#1]#2{%\n  \\let\\@citea\\@empty\n  \\@cite{\\@for\\@citeb:=#2\\do\n    {\\@citea\\def\\@citea{,\\penalty\\@m\\hskip.1pt}%\n     \\edef\\@citeb{\\expandafter\\@firstofone\\@citeb}%\n     \\if@filesw\\immediate\\write\\@auxout{\\string\\citation{\\@citeb}}\\fi\n     \\@ifundefined{b@\\@citeb}{\\mbox{\\reset@font\\bfseries ?}%\n       \\G@refundefinedtrue\n       \\@latex@warning\n         {Citation `\\@citeb' on page \\thepage \\space undefined}}%\n       {\\hbox{\\csname b@\\@citeb\\endcsname}}}}{#1}}\n\n%****************************** FRONTMATTER *\n%\n\\newtoks\\t@glob@notes\n\\newtoks\\t@loc@notes\n\\newcount\\note@cnt\n\\newcounter{author}\n\\newcount\\n@author\n\\def\\n@author@{}\n\\newcounter{address}\n%\n\\newcount\\sv@hyphenpenalty\n%\n\\newcount\\prev@elem \\prev@elem=0\n\\newcount\\cur@elem  \\cur@elem=0\n\\chardef\\e@pretitle=1\n\\chardef\\e@title=1\n\\chardef\\e@subtitle=1\n\\chardef\\e@author=2\n\\chardef\\e@address=3\n%\n\\newif\\if@newelem\n\\newif\\if@firstauthor\n\\newif\\if@preface\n\\newif\\if@hasabstract\n\\newif\\if@haskeywords\n%\n\\newbox\\fm@box\n\\newdimen\\fm@size\n\\newbox\\t@abstract\n\\newbox\\t@keywords\n%\n\\def\\add@tok#1#2{\\global#1\\expandafter{\\the#1#2}}\n\\def\\add@xtok#1#2{\\begingroup\n  \\no@harm\n  \\xdef\\@act{\\global\\noexpand#1{\\the#1#2}}\\@act\n\\endgroup}\n%\n\\def\\tailthanksref[#1]#2{\\noexpand\\pthanksref{#1}}\n\\def\\pthanksref#1{\\global\\advance\\note@cnt\\@ne\\ifnum\\note@cnt>\\@ne\n\\global\\t@loc@notes\\expandafter{\\the\\t@loc@notes\\note@sep}\\fi\n\\global\\t@loc@notes\\expandafter{\\the\\t@loc@notes#1}}\n%\n\\def\\beg@elem{\\global\\t@loc@notes={}\\global\\note@cnt\\z@}\n\\def\\@xnamedef#1{\\expandafter\\xdef\\csname #1\\endcsname}\n\\def\\no@harm{%\n  \\let\\\\=\\relax  \\let\\rm\\relax\n  \\let\\ss=\\relax \\let\\ae=\\relax \\let\\oe=\\relax\n  \\let\\AE=\\relax \\let\\OE=\\relax\n  \\let\\o=\\relax  \\let\\O=\\relax\n  \\let\\i=\\relax  \\let\\j=\\relax\n  \\let\\aa=\\relax \\let\\AA=\\relax\n  \\let\\l=\\relax  \\let\\L=\\relax\n  \\let\\d=\\relax  \\let\\b=\\relax \\let\\c=\\relax\n  \\let\\bar=\\relax\n  \\def\\protect{\\noexpand\\protect\\noexpand}}\n%\n\\def\\proc@elem#1#2{\\begingroup\n    \\no@harm\n    \\def\\thanks##1##{\\@gobble}%\n    \\def\\thanksref##1##{\\@gobble}%\n    \\@xnamedef{@#1}{#2}%\n  \\endgroup\n  \\prev@elem=\\cur@elem\n  \\cur@elem=\\csname e@#1\\endcsname\n  \\expandafter\\elem@nothanksref#2\\thanksref\\relax%\n  \\expandafter\\elem@nothanks#2\\thanks\\relax}\n%\n\\def\\elem@nothanksref#1\\thanksref{\\futurelet\\@peektok\\elem@thanksref}\n\\def\\elem@thanksref{\\ifx\\@peektok\\relax\n  \\else \\expandafter\\elem@morethanksref \\fi}\n\\def\\elem@morethanksref[#1]#2{\\add@thanks{#1}\\elem@nothanksref}\n%\n\\def\\elem@nothanks#1\\thanks{\\futurelet\\@peektok\\elem@thanks}\n\\def\\elem@thanks{\\ifx\\@peektok\\relax\n  \\else \\ifx\\@peektok[ \\expandafter\\expandafter\\expandafter\\elem@morethankse\n  \\else \\expandafter\\expandafter\\expandafter\\elem@morethanks \\fi\\fi}\n%\n\\def\\elem@morethankse[#1]#2{\\thanks@optarg[#1]{#2}\\add@thanks{#1}\\elem@nothanks}\n\\def\\elem@morethanks#1{\\thanks@optarg[]{#1}\\add@thanks{}\\elem@nothanks}\n%\n\\def\\add@thanks#1{%\n  \\global\\advance\\note@cnt\\@ne\n    \\ifnum\\note@cnt>\\@ne \\add@xtok\\t@loc@notes{\\note@sep}\\fi\n    \\ifx.#1.\\add@xtok\\t@loc@notes{\\thefootnote}\\else\n        \\add@xtok\\t@loc@notes{#1}\\fi%\n}\n\\def\\add@addressref#1{%\n  \\global\\advance\\note@cnt\\@ne\n    \\ifnum\\note@cnt>\\@ne \\add@xtok\\t@loc@notes{\\note@sep}\\fi\n    \\add@tok\\t@loc@notes{\\ref{#1}}%\n}\n\\def\\note@sep{,}\n%\n\\def\\thanks@optarg[#1]#2{%\n    \\ifx.#1.\\add@tok\\t@glob@notes{\\footnotetext}%\n    \\else\\add@tok\\t@glob@notes{\\freefootnotetext}\\fi%\n    \\refstepcounter{footnote}%\n    \\ifx.#1.\\add@xtok\\t@glob@notes{[\\the\\c@footnote]}%\n    \\else\\add@xtok\\t@glob@notes{[#1]}\\fi%\n    \\add@tok\\t@glob@notes{{#2}}%\n        \\ignorespaces}%\n%\n% FRONTMATTER\n%\n\\def\\artty#1{}\n%\n\\newdimen\\a@title@skip   \\a@title@skip=12\\p@\n\\newskip\\b@section@skip  \\b@section@skip=12\\p@ plus6\\p@ minus6\\p@%\n\\newskip\\b@pretitle@skip \\b@pretitle@skip=6\\p@\n%\n\\def\\frontmatter{%\n  \\let\\@corresp@note\\relax\n  \\global\\t@glob@notes={}\\global\\c@author\\z@\n  \\global\\c@address\\z@\n  \\global\\n@author=0\\n@author@\\relax\n  \\global\\advance\\n@author\\m@ne\n  \\global\\@firstauthortrue\n  \\global\\@hasabstractfalse\n  \\global\\@prefacefalse\n  \\parindent\\z@\n  \\open@fm \\ignorespaces}\n%\n\\def\\preface{\\@prefacetrue}\n%\n% ENDFRONTMATTER\n%\n\\def\\endfrontmatter{%\n  \\global\\n@author=\\c@author \\@writecount\n  \\global\\@topnum\\z@\n  \\ifx\\@firstpage\\@lastpage\n    \\gdef\\@pagerange{\\@firstpage}\n  \\else\n    \\gdef\\@pagerange{\\@firstpage--\\@lastpage}\n  \\fi\n%  \\thispagestyle{copyright}%\n  \\if@twocolumn\\else\\output@glob@notes\\fi\n  \\if@preface\n    \\@hasabstractfalse\n  \\fi\n  \\if@hasabstract\n    \\normal@text\n    \\vskip 18\\p@\n    \\centering\n    \\leavevmode\\box\\t@abstract\\par\n  \\fi\n  \\if@haskeywords\n    \\normal@text\n    \\if@hasabstract \\vskip6pt\\else\\vskip18pt\\fi    \n    \\centering\n    \\leavevmode\\box\\t@keywords\\par\n  \\fi\n  \\close@fm\n  \\if@twocolumn\\output@glob@notes\\fi\n  \\markboth{\\@runauthor\\@runtitle}{\\@runauthor\\@runtitle}%\n  \\global\\@prefacefalse\n  \\global\\leftskip\\z@\n  \\global\\@rightskip\\z@\n  \\global\\rightskip\\@rightskip\n%  \\global\\c@footnote=0\n  \\let\\title\\relax       \\let\\author\\relax\n  \\let\\address\\relax\n  \\let\\frontmatter\\relax \\let\\endfrontmatter\\relax\n  \\let\\@maketitle\\relax  \\let\\@@maketitle\\relax\n  \\normal@text}\n%\n% Dvieju koloneliu zurnale per visa lapo ploti eina\n% tik pretitle, title ir subtitle. Tam ivedame komanda\n% \\maketitle, kuri uzdaro box'a\n\n  \\def\\two@c@maketitle{%\n    \\global\\let\\close@fm\\relax%\n    \\vskip\\b@section@skip%\n    \\par \\egroup\n    \\emergencystretch=1pc \\twocolumn[\\unvbox\\fm@box]}\n%\n\\if@restonecol\n  \\let\\maketitle\\relax\n\\else\n  \\let\\maketitle\\two@c@maketitle\n\\fi\n%\n\n%\n\\newdimen\\t@xtheight\n\\def\\init@settings{\n\\splittopskip=\\topskip \\splitmaxdepth=\\maxdepth\n\\t@xtheight\\textheight \\advance\\t@xtheight-\\splittopskip}\n%\n\\def\\open@fm{\n  \\global\\setbox\\fm@box=\\vbox\\bgroup\n  \\hsize=\\textwidth\n  \\centering\n  \\sv@hyphenpenalty\\hyphenpenalty\n  \\hyphenpenalty\\@M}\n%\n\n\\def\\close@fm{%\n  \\vskip\\b@section@skip%\n  \\par \\egroup\n  \\if@twocolumn\\else%\n    \\fm@size=\\dp\\fm@box \\advance\\fm@size by \\ht\\fm@box\n    \\@whiledim\\fm@size>\\t@xtheight \\do{%\n      \\global\\setbox\\@tempboxa=\\vsplit\\fm@box to \\t@xtheight\n      \\unvbox\\@tempboxa \\newpage\n      \\fm@size=\\dp\\fm@box 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0\\fi\\the\\@copyear}\n\n%\n\\pubyear{2003}\n%\n\\def\\firstpage#1{\\def\\@tempa{#1}\\ifx\\@tempa\\@empty\\else\n  \\gdef\\@firstpage{#1}\\gdef\\@lastpage{#1}%\n  \\global\\c@page=#1 \\ignorespaces\\fi\n  }\n\\def\\@firstpage{1}\n\\def\\lastpage#1{\\def\\@tempa{#1}\\ifx\\@tempa\\@empty\\else\n  \\gdef\\@lastpage{#1}\\ignorespaces\\fi}\n\\def\\@lastpage{0}\n\\def\\@pagerange{1--0}\n\n% Write the last page:\n\\def\\write@last@page{%\n\\write\\@mainaux{\\string\\global\\string\\@namedef{@lastpage}{\\the\\c@page}}}\n\n\\AtEndDocument{\\write@last@page}\n% SGML\n\\long\\def\\convertas#1#2{#2}\n\\def\\sday#1{#1}\\def\\smonth#1{#1}\\def\\syear#1{#1}\n\\def\\aid#1{\\gdef\\@aid{#1}}\n%\n\\def\\SSDI#1{\\gdef\\@ssdi{#1}} \\def\\@ssdi{000000-00}\n\\def\\issn#1{\\gdef\\@issn{#1}}\n\\def\\price#1{\\gdef\\@price{#1}}\n%\n\\def\\date#1{\\gdef\\@date{#1}}    \\def\\@date{\\today}\n%\n\n\\def\\empty@data{\\@nil}\n%\n\n%***************** BACKMATTER\n\\newcommand\\backmatter{\\goodbreak}\n\n%**************** INICIALIZATION\n\\newcommand\\refname{References}\n\\newcommand\\figurename{Figure}\n\\newcommand\\tablename{Table}\n\\newcommand\\algorithmname{Algorithm}\n\\newcommand\\appendixname{Appendix}\n\\newcommand\\abstractname{Abstract. }\n\\newcommand\\keywordsname{Keywords. }\n\\def\\acknowledgementsname{Acknowledgements}\n%\n\\def\\copyright@sign{\\copyright}\n%\n% DIMENSIONS\n\\def\\@articletypesize{\\large}\n\\def\\pretitle@size{\\LARGE}\n\\def\\title@size{\\huge}\n\\def\\subtitle@size{\\large\\itshape}\n\\def\\authors@size{\\normalsize}\n\\def\\abstract@size{\\footnotesize}\n\\def\\abstract@width{22pc}\n\\def\\abstract@indent{\\noindent}\n\\def\\address@size{\\normalsize\\itshape}\n% Block preparation of contents:\n\\def\\addcontentsline#1#2#3{}\n\\long\\def\\addtocontents#1#2{}\n%\n\\newcommand\\today{}\n\\edef\\today{\\ifcase\\month\\or\n  January\\or February\\or March\\or April\\or May\\or June\\or\n  July\\or August\\or September\\or October\\or November\\or December\\fi\n  \\space\\number\\day, \\number\\year}\n%\n\\@twosidetrue\n\\pagenumbering{arabic}\n\\frenchspacing\n\\init@settings\n\n\\if@twocolumn\\setlength\\tablewidth{\\columnwidth}\n\\else\\setlength\\tablewidth{\\textwidth}\\fi\n%\\pagestyle{headings}\n\\pagestyle{empty}\n\n\\endinput\n%%\n%% End of file `IOS-Book-Article.cls'.\n"
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    "path": "marktoberdorf_paper/DSE/cpp2.json",
    "content": "{ \"name\": \"cpp2\",  \n  \"extend\": \"cpp\",  \n  \"extraKeywords\": [\"function\",\"override\",\"var\",\"instanceof\"]\n}"
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    "path": "marktoberdorf_paper/DSE/dse.bib",
    "content": "@article{King76,\n  author = {James C. King},\n  title = {Symbolic Execution and Program Testing},\n  journal = {Communications of the ACM},\n  volume = {19},\n  number = {7},\n  pages = {385–394}, \n  year = {1976}\n}\n\n@article{Clarke76,\n  author    = {Lori A. Clarke},\n  title     = {A System to Generate Test Data and Symbolically Execute Programs},\n  journal   = {{IEEE} Transactions on Software Engineering},\n  volume    = {2},\n  number    = {3},\n  pages     = {215--222},\n  year      = {1976}\n}\n\n@book{Dijkstra76,\n  author    = {Edsger W. Dijkstra},\n  title     = {A Discipline of Programming},\n  publisher = {Prentice-Hall},\n  year      = {1976}\n}\n\n@article{Korel90,\n  author    = {Bogdan Korel},\n  title     = {Automated Software Test Data Generation},\n  journal   = {{IEEE} Transactions on Software Engineering},\n  volume    = {16},\n  number    = {8},\n  pages     = {870--879},\n  year      = {1990}\n}\n\n@article{Korel92,\n  author    = {Bogdan Korel},\n  title     = {Dynamic Method of Software Test Data Generation},\n  journal   = {Journal of Software Testing, Verification and Reliability},\n  volume    = {2},\n  number    = {4},\n  pages     = {203--213},\n  year      = {1992}\n}\n\n@inproceedings{Gupta00,\n  author    = {Neelam Gupta and\n               Aditya P. Mathur and\n               Mary Lou Soffa},\n  title     = {Generating Test Data for Branch Coverage},\n  booktitle = {Proceedings of the Automate Software Engineering Conference},\n  pages     = {219--228},\n  year      = {2000}\n}\n\n@inproceedings{GodefroidKS05,\n  author    = {Patrice Godefroid and\n               Nils Klarlund and\n               Koushik Sen},\n  title     = {{DART:} directed automated random testing},\n  booktitle = {Proceedings of the {ACM} {SIGPLAN} Conference on Programming\n               Language Design and Implementation},\n  pages     = {213--223},\n  year      = {2005}\n}\n\n@inproceedings{SenACAV06,\n  author    = {Koushik Sen and\n               Gul Agha},\n  title     = {{CUTE} and jCUTE: Concolic Unit Testing and Explicit Path Model-Checking\n               Tools},\n  booktitle = {Proceedings of 18th Computer Aided Verification Conference},\n  pages     = {419--423},\n  year      = {2006}\n}\n\n@inproceedings{CadarE05,\n  author    = {Cristian Cadar and\n               Dawson R. Engler},\n  title     = {Execution Generated Test Cases: How to Make Systems Code Crash Itself},\n  booktitle = {Proceedings of 12th International {SPIN} Workshop},\n  pages     = {2--23},\n  year      = {2005}\n}\n\n@inproceedings{CadarGPDE06,\n  author    = {Cristian Cadar and\n               Vijay Ganesh and\n               Peter M. Pawlowski and\n               David L. Dill and\n               Dawson R. Engler},\n  title     = {{EXE:} automatically generating inputs of death},\n  booktitle = {Proceedings of the 13th {ACM} Conference on Computer and Communications\n               Security},\n  pages     = {322--335},\n  year      = {2006}\n}\n\n@inproceedings{CadarDE08,\n  author    = {Cristian Cadar and\n               Daniel Dunbar and\n               Dawson R. Engler},\n  title     = {{KLEE:} Unassisted and Automatic Generation of High-Coverage Tests\n               for Complex Systems Programs},\n  booktitle = {Proceedings of the 8th {USENIX} Symposium on Operating Systems Design and Implementation},\n  pages     = {209--224},\n  year      = {2008}\n}\n\n@inproceedings{deMouraB08,\n  author    = {Leonardo Mendon{\\c{c}}a de Moura and\n               Nikolaj Bj{\\o}rner},\n  title     = {{Z3:} An Efficient {SMT} Solver},\n  booktitle = {Proceedings of the 14th International Conference of Tools and Algorithms for the Construction and Analysis of Systems},\n  pages     = {337--340},\n  year      = {2008}\n}\n\n@inproceedings{Godefroid11,\n  author    = {Patrice Godefroid},\n  title     = {Higher-order test generation},\n  booktitle = {Proceedings of the {ACM} {SIGPLAN} Conference on Programming\n               Language Design and Implementation},\n  pages     = {258--269},\n  year      = {2011}\n}\n\n@article{GodefroidLM12,\n  author    = {Patrice Godefroid and\n               Michael Y. Levin and\n               David A. Molnar},\n  title     = {{SAGE:} whitebox fuzzing for security testing},\n  journal   = {Communications of the {ACM}},\n  volume    = {55},\n  number    = {3},\n  pages     = {40--44},\n  year      = {2012}\n}\n\n@article{CadarS13,\n  author    = {Cristian Cadar and\n               Koushik Sen},\n  title     = {Symbolic execution for software testing: three decades later},\n  journal   = {Communications of the {ACM}},\n  volume    = {56},\n  number    = {2},\n  pages     = {82--90},\n  year      = {2013}\n}\n"
  },
  {
    "path": "marktoberdorf_paper/DSE/ignores.dic",
    "content": ""
  },
  {
    "path": "marktoberdorf_paper/DSE/out/DSE.tex",
    "content": "\\documentclass{IOS-Book-Article}\n% generated by Madoko, version 0.9.3-beta\n%mdk-data-line={1}\n\n\\usepackage[heading-base=2]{madoko}\n\n\n\\begin{document}\n%mdk-begin-texraw\n%mdk-data-line={25}\n\\begin{frontmatter}              % The preamble begins here.\n%\\pretitle{Pretitle}\n\\title{Deconstructing Dynamic Symbolic Execution}\n%\\runningtitle{IOS Press Style Sample}\n%\\subtitle{Subtitle}\n\\author[A]{\\fnms{Thomas} \\snm{Ball}}\nand \n\\author[B]{\\fnms{Jakub} \\snm{Daniel}}\n\n\\runningauthor{Thomas Ball et al.}\n\\address[A]{Microsoft Research}\n\\address[B]{Charles University}\n\\begin{mdDiv}[class={abstract},elem={abstract},data-line={39}]%\n\\begin{mdP}[data-line={40}]%\n%mdk-data-line={40}\n{}Dynamic symbolic execution (DSE) is a well-known technique\nfor automatically generating tests to achieve higher levels\nof coverage in a program. Two keys ideas of DSE are\nto: (1) seed symbolic execution by executing a program on an\ninitial input; (2) use concrete values from the program\nexecution in place of symbolic expressions whenever symbolic\nreasoning is hard or not desired. We describe\nDSE for a simple core language and then present\na minimalist implementation of DSE for Python (in Python) \nthat follows this basic recipe. The code is available \nat https://www.github.com/thomasjball/PyExZ3/ (tagged %mdk-data-line={50}\n{}{\\textquotedblleft}v1.0{\\textquotedblright}%mdk-data-line={50}\n{}) \nand has been designed to make it easy to experiment with and\nextend.%\n\\end{mdP}%%\n\\end{mdDiv}%\n%mdk-begin-texraw\n%mdk-data-line={56}\n\\begin{keyword}\nSymbolic Execution, Automatic Test Generation, White-box Testing, Automated \nTheorem Provers\n\\end{keyword}\n\\end{frontmatter}\n\\thispagestyle{empty}\n\\pagestyle{empty}\n\\mdHxx[id=sec-intro,label={[1]\\{.heading-label\\}},toc={},data-line={70},caption={[[1]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Introduction},bookmark={1.{\\hspace{0.5em}}Introduction}]{%mdk-data-line={70}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{1}.{\\hspace{0.5em}}}%mdk-data-line={70}\n{}Introduction}%mdk-data-line={73}\n\\defcommand{\\mathkw}[1]{\\textbf{#1}}\n\\begin{mdP}[data-line={78}]%\n%mdk-data-line={78}\n{}Static, path-based symbolic execution explores one control-flow path\nat a time through a (sequential) program %mdk-data-line={79}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={79}\n{}, using an automated theorem\nprover (ATP) to determine if the current path %mdk-data-line={80}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={80}\n{} is feasible%mdk-data-line={80}\n{}{\\mdNbsp}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{clarke76}{}{\\mdSpan[class={bibitem-label}]{4}}, \\mdA[class={bibref,localref},target-element={bibitem}]{king76}{}{\\mdSpan[class={bibitem-label}]{11}}]}%mdk-data-line={80}\n{}. \nIdeally, symbolic execution of a path %mdk-data-line={81}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={81}\n{} through program\n%mdk-data-line={82}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={82}\n{} yields a logic formula %mdk-data-line={82}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\phi_p$}%mdk-data-line={82}\n{} that describes the set of inputs %mdk-data-line={82}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$I$}%mdk-data-line={82}\n{} (possibly empty)\nto program %mdk-data-line={83}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={83}\n{} such that for any %mdk-data-line={83}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$i \\in I$}%mdk-data-line={83}\n{}, the execution %mdk-data-line={83}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P(i)$}%mdk-data-line={83}\n{} follows path %mdk-data-line={83}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={83}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={85}]%\n%mdk-data-line={85}\n{}If the formula %mdk-data-line={85}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\phi_p$}%mdk-data-line={85}\n{} is unsatisfiable then %mdk-data-line={85}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$I$}%mdk-data-line={85}\n{} is empty and so path %mdk-data-line={85}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={85}\n{} is not feasible; \nif the formula is satisfiable then %mdk-data-line={86}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$I$}%mdk-data-line={86}\n{} is not empty and so path %mdk-data-line={86}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={86}\n{} is feasible.\nIn this case, a model of %mdk-data-line={87}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\phi_p$}%mdk-data-line={87}\n{} provides a witness %mdk-data-line={87}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$i \\in I$}%mdk-data-line={87}\n{}.  Thus, a model-generating ATP\ncan be used in conjunction with\nsymbolic execution to automatically generate tests to cover paths\nin a program. Combined with a search strategy, one gets, in the limit,\nan exhaustive white-box testing procedure, for which there are many\napplications%mdk-data-line={92}\n{}{\\mdNbsp}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{cadars13}{}{\\mdSpan[class={bibitem-label}]{2}}, \\mdA[class={bibref,localref},target-element={bibitem}]{cadargpde06}{}{\\mdSpan[class={bibitem-label}]{3}}, \\mdA[class={bibref,localref},target-element={bibitem}]{godefroidlm12}{}{\\mdSpan[class={bibitem-label}]{9}}]}%mdk-data-line={92}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={94}]%\n%mdk-data-line={94}\n{}The formula %mdk-data-line={94}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\phi_p$}%mdk-data-line={94}\n{} is called a %mdk-data-line={94}\n{}\\mdEm{path-condition}%mdk-data-line={94}\n{} of the path %mdk-data-line={94}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={94}\n{}. \nWe will see that a given path %mdk-data-line={95}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={95}\n{} can induce many different path-conditions.\nA path-condition %mdk-data-line={96}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\psi_p$}%mdk-data-line={96}\n{} for path %mdk-data-line={96}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={96}\n{} is %mdk-data-line={96}\n{}\\mdEm{sound}%mdk-data-line={96}\n{} if \nevery input assignment satisfying %mdk-data-line={97}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\psi_p$}%mdk-data-line={97}\n{} defines an\nexecution of program %mdk-data-line={98}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={98}\n{} that follows path %mdk-data-line={98}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={98}\n{}{\\mdNbsp}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{godefroid11}{}{\\mdSpan[class={bibitem-label}]{7}}]}%mdk-data-line={98}\n{}. \nBy its definition, the formula  %mdk-data-line={99}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\phi_p$}%mdk-data-line={99}\n{} is sound and the best representation of %mdk-data-line={99}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={99}\n{}\n(as for all sound path-conditions %mdk-data-line={100}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\psi_p$}%mdk-data-line={100}\n{}, we have that %mdk-data-line={100}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\psi_p \\implies \\phi_p$}%mdk-data-line={100}\n{}).\nIn practice, we attempt to compute sound under-approximations of %mdk-data-line={101}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\phi_p$}%mdk-data-line={101}\n{} \nsuch as %mdk-data-line={102}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\psi_p$}%mdk-data-line={102}\n{}. However, we also find it necessary (and useful) to \ncompute unsound path-conditions.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={105}]%\n%mdk-data-line={105}\n{}A path-condition can be translated into the input\nlanguage of an ATP, such as%mdk-data-line={106}\n{}{\\mdNbsp}\\mdA[data-linkid={z3}]{http://z3.codeplex.org/}{}{Z3}%mdk-data-line={106}\n{}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{demourab08}{}{\\mdSpan[class={bibitem-label}]{5}}]}%mdk-data-line={106}\n{}, which provides an answer\nof %mdk-data-line={107}\n{}{\\textquotedblleft}unsatisfiable{\\textquotedblright}%mdk-data-line={107}\n{}, %mdk-data-line={107}\n{}{\\textquotedblleft}satisfiable{\\textquotedblright}%mdk-data-line={107}\n{} or %mdk-data-line={107}\n{}{\\textquotedblleft}unknown{\\textquotedblright}%mdk-data-line={107}\n{}, due to theoretical or practical\nlimitations in automatically deciding satisfiability of various logics.\nIn the case that the ATP is able to prove %mdk-data-line={109}\n{}{\\textquotedblleft}satisfiable{\\textquotedblright}%mdk-data-line={109}\n{} we can query it for \nsatisfying model in order to generate test inputs. A path-condition \nfor %mdk-data-line={111}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={111}\n{} can be thought of as function from a\nprogram%mdk-data-line={112}\n{}{'}%mdk-data-line={112}\n{}s primary inputs to a Boolean output representing whether\nor not %mdk-data-line={113}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={113}\n{} is executed under a given input. Thus, we are asking\nthe ATP to invert a function when we ask it to decide\nthe satisfiability/unsatisfiability of a path-condition.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={117}]%\n%mdk-data-line={117}\n{}The static translation of a path %mdk-data-line={117}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={117}\n{} through a program %mdk-data-line={117}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={117}\n{} into \nthe most precise path-condition %mdk-data-line={118}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\phi_p$}%mdk-data-line={118}\n{} is not a simple task, as \nprogramming languages and their semantics are very complex.\nCompletely characterizing the set of inputs %mdk-data-line={120}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$I$}%mdk-data-line={120}\n{} that follow\npath %mdk-data-line={121}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={121}\n{} means providing a symbolic interpretation of \nevery operation in the language so that the\nATP can reason about it. For example, consider a method call in Python. \nPython%mdk-data-line={124}\n{}{'}%mdk-data-line={124}\n{}s algorithm for method resolution order (see%mdk-data-line={124}\n{}{\\mdNbsp}\\mdA[data-linkid={mro}]{https://www.python.org/download/releases/2.3/mro/}{}{MRO}%mdk-data-line={124}\n{})\ndepends on the inheritance hierarchy of the program, a directed, \nacyclic graph that can evolve during program execution. Symbolically\nencoding Python%mdk-data-line={127}\n{}{'}%mdk-data-line={127}\n{}s method resolution order is possible but non-trivial.\nThere are other reasons it is hard or undesirable to symbolically \nexecute various operations, as will be explained in detail later.%\n\\end{mdP}%\n\\mdHxxx[id=sec-dynamic-symbolic-execution,label={[1.1]\\{.heading-label\\}},toc={},data-line={131},caption={[[1.1]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Dynamic symbolic execution},bookmark={1.1.{\\hspace{0.5em}}Dynamic symbolic execution}]{%mdk-data-line={131}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{1.1}.{\\hspace{0.5em}}}%mdk-data-line={131}\n{}Dynamic symbolic execution}\\begin{mdP}[data-line={133}]%\n%mdk-data-line={133}\n{}\\mdEm{Dynamic}%mdk-data-line={133}\n{} symbolic execution (DSE) is a form of path-based\nsymbolic execution based on two insights. First, the approach \nstarts by executing program %mdk-data-line={135}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={135}\n{} on\nsome input %mdk-data-line={136}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$i$}%mdk-data-line={136}\n{}, seeding the symbolic execution process\nwith a feasible path%mdk-data-line={137}\n{}{\\mdNbsp}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{gupta00}{}{\\mdSpan[class={bibitem-label}]{10}}, \\mdA[class={bibref,localref},target-element={bibitem}]{korel90}{}{\\mdSpan[class={bibitem-label}]{12}}, \\mdA[class={bibref,localref},target-element={bibitem}]{korel92}{}{\\mdSpan[class={bibitem-label}]{13}}]}%mdk-data-line={137}\n{}. \nSecond,  DSE\nuses concrete values from the execution %mdk-data-line={139}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P(i)$}%mdk-data-line={139}\n{} in place of symbolic expressions \nwhenever symbolic reasoning is not possible or desired%mdk-data-line={140}\n{}{\\mdNbsp}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{cadare05}{}{\\mdSpan[class={bibitem-label}]{1}}, \\mdA[class={bibref,localref},target-element={bibitem}]{godefroidks05}{}{\\mdSpan[class={bibitem-label}]{8}}]}%mdk-data-line={140}\n{}.\nThe major benefit of DSE is to\nsimplify the construction of a symbolic execution tool by\nleveraging concrete execution behavior (given by\nactually running the program).\nAs DSE combines both \nconcrete and symbolic reasoning, it also has been called %mdk-data-line={146}\n{}{\\textquotedblleft}concolic{\\textquotedblright}%mdk-data-line={146}\n{} \nexecution%mdk-data-line={147}\n{}{\\mdNbsp}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{senacav06}{}{\\mdSpan[class={bibitem-label}]{14}}]}%mdk-data-line={147}\n{}.%\n\\end{mdP}%\n\\begin{mdDiv}[class={figure,floating,align-center},id=fig-dse,label={[1]\\{.figure-label\\}},elem={figure},toc-line={[1]\\{.figure-label\\}. Pseudo-code for dynamic symbolic execution},toc={tof},float-env={figure},float-name={Figure},caption={Pseudo-code for dynamic symbolic execution},data-line={149}]%\n\\begin{mdPre}[class={para-block,pre-fenced,pre-fenced3,language-python,lang-python,python,highlighted},language={python},data-line={150},data-line-code={151}]%\n\\mdPrecode[data-line={151}]{{\\preindent{2}}\\mdToken{Identifier,Python}{i}{\\prespace{1}}\\mdToken{Keyword,Python}{=}{\\prespace{1}}\\mdToken{Identifier,Python}{an}{\\prespace{1}}\\mdToken{Identifier,Python}{input}{\\prespace{1}}\\mdToken{Identifier,Python}{to}{\\prespace{1}}\\mdToken{Identifier,Python}{program}{\\prespace{1}}\\mdToken{Constructor,Identifier,Python}{P}\\prebr{}\n{\\preindent{2}}\\mdToken{Keyword,Python}{while}{\\prespace{1}}\\mdToken{Identifier,Python}{defined}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{i}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{5}}\\mdToken{Identifier,Python}{p}{\\prespace{1}}\\mdToken{Keyword,Python}{=}{\\prespace{1}}\\mdToken{Identifier,Python}{path}{\\prespace{1}}\\mdToken{Identifier,Python}{covered}{\\prespace{1}}\\mdToken{Identifier,Python}{by}{\\prespace{1}}\\mdToken{Identifier,Python}{execution}{\\prespace{1}}\\mdToken{Constructor,Identifier,Python}{P}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{i}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\prebr{}\n{\\preindent{5}}\\mdToken{Identifier,Python}{cond}{\\prespace{1}}\\mdToken{Keyword,Python}{=}{\\prespace{1}}\\mdToken{Identifier,Python}{pathCondition}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{p}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\prebr{}\n{\\preindent{5}}\\mdToken{Identifier,Python}{s}{\\prespace{1}}\\mdToken{Keyword,Python}{=}{\\prespace{1}}\\mdToken{Constructor,Identifier,Python}{ATP}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Namespace,Identifier,Python}{Not}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{cond}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\prebr{}\n{\\preindent{5}}\\mdToken{Identifier,Python}{i}{\\prespace{1}}\\mdToken{Keyword,Python}{=}{\\prespace{1}}\\mdToken{Identifier,Python}{s}\\mdToken{Delimiter,Python}{.}\\mdToken{Identifier,Python}{model}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%\n\\end{mdPre}%\n\\mdHr[class={figureline,madoko},data-line={158}]{}\\begin{mdDiv}[data-line={159}]%\n%mdk-data-line={159}\n{}\\mdSpan[class={figure-caption}]{\\mdSpan[class={caption-before}]{\\mdStrong{Figure{\\mdNbsp}\\mdSpan[class={figure-label}]{1}.} }Pseudo-code for dynamic symbolic execution}%mdk-data-line={159}\n{}%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[class={indent},data-line={160}]%\n%mdk-data-line={160}\n{}The pseudo-code of Figure%mdk-data-line={160}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-dse}{}{\\mdSpan[class={figure-label}]{1}}%mdk-data-line={160}\n{} shows the high level process\nof DSE. The variable %mdk-data-line={161}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{i}}%mdk-data-line={161}\n{} represents an input\nto program %mdk-data-line={162}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Constructor,Identifier,Python}{P}}%mdk-data-line={162}\n{}. Execution of program %mdk-data-line={162}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Constructor,Identifier,Python}{P}}%mdk-data-line={162}\n{} on the input %mdk-data-line={162}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{i}}%mdk-data-line={162}\n{}\ntraces  a path %mdk-data-line={163}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{p}}%mdk-data-line={163}\n{}, from which\na logical formula %mdk-data-line={164}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{pathCondition}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{p}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={164}\n{} is constructed.\nFinally, the ATP is called with the negation of the path-condition\nto find a new input (that hopefully will cover a new path).  This\npseudo-code elides a number of details that we will deal with later.%\n\\end{mdP}%\n\\begin{mdDiv}[class={figure,floating,align-center},id=fig-easy-dse,label={[2]\\{.figure-label\\}},elem={figure},toc-line={[2]\\{.figure-label\\}. Easy example: computing the maximum of four numbers in Python.},toc={tof},float-env={figure},float-name={Figure},caption={Easy example: computing the maximum of four numbers in Python.},data-line={169}]%\n\\begin{mdPre}[class={para-block,pre-fenced,pre-fenced3,language-python,lang-python,python,highlighted},language={python},data-line={170},data-line-code={171}]%\n\\mdPrecode[data-line={171}]{\\mdToken{Keyword,Python}{def}{\\prespace{1}}\\mdToken{Identifier,Python}{max2}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{s}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{t}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{4}}\\mdToken{Keyword,Python}{if}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{s}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textless}}{\\prespace{1}}\\mdToken{Identifier,Python}{t}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{8}}\\mdToken{Keyword,Python}{return}{\\prespace{1}}\\mdToken{Identifier,Python}{t}\\prebr{}\n{\\preindent{4}}\\mdToken{Keyword,Python}{else}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{8}}\\mdToken{Keyword,Python}{return}{\\prespace{1}}\\mdToken{Identifier,Python}{s}\\prebr{}\n\\prebr{}\n\\mdToken{Keyword,Python}{def}{\\prespace{1}}\\mdToken{Identifier,Python}{max4}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{4}}\\mdToken{Keyword,Python}{return}{\\prespace{1}}\\mdToken{Identifier,Python}{max2}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{max2}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{max2}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%\n\\end{mdPre}%\n\\mdHr[class={figureline,madoko},data-line={180}]{}\\begin{mdDiv}[data-line={181}]%\n%mdk-data-line={181}\n{}\\mdSpan[class={figure-caption}]{\\mdSpan[class={caption-before}]{\\mdStrong{Figure{\\mdNbsp}\\mdSpan[class={figure-label}]{2}.} }Easy example: computing the maximum of four numbers in Python.}%mdk-data-line={181}\n{}%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[class={indent,para-continue},data-line={182}]%\n%mdk-data-line={182}\n{}Consider the  Python function %mdk-data-line={182}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{max4}}%mdk-data-line={182}\n{} in Figure%mdk-data-line={182}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-easy-dse}{}{\\mdSpan[class={figure-label}]{2}}%mdk-data-line={182}\n{},\nwhich computes the maximum of four numbers via three calls to\nthe function %mdk-data-line={184}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{max2}}%mdk-data-line={184}\n{}. Suppose we execute %mdk-data-line={184}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{max4}}%mdk-data-line={184}\n{} with values\nof zero for all four arguments.  In this case, the \nexecution path %mdk-data-line={186}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={186}\n{} contains three comparisons (in the order %mdk-data-line={186}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textless}}{\\prespace{1}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={186}\n{}, \n%mdk-data-line={187}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textless}}{\\prespace{1}}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={187}\n{}, %mdk-data-line={187}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textless}}{\\prespace{1}}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={187}\n{}), all of which evaluate false.\nThus, the path-condition for path %mdk-data-line={188}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={188}\n{} is %mdk-data-line={188}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Keyword,Python}{not}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Keyword,Python}{not}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Keyword,Python}{not}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textless}}{\\prespace{1}}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={188}\n{}.\nNegating this condition yields %mdk-data-line={189}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{or}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{or}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={189}\n{}.\nTaking the execution ordering of the three comparisons into account, we\nderive three expressions from the negated path-condition to generate\nnew inputs that will explore execution prefixes of path %mdk-data-line={192}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={192}\n{} of increasing length:%\n\\end{mdP}%\n\\begin{mdUl}[class={ul,list-star,compact},elem={ul},data-line={194}]%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(1)]\\{.ul-li-label\\}},elem={li},data-line={194}]%\n%mdk-data-line={194}\n{}\\mdEm{length 0}%mdk-data-line={194}\n{}: %mdk-data-line={194}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={194}\n{}%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(2)]\\{.ul-li-label\\}},elem={li},data-line={195}]%\n%mdk-data-line={195}\n{}\\mdEm{length 1}%mdk-data-line={195}\n{}: %mdk-data-line={195}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Keyword,Python}{not}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={195}\n{}%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(3)]\\{.ul-li-label\\}},elem={li},data-line={196}]%\n%mdk-data-line={196}\n{}\\mdEm{length 2}%mdk-data-line={196}\n{}: %mdk-data-line={196}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Keyword,Python}{not}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Keyword,Python}{not}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={196}\n{}%\n\\end{mdLi}%%\n\\end{mdUl}%\n\\begin{mdP}[class={para-continue},data-line={198}]%\n%mdk-data-line={198}\n{}The purpose of taking execution order into account should be clear, as the\ncomparison %mdk-data-line={199}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={199}\n{} only executes in the case where %mdk-data-line={199}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Keyword,Python}{not}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Keyword,Python}{not}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={199}\n{}\nholds. Integer solutions to the above three systems of constraints are:%\n\\end{mdP}%\n\\begin{mdUl}[class={ul,list-star,compact},elem={ul},data-line={202}]%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(1)]\\{.ul-li-label\\}},elem={li},data-line={202}]%\n%mdk-data-line={202}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{a}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{b}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{2}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{c}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{d}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}}%mdk-data-line={202}\n{}%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(2)]\\{.ul-li-label\\}},elem={li},data-line={203}]%\n%mdk-data-line={203}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{a}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{b}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{c}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{d}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{3}}%mdk-data-line={203}\n{}%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(3)]\\{.ul-li-label\\}},elem={li},data-line={204}]%\n%mdk-data-line={204}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{a}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{b}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{c}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{2}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{d}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}}%mdk-data-line={204}\n{}%\n\\end{mdLi}%%\n\\end{mdUl}%\n\\begin{mdP}[data-line={206}]%\n%mdk-data-line={206}\n{}In the three cases above, we sought solutions that kept as many of\nthe variables as possible equal to the original input (in which \nall variables are equal to 0). Execution of the %mdk-data-line={208}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{max4}}%mdk-data-line={208}\n{} function \non the input corresponding to the first solution produces the path-condition\n%mdk-data-line={210}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Keyword,Python}{not}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Keyword,Python}{not}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{b}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textless}}{\\prespace{1}}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={210}\n{}, from which we can produce more\ninputs.   For this (loop-free function), there are a finite number\nof path-conditions. We leave it as an exercise to the reader to \nenumerate them all.%\n\\end{mdP}%\n\\mdHxxx[id=sec-leveraging-concrete-values-in-dse,label={[1.2]\\{.heading-label\\}},toc={},data-line={215},caption={[[1.2]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Leveraging concrete values in DSE},bookmark={1.2.{\\hspace{0.5em}}Leveraging concrete values in DSE}]{%mdk-data-line={215}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{1.2}.{\\hspace{0.5em}}}%mdk-data-line={215}\n{}Leveraging concrete values in DSE}\\begin{mdP}[data-line={217}]%\n%mdk-data-line={217}\n{}We now consider several situations where we can make use of concrete\nvalues in DSE. In the realm of (unbounded-precision) integer arithmetic \n(e.g., bignum integer arithmetic, as in Python 3.0 onwards),\nit is easy  to come up with  tiny programs that will be %mdk-data-line={220}\n{}\\mdEm{very difficult}%mdk-data-line={220}\n{},\nif not %mdk-data-line={221}\n{}\\mdEm{impossible}%mdk-data-line={221}\n{}, \nfor any symbolic execution tool to deal with, such as the function %mdk-data-line={222}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{fermat3}}%mdk-data-line={222}\n{}\nin Figure%mdk-data-line={223}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-fermat3}{}{\\mdSpan[class={figure-label}]{3}}%mdk-data-line={223}\n{}.%mdk-data-line={223}\n{} %mdk-data-line={223}\n{}%\n\\end{mdP}%\n\\begin{mdDiv}[class={figure,floating,align-center},id=fig-fermat3,label={[3]\\{.figure-label\\}},elem={figure},toc-line={[3]\\{.figure-label\\}. Hard example for symbolic execution},toc={tof},float-env={figure},float-name={Figure},caption={Hard example for symbolic execution},data-line={226}]%\n\\begin{mdPre}[class={para-block,pre-fenced,pre-fenced3,language-python,lang-python,python,highlighted},language={python},data-line={227},data-line-code={228}]%\n\\mdPrecode[data-line={228}]{\\mdToken{Keyword,Python}{def}{\\prespace{1}}\\mdToken{Identifier,Python}{fermat3}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{y}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{z}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{3}}\\mdToken{Keyword,Python}{if}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textgreater}}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{y}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textgreater}}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{z}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textgreater}}{\\prespace{1}}\\mdToken{Number,Python}{0}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{6}}\\mdToken{Keyword,Python}{if}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{x}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{x}{\\prespace{1}}\\mdToken{Operator,Python}{+}{\\prespace{1}}\\mdToken{Identifier,Python}{y}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{y}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{y}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{z}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{z}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{z}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{10}}\\mdToken{Keyword,Python}{return}{\\prespace{1}}\\mdToken{String,Delim,Python,BracketOpen}{{\"}}\\mdToken{String,Python}{Fermat\\prespace{1}and\\prespace{1}Wiles\\prespace{1}were\\prespace{1}wrong!?!}\\mdToken{String,Delim,Python,BracketClose}{{\"}}\\prebr{}\n{\\preindent{3}}\\mdToken{Keyword,Python}{return}{\\prespace{1}}\\mdToken{Number,Python}{0}}%\n\\end{mdPre}%\n\\mdHr[class={figureline,madoko},data-line={234}]{}\\begin{mdDiv}[data-line={235}]%\n%mdk-data-line={235}\n{}\\mdSpan[class={figure-caption}]{\\mdSpan[class={caption-before}]{\\mdStrong{Figure{\\mdNbsp}\\mdSpan[class={figure-label}]{3}.} }Hard example for symbolic execution}%mdk-data-line={235}\n{}%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[class={indent},data-line={237}]%\n%mdk-data-line={237}\n{}Fermat%mdk-data-line={237}\n{}{'}%mdk-data-line={237}\n{}s Last Theorem, proved\nby Andrew Wiles in the late 20th century, states that no \nthree positive integers %mdk-data-line={239}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x$}%mdk-data-line={239}\n{}, %mdk-data-line={239}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$y$}%mdk-data-line={239}\n{}, and %mdk-data-line={239}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$z$}%mdk-data-line={239}\n{} can satisfy the equation\n%mdk-data-line={240}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x^n + y^n = z^n$}%mdk-data-line={240}\n{} for any integer value of %mdk-data-line={240}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$n$}%mdk-data-line={240}\n{} greater than two.\nThe function %mdk-data-line={241}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{fermat3}}%mdk-data-line={241}\n{} encodes this statement for %mdk-data-line={241}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$n=3$}%mdk-data-line={241}\n{}.   It\nis not reasonable to have a computer waste time trying to find\na solution that would cause %mdk-data-line={243}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{fermat3}}%mdk-data-line={243}\n{} to print the string \n%mdk-data-line={244}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{String,Delim,Python,BracketOpen}{{\"}}\\mdToken{String,Python}{Fermat\\prespace{1}and\\prespace{1}Wiles\\prespace{1}were\\prespace{1}wrong!?!}\\mdToken{String,Delim,Python,BracketClose}{{\"}}}%mdk-data-line={244}\n{}.  In cases of complex (non-linear)\narithmetic operations,\nsuch as %mdk-data-line={246}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{x}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{x}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{x}}%mdk-data-line={246}\n{}, we might choose to handle the operation concretely.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={248}]%\n%mdk-data-line={248}\n{}There are a number of ways to deal with the above issue: one is \nto recognize all non-linear terms in a symbolic expression and replace\nthem with their concrete counterparts during execution. For the %mdk-data-line={250}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{fermat3}}%mdk-data-line={250}\n{} \nexample, this would mean that during DSE the symbolic expression \n%mdk-data-line={252}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{x}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{x}{\\prespace{1}}\\mdToken{Operator,Python}{+}{\\prespace{1}}\\mdToken{Identifier,Python}{y}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{y}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{y}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{z}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{z}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{z}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={252}\n{} would be reduced to the constant %mdk-data-line={252}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{False}}%mdk-data-line={252}\n{}\nby evaluation on the concrete values of variables %mdk-data-line={253}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{x}}%mdk-data-line={253}\n{}, %mdk-data-line={253}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{y}}%mdk-data-line={253}\n{} and %mdk-data-line={253}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{z}}%mdk-data-line={253}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={255}]%\n%mdk-data-line={255}\n{}Besides difficult operations (such as non-linear arithmetic),\nother examples of code that we might treat\nconcretely instead of symbolically include\nfunctions that are hard to invert, such as cryptographic hash functions,\nor low-level functions that we do not wish to test (such as \noperating system functions).  Consider\nthe code in Figure%mdk-data-line={261}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-hash}{}{\\mdSpan[class={figure-label}]{4}}%mdk-data-line={261}\n{}, which applies the function\n%mdk-data-line={262}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{unknown}}%mdk-data-line={262}\n{} to argument %mdk-data-line={262}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{x}}%mdk-data-line={262}\n{} and compares it to argument %mdk-data-line={262}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{y}}%mdk-data-line={262}\n{}.\nBy using the name %mdk-data-line={263}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{unknown}}%mdk-data-line={263}\n{} we simply mean to say that we \nwish to model this function as a black box, with no knowledge\nof how it operates internally.%\n\\end{mdP}%\n\\begin{mdDiv}[class={figure,floating,align-center},id=fig-hash,label={[4]\\{.figure-label\\}},elem={figure},toc-line={[4]\\{.figure-label\\}. Another hard example for symbolic execution},toc={tof},float-env={figure},float-name={Figure},caption={Another hard example for symbolic execution},data-line={267}]%\n\\begin{mdPre}[class={para-block,pre-fenced,pre-fenced3,language-python,lang-python,python,highlighted},language={python},data-line={268},data-line-code={269}]%\n\\mdPrecode[data-line={269}]{\\mdToken{Keyword,Python}{def}{\\prespace{1}}\\mdToken{Identifier,Python}{dart}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{y}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{2}}\\mdToken{Keyword,Python}{if}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{unknown}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{y}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{5}}\\mdToken{Keyword,Python}{return}{\\prespace{1}}\\mdToken{Number,Python}{1}\\prebr{}\n{\\preindent{2}}\\mdToken{Keyword,Python}{return}{\\prespace{1}}\\mdToken{Number,Python}{0}}%\n\\end{mdPre}%\n\\mdHr[class={figureline,madoko},data-line={274}]{}\\begin{mdDiv}[data-line={275}]%\n%mdk-data-line={275}\n{}\\mdSpan[class={figure-caption}]{\\mdSpan[class={caption-before}]{\\mdStrong{Figure{\\mdNbsp}\\mdSpan[class={figure-label}]{4}.} }Another hard example for symbolic execution}%mdk-data-line={275}\n{}%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[class={indent},data-line={276}]%\n%mdk-data-line={276}\n{}In such a case, we can use DSE to execute the function %mdk-data-line={276}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{unknown}}%mdk-data-line={276}\n{}\non a specific input (say %mdk-data-line={277}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Number,Python}{5013}}%mdk-data-line={277}\n{}) and observe its output\n(say %mdk-data-line={278}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Number,Python}{42}}%mdk-data-line={278}\n{}). That is, rather than execute %mdk-data-line={278}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{unknown}}%mdk-data-line={278}\n{} symbolically\nand invoke an ATP to invert the function%mdk-data-line={279}\n{}{'}%mdk-data-line={279}\n{}s path-condition, we\nsimply treat the call to %mdk-data-line={280}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{unknown}}%mdk-data-line={280}\n{} concretely, substituting\nits return value (in this case %mdk-data-line={281}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Number,Python}{42}}%mdk-data-line={281}\n{}) for the specialized expression \n%mdk-data-line={282}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{unknown}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Number,Python}{5013}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{y}}%mdk-data-line={282}\n{} to get the predicate %mdk-data-line={282}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Number,Python}{42}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{y}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={282}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={284}]%\n%mdk-data-line={284}\n{}Adding the constraint %mdk-data-line={284}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{5013}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={284}\n{} yields the sound but\nrather specific path-condition \n%mdk-data-line={286}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{5013}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Number,Python}{42}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{y}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={286}\n{}.\nNote that the path-condition %mdk-data-line={287}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Number,Python}{42}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{y}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={287}\n{} is not sound, as it admits\nany value for the variable %mdk-data-line={288}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{x}}%mdk-data-line={288}\n{}, which likely includes many values\nfor which %mdk-data-line={289}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{unknown}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{y}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={289}\n{} is false.%\n\\end{mdP}%\n\\mdHxxx[id=sec-overview,label={[1.3]\\{.heading-label\\}},toc={},data-line={291},caption={[[1.3]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Overview},bookmark={1.3.{\\hspace{0.5em}}Overview}]{%mdk-data-line={291}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{1.3}.{\\hspace{0.5em}}}%mdk-data-line={291}\n{}Overview}\\begin{mdP}[class={para-continue},data-line={293}]%\n%mdk-data-line={293}\n{}This introduction elides many important \nissues that arise in implementing DSE for a real language, which we will \nfocus on in the remainder of the paper. These include how to:%\n\\end{mdP}%\n\\begin{mdUl}[class={ul,list-star,compact},elem={ul},data-line={297}]%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(1)]\\{.ul-li-label\\}},elem={li},data-line={297}]%\n%mdk-data-line={297}\n{}Identify the code under test %mdk-data-line={297}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={297}\n{} and the symbolic inputs to %mdk-data-line={297}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={297}\n{};%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(2)]\\{.ul-li-label\\}},elem={li},data-line={298}]%\n%mdk-data-line={298}\n{}Trace the control flow path %mdk-data-line={298}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={298}\n{} taken by execution %mdk-data-line={298}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P(i)$}%mdk-data-line={298}\n{};%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(3)]\\{.ul-li-label\\}},elem={li},data-line={299}]%\n%mdk-data-line={299}\n{}Reinterpret program operations to compute symbolic expressions;%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(4)]\\{.ul-li-label\\}},elem={li},data-line={300}]%\n%mdk-data-line={300}\n{}Generate a path-condition from %mdk-data-line={300}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={300}\n{} and the symbolic expressions;%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(5)]\\{.ul-li-label\\}},elem={li},data-line={301}]%\n%mdk-data-line={301}\n{}Generate a new input %mdk-data-line={301}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$i'$}%mdk-data-line={301}\n{} by negating (part of) the path-condition, translating\nthe path-condition to the input language of an ATP, invoking the ATP, and\nlifting a satisfying model (if any) back up to the source level;%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(6)]\\{.ul-li-label\\}},elem={li},data-line={304}]%\n%mdk-data-line={304}\n{}Guide the search to expose new paths.%\n\\end{mdLi}%%\n\\end{mdUl}%\n\\begin{mdP}[data-line={306}]%\n%mdk-data-line={306}\n{}The rest of this paper is organized as follows. Section%mdk-data-line={306}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-semantics}{}{\\mdSpan[class={heading-label}]{2}}%mdk-data-line={306}\n{} \ndescribes an instrumented typing discipline where we lift each type (representing \na set of concrete values) to a symbolic type (representing\na set of pairs of concrete and symbolic values).\nSection%mdk-data-line={310}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-sp2dse}{}{\\mdSpan[class={heading-label}]{3}}%mdk-data-line={310}\n{} shows how strongest postconditions defines a symbolic\nsemantics for a small programming language and how strongest postconditions\ncan be refined to model DSE.\nSection%mdk-data-line={313}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-impl}{}{\\mdSpan[class={heading-label}]{4}}%mdk-data-line={313}\n{} describes an implementation of DSE for the Python language\nin the Python language that follows the instrumented semantics pattern closely\n(full implementation and tests available at%mdk-data-line={315}\n{}{\\mdNbsp}\\mdA[data-linkid={pyexz3}]{https://github.com/thomasjball/PyExZ3/}{}{PyExZ3}%mdk-data-line={315}\n{}, tagged %mdk-data-line={315}\n{}{\\textquotedblleft}v1.0{\\textquotedblright}%mdk-data-line={315}\n{}).\nSection%mdk-data-line={316}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-int2z3}{}{\\mdSpan[class={heading-label}]{5}}%mdk-data-line={316}\n{} describes the symbolic encoding of Python integer \noperations using two decision procedures of Z3: linear arithmetic with\nuninterpreted functions in place of non-linear operations;\nfixed-width bit-vectors with precise encodings of most operations.\nSection%mdk-data-line={320}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-extensions}{}{\\mdSpan[class={heading-label}]{6}}%mdk-data-line={320}\n{} offers a number of ideas for projects\nto extend the capabilities of%mdk-data-line={321}\n{}{\\mdNbsp}\\mdA[data-linkid={pyexz3}]{https://github.com/thomasjball/PyExZ3/}{}{PyExZ3}%mdk-data-line={321}\n{}.%\n\\end{mdP}%\n\\mdHxx[id=sec-semantics,label={[2]\\{.heading-label\\}},toc={},data-line={323},caption={[[2]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Instrumented Types},bookmark={2.{\\hspace{0.5em}}Instrumented Types}]{%mdk-data-line={323}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{2}.{\\hspace{0.5em}}}%mdk-data-line={323}\n{}Instrumented Types}\\begin{mdP}[data-line={325}]%\n%mdk-data-line={325}\n{}We are given a universe of classes/types %mdk-data-line={325}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U$}%mdk-data-line={325}\n{}; a type %mdk-data-line={325}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T \\in U$}%mdk-data-line={325}\n{} carries\nalong a  set of operations that apply to values of type %mdk-data-line={326}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={326}\n{},\nwhere an operation %mdk-data-line={327}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T$}%mdk-data-line={327}\n{}  takes an argument list of typed \nvalues as input (the first being of type %mdk-data-line={328}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={328}\n{}) and produces a single \ntyped value as output. Nullary (static) operations of type %mdk-data-line={329}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={329}\n{} can be \nused to create values of type %mdk-data-line={330}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={330}\n{} (such as constants, objects, etc.)%\n\\end{mdP}%\n\\begin{mdP}[class={indent,para-continue},data-line={332}]%\n%mdk-data-line={332}\n{}A program %mdk-data-line={332}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={332}\n{} has typed input variables\n%mdk-data-line={333}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$v_1 : T_1 \\ldots v_k : T_k$}%mdk-data-line={333}\n{} and a body from the language of statements %mdk-data-line={333}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$S$}%mdk-data-line={333}\n{}:%\n\\end{mdP}%\n\\begin{mdDiv}[class={mathpre,para-block,input-mathpre},elem={mathpre},data-line={335}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={336}\n\\begin{mdMathprearray}%mdk\n\\mathid{S}\\mdMathspace{1}\\rightarrow   &\\mdMathspace{1}\\mathid{v}\\mdMathspace{1}:=\\mdMathspace{1}\\mathid{E}\\mdMathbr{}\n\\mdMathindent{3}|\\mdMathspace{1}&\\mdMathspace{1}\\mathkw{skip}\\mdMathspace{1}\\mdMathbr{}\n\\mdMathindent{3}|\\mdMathspace{1}&\\mdMathspace{1}\\mathid{S}_1\\mdMathspace{1};\\mdMathspace{1}\\mathid{S}_2\\mdMathspace{1}\\mdMathbr{}\n\\mdMathindent{3}|\\mdMathspace{1}&\\mdMathspace{1}\\mathkw{if}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{S}_1\\mdMathspace{1}\\mathkw{else}\\mdMathspace{1}\\mathid{S}_2\\mdMathspace{1}\\mathkw{end}\\mdMathbr{}\n\\mdMathindent{3}|\\mdMathspace{1}&\\mdMathspace{1}\\mathkw{while}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{do}\\mdMathspace{1}\\mathid{S}\\mdMathspace{1}\\mathkw{end}\n\\end{mdMathprearray}%mdk\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[data-line={343}]%\n%mdk-data-line={343}\n{}The language of expressions (%mdk-data-line={343}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={343}\n{}) is defined by the application of operations\nto values, where constants (nullary operations) and \nprogram variables form the leaves \nof the expression tree and non-nullary operators \nform the interior nodes of the tree.\nFor now, we will consider all values to be immutable.\nThat is, the only source of mutation in the language is the \nassignment statement.%\n\\end{mdP}%\n\\begin{mdP}[class={indent,para-continue},data-line={352}]%\n%mdk-data-line={352}\n{}To introduce symbolic execution into the picture,\nwe can imagine that a type %mdk-data-line={353}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T \\in U$}%mdk-data-line={353}\n{} has\n(one or more) counterparts in a symbolic universe %mdk-data-line={354}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={354}\n{}. A type %mdk-data-line={354}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T' \\in U'$}%mdk-data-line={354}\n{}\nis a subtype of %mdk-data-line={355}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T \\in U$}%mdk-data-line={355}\n{} with two purposes:%\n\\end{mdP}%\n\\begin{mdUl}[class={ul,list-star,loose},elem={ul},data-line={357}]%\n\\begin{mdLi}[class={li,ul-li,list-star-li,loose-li},label={[(1)]\\{.ul-li-label\\}},elem={li},data-line={357}]%\n\\begin{mdP}[data-line={357}]%\n%mdk-data-line={357}\n{}First, a value of type %mdk-data-line={357}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}%mdk-data-line={357}\n{} represents a pair of values: \na concrete value %mdk-data-line={358}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={358}\n{} of (super)type %mdk-data-line={358}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={358}\n{} and a symbolic expression %mdk-data-line={358}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$e$}%mdk-data-line={358}\n{}. \nA symbolic expression is a tree\nwhose leaves are either nullary operators (i.e., constants) of a type in %mdk-data-line={360}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U$}%mdk-data-line={360}\n{}\nor are Skolem constants representing the (symbolic) inputs (%mdk-data-line={361}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$v_1 \\ldots v_k$}%mdk-data-line={361}\n{})\nto the program %mdk-data-line={362}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={362}\n{}, and whose interior nodes represent operations \nfrom types in %mdk-data-line={363}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U$}%mdk-data-line={363}\n{}. We refer to Skolem constants as %mdk-data-line={363}\n{}{\\textquotedblleft}symbolic constants{\\textquotedblright}%mdk-data-line={363}\n{}\nfrom this point on. Note that symbolic expressions do not contain\nreferences to program variables.%\n\\end{mdP}%%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,loose-li},label={[(2)]\\{.ul-li-label\\}},elem={li},data-line={367}]%\n\\begin{mdP}[data-line={367}]%\n%mdk-data-line={367}\n{}Second, the type %mdk-data-line={367}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}%mdk-data-line={367}\n{} redefines some of the operations %mdk-data-line={367}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T$}%mdk-data-line={367}\n{},\nnamely those for which we wish to compute symbolic expressions.\nAn operation %mdk-data-line={369}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T'$}%mdk-data-line={369}\n{} has the same parameter list as %mdk-data-line={369}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T$}%mdk-data-line={369}\n{}, allowing it\nto take inputs with types from both %mdk-data-line={370}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U$}%mdk-data-line={370}\n{} and %mdk-data-line={370}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={370}\n{}. The return type\nof %mdk-data-line={371}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T'$}%mdk-data-line={371}\n{} generally is from %mdk-data-line={371}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={371}\n{} (though it can be from %mdk-data-line={371}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U$}%mdk-data-line={371}\n{}). \nThus, %mdk-data-line={372}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T'$}%mdk-data-line={372}\n{} is a proper function subtype of %mdk-data-line={372}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T$}%mdk-data-line={372}\n{}. \nThe purpose of %mdk-data-line={373}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T'$}%mdk-data-line={373}\n{} is to:\n(1) perform operation %mdk-data-line={374}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T$}%mdk-data-line={374}\n{} on the concrete \nvalues associated with its inputs; \n(2) build a symbolic expression tree rooted at operation %mdk-data-line={376}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o$}%mdk-data-line={376}\n{} \nwhose children are the trees associated with the inputs to %mdk-data-line={377}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o$}%mdk-data-line={377}\n{}.%\n\\end{mdP}%%\n\\end{mdLi}%%\n\\end{mdUl}%\n\\begin{mdP}[data-line={379}]%\n%mdk-data-line={379}\n{}Figure%mdk-data-line={379}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-subtype}{}{\\mdSpan[class={figure-label}]{5}}%mdk-data-line={379}\n{} presents pseudo code for the instrumentation\nof a type %mdk-data-line={380}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={380}\n{} via a type %mdk-data-line={380}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}%mdk-data-line={380}\n{}.\nThe class %mdk-data-line={381}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdToken{Type,Identifier,Cpp}{Symbolic}}%mdk-data-line={381}\n{} is used to hold an expression tree (%mdk-data-line={381}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdToken{Type,Identifier,Cpp}{Expr}}%mdk-data-line={381}\n{}).\nGiven a class %mdk-data-line={382}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T \\in U$}%mdk-data-line={382}\n{}, a symbolic type %mdk-data-line={382}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T' \\in U'$}%mdk-data-line={382}\n{} is defined by inheriting \nfrom both %mdk-data-line={383}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={383}\n{} and %mdk-data-line={383}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdToken{Type,Identifier,Cpp}{Symbolic}}%mdk-data-line={383}\n{}. This ensures that a %mdk-data-line={383}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}%mdk-data-line={383}\n{} can be used\nwherever a %mdk-data-line={384}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={384}\n{} is expected.%\n\\end{mdP}%\n\\begin{mdDiv}[class={figure,floating,align-center},id=fig-subtype,label={[5]\\{.figure-label\\}},elem={figure},toc-line={[5]\\{.figure-label\\}. Type instrumentation to carry both concrete values and symbolic expressions.},toc={tof},float-env={figure},float-name={Figure},caption={Type instrumentation to carry both concrete values and symbolic expressions.},data-line={386}]%\n\\begin{mdPre}[class={para-block,pre-fenced,pre-fenced3,language-cpp2,lang-cpp2,cpp2,highlighted},data-line={387},data-line-code={388},language={cpp2}]%\n\\mdPrecode[data-line={388}]{{\\preindent{2}}\\mdToken{Keyword,Cpp}{class}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}}}{\\prespace{1}}\\mdToken{Operator,Cpp}{:}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T$}}}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{Symbolic}{\\prespace{1}}\\mdToken{Delimiter,Curly,Cpp,BracketOpen}{\\{}\\prebr{}\n{\\preindent{4}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}}}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{c}\\mdToken{Operator,Cpp}{:}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T$}}}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Identifier,Cpp}{e}\\mdToken{Operator,Cpp}{:}\\mdToken{Type,Identifier,Cpp}{Expr}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}{\\prespace{1}}\\mdToken{Operator,Cpp}{:}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T$}}}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{c}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Source,Cpp}{S}\\mdToken{Identifier,Cpp}{ymbolic}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{e}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}{\\prespace{1}}\\mdToken{Delimiter,Curly,Cpp,BracketOpen}{\\{}\\mdToken{Delimiter,Curly,Cpp,BracketClose}{\\}}\\prebr{}\n\\prebr{}\n{\\preindent{4}}\\mdToken{Keyword,Extra,Cpp}{override}{\\prespace{1}}\\mdToken{Identifier,Cpp}{o}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Keyword,Cpp}{this}\\mdToken{Operator,Cpp}{:}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T$}}}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Identifier,Cpp}{f1}\\mdToken{Operator,Cpp}{:}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T_1$}}}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}{\\prespace{1}}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Identifier,Cpp}{fk}\\mdToken{Operator,Cpp}{:}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T_k$}}}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}{\\prespace{1}}\\mdToken{Operator,Cpp}{:}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$R'$}}}{\\prespace{1}}\\mdToken{Delimiter,Curly,Cpp,BracketOpen}{\\{}\\prebr{}\n{\\preindent{6}}\\mdToken{Keyword,Extra,Cpp}{var}{\\prespace{1}}\\mdToken{Identifier,Cpp}{c}{\\prespace{1}}:={\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T$}}}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Identifier,Cpp}{o}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Keyword,Cpp}{this}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Identifier,Cpp}{f1}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}{\\prespace{1}}\\mdToken{Delimiter,Cpp}{,}\\mdToken{Identifier,Cpp}{fk}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}\\prebr{}\n{\\preindent{6}}\\mdToken{Keyword,Extra,Cpp}{var}{\\prespace{1}}\\mdToken{Identifier,Cpp}{e}{\\prespace{1}}:={\\prespace{1}}\\mdToken{Keyword,Cpp}{new}{\\prespace{1}}\\mdToken{Source,Cpp}{E}\\mdToken{Identifier,Cpp}{xpr}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T$}}}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Identifier,Cpp}{o}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Identifier,Cpp}{expr}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{self}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Identifier,Cpp}{expr}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{f1}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Identifier,Cpp}{expr}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{fk}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}\\prebr{}\n{\\preindent{6}}\\mdToken{Keyword,Cpp}{return}{\\prespace{1}}\\mdToken{Keyword,Cpp}{new}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$R'$}}}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{c}\\mdToken{Delimiter,Cpp}{,}\\mdToken{Identifier,Cpp}{e}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}{\\prespace{1}}\\prebr{}\n{\\preindent{4}}\\mdToken{Delimiter,Curly,Cpp,BracketClose}{\\}}\\prebr{}\n{\\preindent{4}}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}\\prebr{}\n{\\preindent{2}}\\mdToken{Delimiter,Curly,Cpp,BracketClose}{\\}}\\prebr{}\n{\\preindent{2}}\\prebr{}\n{\\preindent{2}}\\mdToken{Keyword,Cpp}{class}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$R'$}}}{\\prespace{1}}\\mdToken{Operator,Cpp}{:}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$R$}}}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{Symbolic}{\\prespace{1}}\\mdToken{Delimiter,Curly,Cpp,BracketOpen}{\\{}{\\prespace{1}}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}{\\prespace{1}}\\mdToken{Delimiter,Curly,Cpp,BracketClose}{\\}}\\prebr{}\n{\\preindent{2}}\\prebr{}\n{\\preindent{2}}\\mdToken{Keyword,Extra,Cpp}{function}{\\prespace{1}}\\mdToken{Identifier,Cpp}{expr}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{v}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}{\\prespace{1}}\\mdToken{Operator,Cpp}{=}{\\prespace{1}}\\mdToken{Identifier,Cpp}{v}{\\prespace{1}}\\mdToken{Keyword,Extra,Cpp}{instanceof}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{Symbolic}{\\prespace{1}}\\mdToken{Operator,Cpp}{?}{\\prespace{1}}\\mdToken{Identifier,Cpp}{v}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Identifier,Cpp}{getExpr}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}{\\prespace{1}}\\mdToken{Operator,Cpp}{:}{\\prespace{1}}\\mdToken{Identifier,Cpp}{v}}%\n\\end{mdPre}%\n\\mdHr[class={figureline,madoko},data-line={403}]{}\\begin{mdDiv}[data-line={404}]%\n%mdk-data-line={404}\n{}\\mdSpan[class={figure-caption}]{\\mdSpan[class={caption-before}]{\\mdStrong{Figure{\\mdNbsp}\\mdSpan[class={figure-label}]{5}.} }Type instrumentation to carry both concrete values and symbolic expressions.}%mdk-data-line={404}\n{}%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[class={indent,para-continue},data-line={405}]%\n%mdk-data-line={405}\n{}A type such as %mdk-data-line={405}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}%mdk-data-line={405}\n{}  only can be constructed by providing a concrete value %mdk-data-line={405}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={405}\n{} of \ntype %mdk-data-line={406}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={406}\n{} and a symbolic expression %mdk-data-line={406}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$e$}%mdk-data-line={406}\n{} to the constructor for %mdk-data-line={406}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}%mdk-data-line={406}\n{}.\nThis will be done in exactly two places:%\n\\end{mdP}%\n\\begin{mdUl}[class={ul,list-star,loose},elem={ul},data-line={409}]%\n\\begin{mdLi}[class={li,ul-li,list-star-li,loose-li},label={[(1)]\\{.ul-li-label\\}},elem={li},data-line={409}]%\n\\begin{mdP}[data-line={409}]%\n%mdk-data-line={409}\n{}by the creation of symbolic constants associated with the primary\ninputs (%mdk-data-line={410}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$v_1 \\ldots v_k$}%mdk-data-line={410}\n{}) to the program;%\n\\end{mdP}%%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,loose-li},label={[(2)]\\{.ul-li-label\\}},elem={li},data-line={412}]%\n\\begin{mdP}[data-line={412}]%\n%mdk-data-line={412}\n{}by the instrumented operations as shown  in Figure%mdk-data-line={412}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-subtype}{}{\\mdSpan[class={figure-label}]{5}}%mdk-data-line={412}\n{}.%\n\\end{mdP}%%\n\\end{mdLi}%%\n\\end{mdUl}%\n\\begin{mdP}[data-line={414}]%\n%mdk-data-line={414}\n{}An instrumented operation %mdk-data-line={414}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o$}%mdk-data-line={414}\n{} on arguments (%mdk-data-line={414}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdToken{Keyword,Cpp}{this}}%mdk-data-line={414}\n{}, %mdk-data-line={414}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{f1}}%mdk-data-line={414}\n{}, %mdk-data-line={414}\n{}{\\dots}%mdk-data-line={414}\n{}, %mdk-data-line={414}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{fk}}%mdk-data-line={414}\n{})\nfirst invokes its corresponding underlying\noperator %mdk-data-line={416}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T.o$}%mdk-data-line={416}\n{} on arguments (%mdk-data-line={416}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdToken{Keyword,Cpp}{this}}%mdk-data-line={416}\n{}, %mdk-data-line={416}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{f1}}%mdk-data-line={416}\n{}, %mdk-data-line={416}\n{}{\\dots}%mdk-data-line={416}\n{}, %mdk-data-line={416}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{fk}}%mdk-data-line={416}\n{}) to get concrete value %mdk-data-line={416}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{c}}%mdk-data-line={416}\n{}.\nIt then constructs a new expression tree %mdk-data-line={417}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{e}}%mdk-data-line={417}\n{}\nrooted at operator %mdk-data-line={418}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T.o$}%mdk-data-line={418}\n{}, whose children are the result of\nmapping the function %mdk-data-line={419}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{expr}}%mdk-data-line={419}\n{} over (%mdk-data-line={419}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdToken{Keyword,Cpp}{this}}%mdk-data-line={419}\n{}, %mdk-data-line={419}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{f1}}%mdk-data-line={419}\n{}, %mdk-data-line={419}\n{}{\\dots}%mdk-data-line={419}\n{}, %mdk-data-line={419}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{fk}}%mdk-data-line={419}\n{}). \nThe helper function %mdk-data-line={420}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{expr}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{v}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={420}\n{}\nevaluates to an expression tree in the case that %mdk-data-line={421}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{v}}%mdk-data-line={421}\n{} is of %mdk-data-line={421}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdToken{Type,Identifier,Cpp}{Symbolic}}%mdk-data-line={421}\n{} type\n(representing a type in %mdk-data-line={422}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={422}\n{}) and evaluates to %mdk-data-line={422}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{v}}%mdk-data-line={422}\n{} itself, a concrete value\nof some type in %mdk-data-line={423}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U$}%mdk-data-line={423}\n{}, otherwise.\nFinally, having computed the values %mdk-data-line={424}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{c}}%mdk-data-line={424}\n{} and %mdk-data-line={424}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{e}}%mdk-data-line={424}\n{}, the instrumented operator \nreturns %mdk-data-line={425}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$R'$}}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{c}\\mdToken{Delimiter,Cpp}{,}\\mdToken{Identifier,Cpp}{e}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}}%mdk-data-line={425}\n{}, where %mdk-data-line={425}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$R$}%mdk-data-line={425}\n{} is the return type of operator %mdk-data-line={425}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T.o$}%mdk-data-line={425}\n{},\nand %mdk-data-line={426}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$R'$}%mdk-data-line={426}\n{} is a subtype of %mdk-data-line={426}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$R$}%mdk-data-line={426}\n{} from universe %mdk-data-line={426}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={426}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={428}]%\n%mdk-data-line={428}\n{}Looked at another way, the universe %mdk-data-line={428}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={428}\n{} represents the %mdk-data-line={428}\n{}{\\textquotedblleft}tainting{\\textquotedblright}%mdk-data-line={428}\n{} of\ntypes from %mdk-data-line={429}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U$}%mdk-data-line={429}\n{}. Tainted values flow from program inputs to the \noperands of operators. If an operator has been redefined\n(as above) then the taint propagates from its inputs to its outputs.\nOn the other hand, if the operator has not been redefined, then it will\nnot propagate the taint. In the context of DSE, %mdk-data-line={433}\n{}{\\textquotedblleft}taint{\\textquotedblright}%mdk-data-line={433}\n{} means\nthat the instrumented semantics carries along a symbolic expression tree %mdk-data-line={434}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$e$}%mdk-data-line={434}\n{}\nalong with a concrete value %mdk-data-line={435}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={435}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={437}]%\n%mdk-data-line={437}\n{}The choice of types from the universe %mdk-data-line={437}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={437}\n{} determines how symbolic\nexpressions are constructed. For each %mdk-data-line={438}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T \\in U$}%mdk-data-line={438}\n{}, the %mdk-data-line={438}\n{}{\\textquotedblleft}most symbolic{\\textquotedblright}%mdk-data-line={438}\n{}\n(least concrete) choice is the %mdk-data-line={439}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}%mdk-data-line={439}\n{} that redefines every operator of %mdk-data-line={439}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={439}\n{}\n(as shown in Figure%mdk-data-line={440}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-subtype}{}{\\mdSpan[class={figure-label}]{5}}%mdk-data-line={440}\n{}).\nThe %mdk-data-line={441}\n{}{\\textquotedblleft}least symbolic{\\textquotedblright}%mdk-data-line={441}\n{} (most concrete) choice is %mdk-data-line={441}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T' = T$}%mdk-data-line={441}\n{} which\nredefines no operators.  Let %mdk-data-line={442}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$symbolic(T)$}%mdk-data-line={442}\n{} be the\nset of types in %mdk-data-line={443}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={443}\n{} that are subtypes of %mdk-data-line={443}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={443}\n{}. The types\nin %mdk-data-line={444}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$symbolic(T)$}%mdk-data-line={444}\n{} are partially ordered by subset inclusion on the \nset of operators from %mdk-data-line={445}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={445}\n{} they redefine.%\n\\end{mdP}%\n\n\\mdHxx[id=sec-sp2dse,label={[3]\\{.heading-label\\}},toc={},data-line={449},caption={[[3]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}From Strongest Postconditions to DSE},bookmark={3.{\\hspace{0.5em}}From Strongest Postconditions to DSE}]{%mdk-data-line={449}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{3}.{\\hspace{0.5em}}}%mdk-data-line={449}\n{}From Strongest Postconditions to DSE}\\begin{mdP}[data-line={451}]%\n%mdk-data-line={451}\n{}The previous section showed how symbolic expressions can be computed via a\nset of instrumented types, where the expressions are computed as a side-effect \nof the execution of program operations.\nThis section shows how these symbolic expressions can be used to form a \n%mdk-data-line={455}\n{}\\mdEm{path-condition}%mdk-data-line={455}\n{} (which then can be compiled into a logic formula and \npassed to an automated theorem prover to find new inputs to drive a \nprogram%mdk-data-line={457}\n{}{'}%mdk-data-line={457}\n{}s execution along new paths). We derive a %mdk-data-line={457}\n{}\\mdEm{path-condition}%mdk-data-line={457}\n{}\ndirectly from the %mdk-data-line={458}\n{}\\mdEm{strongest postcondition}%mdk-data-line={458}\n{} (symbolic) semantics of our\nprogramming language, refining it to model the basic operations\nof an interpreter.%\n\\end{mdP}%\n\\mdHxxx[id=sec-strongest-postconditions,label={[3.1]\\{.heading-label\\}},toc={},data-line={462},caption={[[3.1]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Strongest Postconditions},bookmark={3.1.{\\hspace{0.5em}}Strongest Postconditions}]{%mdk-data-line={462}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{3.1}.{\\hspace{0.5em}}}%mdk-data-line={462}\n{}Strongest Postconditions}\\begin{mdP}[class={para-continue},data-line={464}]%\n%mdk-data-line={464}\n{}The strongest postcondition transformer %mdk-data-line={464}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$SP$}%mdk-data-line={464}\n{}{\\mdNbsp}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{dijkstra76}{}{\\mdSpan[class={bibitem-label}]{6}}]}%mdk-data-line={464}\n{} is defined over\na predicate %mdk-data-line={465}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={465}\n{} representing a set of \npre-states and a statement %mdk-data-line={466}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$S$}%mdk-data-line={466}\n{} from our language. The transformer %mdk-data-line={466}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$SP(P,S)$}%mdk-data-line={466}\n{}\nyields a predicate %mdk-data-line={467}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$Q$}%mdk-data-line={467}\n{} such that for any state %mdk-data-line={467}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$s$}%mdk-data-line={467}\n{} satisfying\npredicate %mdk-data-line={468}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={468}\n{}, the execution of statement %mdk-data-line={468}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$S$}%mdk-data-line={468}\n{} from state %mdk-data-line={468}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$s$}%mdk-data-line={468}\n{},\nif it does not go wrong or diverge, yields a state %mdk-data-line={469}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$s'$}%mdk-data-line={469}\n{} satisfying predicate %mdk-data-line={469}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$Q$}%mdk-data-line={469}\n{}. \nThe strongest postcondition for the statements in our language\nis defined by the following five rules:%\n\\end{mdP}%\n\\begin{mdDiv}[class={mathpre,para-block,input-mathpre},elem={mathpre},data-line={473}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={474}\n\\begin{mdMathprearray}%mdk\n1.\\mdMathspace{2}&\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mdMathspace{1}\\mathid{x}\\mdMathspace{1}:=\\mdMathspace{1}\\mathid{E})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{1}\\exists \\mathid{y}\\mdMathspace{1}.\\mdMathspace{1}(\\mathid{x}\\mdMathspace{1}=\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}[\\mdMathspace{1}\\mathid{x}\\mdMathspace{1}\\rightarrow \\mathid{y}\\mdMathspace{1}])\\mdMathspace{1}\\wedge \\mathid{P}\\mdMathspace{1}[\\mdMathspace{1}\\mathid{x}\\mdMathspace{1}\\rightarrow \\mathid{y}\\mdMathspace{1}]\\mdMathbr{}\n2.\\mdMathspace{2}&\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mdMathspace{1}\\mathkw{skip})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{1}\\mathid{P}\\mdMathbr{}\n3.\\mdMathspace{2}&\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mathid{S}_{1};\\mathid{S}_{2})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{1}\\mathid{SP}(\\mathid{SP}(\\mathid{P},\\mathid{S}_{1}),\\mdMathspace{1}\\mathid{S}_{2})\\mdMathbr{}\n4.\\mdMathspace{2}&\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mathkw{if}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{S}_1\\mdMathspace{1}\\mathkw{else}\\mdMathspace{1}\\mathid{S}_2\\mdMathspace{1}\\mathkw{end})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathbr{}\n\\mdMathindent{4}&\\mdMathspace{7}\\mathid{SP}(\\mathid{P}\\wedge \\mathid{E},\\mathid{S}_1)\\mdMathspace{1}\\vee  \\mathid{SP}(\\mathid{P}\\mdMathspace{1}\\wedge \\neg \\mathid{E},\\mathid{S}_2)\\mdMathbr{}\n5.\\mdMathspace{2}&\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mathkw{while}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{do}\\mdMathspace{1}\\mathid{S}\\mdMathspace{1}\\mathkw{end})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathbr{}\n\\mdMathindent{4}&\\mdMathspace{6}\\mathid{SP}(\\mathid{P},\\mathkw{if}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{S};\\mdMathspace{1}\\mathkw{while}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{do}\\mdMathspace{1}\\mathid{S}\\mdMathspace{1}\\mathkw{end}\\mdMathspace{1}\\mathkw{else}\\mdMathspace{1}\\mathkw{skip}\\mdMathspace{1}\\mathkw{end})\n\\end{mdMathprearray}%mdk\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[data-line={483}]%\n%mdk-data-line={483}\n{}Rule (1) defines the strongest postcondition for \nthe assignment statement. The assignment is modeled\nlogically by the equality %mdk-data-line={485}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x = E$}%mdk-data-line={485}\n{} where any free occurrence of %mdk-data-line={485}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x$}%mdk-data-line={485}\n{} in %mdk-data-line={485}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={485}\n{}\nis replaced by the existentially quantified variable %mdk-data-line={486}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$y$}%mdk-data-line={486}\n{}, which represents\nthe value of %mdk-data-line={487}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x$}%mdk-data-line={487}\n{} in the pre-state. The same substitution (%mdk-data-line={487}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$[x \\rightarrow y ]$}%mdk-data-line={487}\n{})\nis applied to the pre-state predicate %mdk-data-line={488}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={488}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={490}]%\n%mdk-data-line={490}\n{}Rules (2)-(5) define the strongest postcondition for the four control-flow\nstatements. The rules for the %mdk-data-line={491}\n{}\\mdStrong{skip}%mdk-data-line={491}\n{} statement and sequencing (;) are\nstraightforward. \nOf particular interest, note that the rule for the %mdk-data-line={493}\n{}\\mdStrong{if-then-else}%mdk-data-line={493}\n{} \nstatement splits cases on the expression %mdk-data-line={494}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={494}\n{}. \nIt is here that DSE will choose one of the cases for us, as the concrete\nexecution will evaluate %mdk-data-line={496}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={496}\n{} either to be true or false. This gives rise\nto the path-condition (either %mdk-data-line={497}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P \\wedge E$}%mdk-data-line={497}\n{} or %mdk-data-line={497}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P \\wedge \\neg E$}%mdk-data-line={497}\n{}). The\nrecursive rule for the %mdk-data-line={498}\n{}\\mdStrong{while}%mdk-data-line={498}\n{} loop unfolds as many times as the\nexpression %mdk-data-line={499}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={499}\n{} evaluates true, adding to the path-condition.%\n\\end{mdP}%\n\\mdHxxx[id=sp-refined,label={[3.2]\\{.heading-label\\}},toc={},data-line={501},caption={[[3.2]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}From \\$SP\\$ to DSE},bookmark={3.2.{\\hspace{0.5em}}From \\$SP\\$ to DSE}]{%mdk-data-line={501}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{3.2}.{\\hspace{0.5em}}}%mdk-data-line={501}\n{}From %mdk-data-line={501}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$SP$}%mdk-data-line={501}\n{} to DSE}\\begin{mdP}[data-line={503}]%\n%mdk-data-line={503}\n{}Assume that an execution begins with the assignment of initial values %mdk-data-line={503}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c_1 \\ldots c_k$}%mdk-data-line={503}\n{} to the program %mdk-data-line={503}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={503}\n{}{'}%mdk-data-line={503}\n{}s inputs\n%mdk-data-line={504}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$V = \\{ v_1 : T_1 \\ldots v_k : T_k \\}$}%mdk-data-line={504}\n{}.  To seed symbolic execution, some of the types %mdk-data-line={504}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T_i$}%mdk-data-line={504}\n{} are\nreplaced by symbolic counterparts %mdk-data-line={505}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'_i$}%mdk-data-line={505}\n{}, in which case\n%mdk-data-line={506}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$v_i$}%mdk-data-line={506}\n{} is initialized to the value %mdk-data-line={506}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$sc_i = T'_i (c_i,SC(v_i))$}%mdk-data-line={506}\n{} instead of the value %mdk-data-line={506}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c_i$}%mdk-data-line={506}\n{},\nwhere %mdk-data-line={507}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$SC(v_i)$}%mdk-data-line={507}\n{} is the symbolic constant representing the initial value of variable %mdk-data-line={507}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$v_i$}%mdk-data-line={507}\n{}.\nThe symbolic constant %mdk-data-line={508}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$SC(v_i)$}%mdk-data-line={508}\n{} can be thought of as representing any value of\ntype %mdk-data-line={509}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T_i$}%mdk-data-line={509}\n{}, which includes the value %mdk-data-line={509}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c_i$}%mdk-data-line={509}\n{}.%mdk-data-line={509}\n{} %mdk-data-line={509}\n{}%\n\\end{mdP}%\n\\begin{mdP}[class={indent,para-continue},data-line={511}]%\n%mdk-data-line={511}\n{}Let %mdk-data-line={511}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$V_s$}%mdk-data-line={511}\n{} and %mdk-data-line={511}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$V_c$}%mdk-data-line={511}\n{} partition the variables of %mdk-data-line={511}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$V$}%mdk-data-line={511}\n{} into those variables\nthat are treated symbolically (%mdk-data-line={512}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$V_s$}%mdk-data-line={512}\n{}) and those that are treated concretely \n(%mdk-data-line={513}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$V_c$}%mdk-data-line={513}\n{}). The initial state of the program is characterized by the formula%\n\\end{mdP}%\n\\begin{mdDiv}[class={equation,para-block},label={[(1)]\\{.equation-label\\}},elem={equation},line-adjust={0},data-line={515}]%\n%mdk-data-line={515}\n{}\\mdSpan[class={equation-before}]{\\mdSpan[class={equation-label}]{(1)}}%mdk-data-line={515}\n{}\n\\begin{mdDiv}[class={mathdisplay,para-block,input-math},elem={mathdisplay},color={},math-needpdf={},line-adjust={0},data-line={516}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={516}\nInit = (\\bigwedge_{v_i \\in V_s} v_i = sc_i) ~\\wedge~ (~\\bigwedge_{v_i \\in V_c} v_i = c_i)\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[data-line={519}]%\n%mdk-data-line={519}\n{}Thus, we see that the initial value of every input variable is characterized by \na symbolic constant %mdk-data-line={520}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$sc_i$}%mdk-data-line={520}\n{} or constant %mdk-data-line={520}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c_i$}%mdk-data-line={520}\n{}.  We assume that every non-input\nvariable in the program is initialized before being used.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={523}]%\n%mdk-data-line={523}\n{}The strongest postcondition is formulated to deal with open programs,\nprograms in which some variables are used before being assigned to. \nThis surfaces in Rule (1) for assignment, which uses existential quantification\nto refer to the value of variable %mdk-data-line={526}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x$}%mdk-data-line={526}\n{} in the pre-state.%\n\\end{mdP}%\n\\begin{mdP}[class={indent,para-continue},data-line={528}]%\n%mdk-data-line={528}\n{}By construction,\nwe have that every variable is defined before being used.  This means\nthat the precondition %mdk-data-line={530}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={530}\n{} can be reformulated as a pair %mdk-data-line={530}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$<\\sigma,P_c>$}%mdk-data-line={530}\n{},\nwhere %mdk-data-line={531}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma$}%mdk-data-line={531}\n{} is a store mapping variables to values and %mdk-data-line={531}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P_c$}%mdk-data-line={531}\n{} is\nthe path-condition, a list of symbolic expressions (predicates) \ncorresponding\nto the expressions %mdk-data-line={534}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={534}\n{} evaluated in the context of an %mdk-data-line={534}\n{}\\mdStrong{if-then-else}%mdk-data-line={534}\n{} \nstatement. Initially, we have that :%\n\\end{mdP}%\n\\begin{mdDiv}[class={equation,para-block},label={[(2)]\\{.equation-label\\}},elem={equation},line-adjust={0},data-line={537}]%\n%mdk-data-line={537}\n{}\\mdSpan[class={equation-before}]{\\mdSpan[class={equation-label}]{(2)}}%mdk-data-line={537}\n{}\n\\begin{mdDiv}[class={mathdisplay,para-block,input-math},elem={mathdisplay},color={},math-needpdf={},line-adjust={0},data-line={538}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={538}\n\\sigma = \\{ (v_i,sc_i) | v_i \\in V_s \\} \\cup \\{ (v_i,c_i) | v_i \\in V_c \\}\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[class={para-continue},data-line={541}]%\n%mdk-data-line={541}\n{}representing the initial condition %mdk-data-line={541}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$Init$}%mdk-data-line={541}\n{}, and %mdk-data-line={541}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P_c = []$}%mdk-data-line={541}\n{}, the empty list.\nWe use %mdk-data-line={542}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma'$}%mdk-data-line={542}\n{} to refer to the formula that the store %mdk-data-line={542}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma$}%mdk-data-line={542}\n{}\ninduces:%\n\\end{mdP}%\n\\begin{mdDiv}[class={equation,para-block},label={[(3)]\\{.equation-label\\}},elem={equation},line-adjust={0},data-line={545}]%\n%mdk-data-line={545}\n{}\\mdSpan[class={equation-before}]{\\mdSpan[class={equation-label}]{(3)}}%mdk-data-line={545}\n{}\n\\begin{mdDiv}[class={mathdisplay,para-block,input-math},elem={mathdisplay},color={},math-needpdf={},line-adjust={0},data-line={546}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={546}\n\\sigma ' =  \\bigwedge_{(v,V) \\in \\sigma} (v = V)\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[data-line={549}]%\n%mdk-data-line={549}\n{}Thus, the pair %mdk-data-line={549}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$<\\sigma,P_c>$}%mdk-data-line={549}\n{} represents the predicate \n%mdk-data-line={550}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P = \\sigma' \\wedge (\\bigwedge_{c \\in P_c} c)$}%mdk-data-line={550}\n{}.\nA store %mdk-data-line={551}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma$}%mdk-data-line={551}\n{} supports two operations: %mdk-data-line={551}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma[x]$}%mdk-data-line={551}\n{} which denotes the\nvalue that %mdk-data-line={552}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x$}%mdk-data-line={552}\n{} maps to under %mdk-data-line={552}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma$}%mdk-data-line={552}\n{}; %mdk-data-line={552}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma[x \\mapsto V]$}%mdk-data-line={552}\n{}, which \nproduces a new store in which %mdk-data-line={553}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x$}%mdk-data-line={553}\n{} maps to value %mdk-data-line={553}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$V$}%mdk-data-line={553}\n{} and is everywhere \nelse the same as %mdk-data-line={554}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma$}%mdk-data-line={554}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent,para-continue},data-line={556}]%\n%mdk-data-line={556}\n{}Now, we can redefine strongest postcondition for assignment to eliminate the\nuse of existential quantification and model the operation of an interpreter,\nby separating out the notion of the store:%\n\\end{mdP}%\n\\begin{mdDiv}[class={mathpre,para-block,input-mathpre},elem={mathpre},data-line={560}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={561}\n\\begin{mdMathprearray}%mdk\n1.\\mdMathspace{1}&\\mdMathspace{1}\\mathid{SP}(<\\sigma,\\mdMathspace{1}\\mathid{P}_\\mathid{c}>,\\mdMathspace{1}\\mathid{x}\\mdMathspace{1}:=\\mdMathspace{1}\\mathid{E})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{1}<\\sigma[\\mathid{x}\\mdMathspace{1}\\mapsto \\mathid{eval}(\\sigma,\\mathid{E})],\\mdMathspace{1}\\mathid{P}_\\mathid{c}>\\\\\n\\end{mdMathprearray}%mdk\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[data-line={564}]%\n%mdk-data-line={564}\n{}where %mdk-data-line={564}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$eval(\\sigma,E)$}%mdk-data-line={564}\n{} evaluates expression %mdk-data-line={564}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={564}\n{} under the store %mdk-data-line={564}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma$}%mdk-data-line={564}\n{}\n(where every occurrence of a free variable\n%mdk-data-line={566}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$v$}%mdk-data-line={566}\n{} in %mdk-data-line={566}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={566}\n{} is replaced by the value %mdk-data-line={566}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma[v]$}%mdk-data-line={566}\n{}). \nThis is the standard substitution rule of a standard operational semantics.%\n\\end{mdP}%\n\\begin{mdP}[class={indent,para-continue},data-line={569}]%\n%mdk-data-line={569}\n{}We also redefine the rule for the %mdk-data-line={569}\n{}\\mdStrong{if-then-else}%mdk-data-line={569}\n{} statement so\nthat it chooses which branch to take and appends the appropriate\nsymbolic expression (predicate) to the path-condition %mdk-data-line={571}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P_c$}%mdk-data-line={571}\n{}:%\n\\end{mdP}%\n\\begin{mdDiv}[class={mathpre,para-block,input-mathpre},elem={mathpre},data-line={573}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={574}\n\\begin{mdMathprearray}%mdk\n4.\\mdMathspace{1}&\\mdMathspace{1}\\mathid{SP}(<\\sigma,\\mdMathspace{1}\\mathid{P}_\\mathid{c}>,\\mdMathspace{1}\\mathkw{if}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{S}_1\\mdMathspace{1}\\mathkw{else}\\mdMathspace{1}\\mathid{S}_2\\mdMathspace{1}\\mathkw{end})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{2}\\mdMathbr{}\n\\mdMathindent{3}&\\mdMathspace{3}\\mathkw{let}\\mdMathspace{1}\\mathid{choice}\\mdMathspace{1}=\\mdMathspace{1}\\mathid{eval}(\\sigma,\\mathid{E})\\mdMathspace{1}\\mathkw{in}\\mdMathbr{}\n\\mdMathindent{3}&\\mdMathspace{3}\\mathkw{if}\\mdMathspace{1}\\mathid{choice}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{SP}(<\\sigma,\\mdMathspace{1}\\mathid{P}_\\mathid{c}\\mdMathspace{1}::\\mdMathspace{1}\\mathid{expr}(\\mathid{choice})\\mdMathspace{1}>,\\mathid{S}_1)\\mdMathspace{1}\\mdMathbr{}\n\\mdMathindent{3}&\\mdMathspace{3}\\mathkw{else}\\mdMathspace{1}\\mathid{SP}(<\\sigma,\\mdMathspace{1}\\mathid{P}_\\mathid{c}\\mdMathspace{1}::\\mdMathspace{1}\\neg \\mathid{expr}(\\mathid{choice})\\mdMathspace{1}>,\\mathid{S}_2)\n\\end{mdMathprearray}%mdk\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[data-line={580}]%\n%mdk-data-line={580}\n{}The other strongest postcondition rules remain unchanged.%\n\\end{mdP}%\n\\mdHxxx[id=sec-summing-it-up,label={[3.3]\\{.heading-label\\}},toc={},data-line={582},caption={[[3.3]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Summing it up},bookmark={3.3.{\\hspace{0.5em}}Summing it up}]{%mdk-data-line={582}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{3.3}.{\\hspace{0.5em}}}%mdk-data-line={582}\n{}Summing it up}\\begin{mdP}[data-line={584}]%\n%mdk-data-line={584}\n{}We have shown how the symbolic predicate transformer %mdk-data-line={584}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$SP$}%mdk-data-line={584}\n{} can \nbe refined into a symbolic interpreter operating over the\nsymbolic types defined in the previous section.\nIn the case when every input variable is symbolic and every\noperator is redefined, the path-condition is equivalent to the\n%mdk-data-line={589}\n{}\\mdEm{strongest postcondition}%mdk-data-line={589}\n{} of the execution path %mdk-data-line={589}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={589}\n{}. \nThis guarantees that the path-condition for %mdk-data-line={590}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={590}\n{} is %mdk-data-line={590}\n{}\\mdEm{sound}%mdk-data-line={590}\n{}.\nIn the case where a subset of the input variables are symbolic\nand/or not all operators are redefined, the path-condition of %mdk-data-line={592}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={592}\n{} \nis not guaranteed to be sound. We leave it as an exercise to the\nreader to establish sufficient conditions under which the \nuse of concrete values in place of symbolic expressions is \nguaranteed to result in sound path-conditions.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={598}]%\n%mdk-data-line={598}\n{}This section does not address the compilation of a symbolic\nexpression to the (logic) language of an underlying ATP, nor the\nlifting of a satisfying assignment to a formula back to the\nlevel of the source language. This is best done for a particular\nsource language and ATP, as detailed in the next section.%\n\\end{mdP}%\n\\mdHxx[id=sec-impl,label={[4]\\{.heading-label\\}},toc={},data-line={605},caption={[[4]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Architecture of PyExZ3},bookmark={4.{\\hspace{0.5em}}Architecture of PyExZ3}]{%mdk-data-line={605}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{4}.{\\hspace{0.5em}}}%mdk-data-line={605}\n{}Architecture of PyExZ3}\\begin{mdP}[data-line={607}]%\n%mdk-data-line={607}\n{}In this section we present the high-level architecture\nof a simple DSE tool for the Python language, written in Python, called%mdk-data-line={608}\n{}{\\mdNbsp}\\mdA[data-linkid={pyexz3}]{https://github.com/thomasjball/PyExZ3/}{}{PyExZ3}%mdk-data-line={608}\n{}. \nFigure%mdk-data-line={609}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-arch}{}{\\mdSpan[class={figure-label}]{6}}%mdk-data-line={609}\n{}\nshows the class diagram (dashed edges are %mdk-data-line={610}\n{}{\\textquotedblleft}has-a{\\textquotedblright}%mdk-data-line={610}\n{} relationships; solid edges\nare %mdk-data-line={611}\n{}{\\textquotedblleft}is-a{\\textquotedblright}%mdk-data-line={611}\n{} relationships) of the tool.%\n\\end{mdP}%\n\\begin{mdDiv}[class={figure,floating,align-center},id=fig-arch,label={[6]\\{.figure-label\\}},elem={figure},toc-line={[6]\\{.figure-label\\}. Classes in PyExZ3},toc={tof},float-env={figure},float-name={Figure},caption={Classes in PyExZ3},page-align={here},data-line={613}]%\n\\begin{mdP}[data-line={614}]%\n%mdk-data-line={614}\n{}\\mdImg[width={1.00\\linewidth},data-linkid={arch}]{arch.png}%mdk-data-line={614}\n{}%\n\\end{mdP}%\n\\mdHr[class={figureline,madoko},data-line={615}]{}\\begin{mdDiv}[data-line={616}]%\n%mdk-data-line={616}\n{}\\mdSpan[class={figure-caption}]{\\mdSpan[class={caption-before}]{\\mdStrong{Figure{\\mdNbsp}\\mdSpan[class={figure-label}]{6}.} }Classes in PyExZ3}%mdk-data-line={616}\n{}%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\mdHxxx[id=sec-loading-the-code-under-test,label={[4.1]\\{.heading-label\\}},toc={},data-line={619},caption={[[4.1]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Loading the code under test},bookmark={4.1.{\\hspace{0.5em}}Loading the code under test}]{%mdk-data-line={619}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{4.1}.{\\hspace{0.5em}}}%mdk-data-line={619}\n{}Loading the code under test}\\begin{mdP}[data-line={621}]%\n%mdk-data-line={621}\n{}The %mdk-data-line={621}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Loader}}%mdk-data-line={621}\n{} class takes as input the name of a Python file (e.g., %mdk-data-line={621}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}\\mdToken{Delimiter,Python}{.}\\mdToken{Identifier,Python}{py}}%mdk-data-line={621}\n{}) \nto import. The loader expects to find a function named\n%mdk-data-line={623}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}}%mdk-data-line={623}\n{} inside the file %mdk-data-line={623}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}\\mdToken{Delimiter,Python}{.}\\mdToken{Identifier,Python}{py}}%mdk-data-line={623}\n{}, which will serve as the starting point\nfor symbolic execution. The %mdk-data-line={624}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{FunctionInvocation}}%mdk-data-line={624}\n{} class\nwraps this starting point. By default, each parameter to %mdk-data-line={625}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}}%mdk-data-line={625}\n{} is\na %mdk-data-line={626}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicInteger}}%mdk-data-line={626}\n{} unless there is decorator %mdk-data-line={626}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Python}{@}\\mdToken{Identifier,Python}{symbolic}}%mdk-data-line={626}\n{} specifying\nthe type to use for a particular argument.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={629}]%\n%mdk-data-line={629}\n{}The loader provides the capability to reload the\nmodule %mdk-data-line={630}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}\\mdToken{Delimiter,Python}{.}\\mdToken{Identifier,Python}{py}}%mdk-data-line={630}\n{} so that the function %mdk-data-line={630}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}}%mdk-data-line={630}\n{} can be \nreexecuted within the same process from the same initial\nstate with different inputs (see the class %mdk-data-line={632}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{ExplorationEngine}}%mdk-data-line={632}\n{})\nvia the %mdk-data-line={633}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{FunctionInvocation}}%mdk-data-line={633}\n{} class.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={635}]%\n%mdk-data-line={635}\n{}Finally, the loader looks for specially named functions %mdk-data-line={635}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{expected\\_result}}%mdk-data-line={635}\n{}\n(%mdk-data-line={636}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{expected\\_result\\_set}}%mdk-data-line={636}\n{}) in file %mdk-data-line={636}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}\\mdToken{Delimiter,Python}{.}\\mdToken{Identifier,Python}{py}}%mdk-data-line={636}\n{} to use as a test oracle after\nthe path exploration (by %mdk-data-line={637}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{ExplorationEngine}}%mdk-data-line={637}\n{}) has completed. These\nfunctions are expected to return a list of values to check\nagainst the list of return values collected from the executions of \nthe %mdk-data-line={640}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}}%mdk-data-line={640}\n{} function.\nThe presence of the function %mdk-data-line={641}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{expected\\_result}}%mdk-data-line={641}\n{} (%mdk-data-line={641}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{expected\\_result\\_set}}%mdk-data-line={641}\n{}) \nyields a comparison of the two lists as bags (sets). We use such weaker\ntests, rather than list equality, because the order in which paths\nare explored by the %mdk-data-line={644}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{ExplorationEngine}}%mdk-data-line={644}\n{} can easily change due to small\ndifferences in the input programs.%\n\\end{mdP}%\n\\mdHxxx[id=sec-symbolic-types,label={[4.2]\\{.heading-label\\}},toc={},data-line={647},caption={[[4.2]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Symbolic types},bookmark={4.2.{\\hspace{0.5em}}Symbolic types}]{%mdk-data-line={647}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{4.2}.{\\hspace{0.5em}}}%mdk-data-line={647}\n{}Symbolic types}\\begin{mdP}[data-line={649}]%\n%mdk-data-line={649}\n{}Python supports multiple inheritance and, more importantly,\nallows user-defined classes to inherit \nfrom its built-in types (such as %mdk-data-line={651}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{object}}%mdk-data-line={651}\n{} and %mdk-data-line={651}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{int}}%mdk-data-line={651}\n{}).\nWe use these two features two implement\nsymbolic versions of Python objects and integers, \nfollowing the instrumented type approach defined in Section%mdk-data-line={654}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-semantics}{}{\\mdSpan[class={heading-label}]{2}}%mdk-data-line={654}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={656}]%\n%mdk-data-line={656}\n{}The abstract class %mdk-data-line={656}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicType}}%mdk-data-line={656}\n{} contains the \nsymbolic expression tree and provides basic functions for constructing \nand accessing the tree.  This class does double duty, as it is used\nto represent the (typed) symbolic constants associated with the parameters to the \nfunction, as well as the expression trees (per Section%mdk-data-line={660}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-semantics}{}{\\mdSpan[class={heading-label}]{2}}%mdk-data-line={660}\n{}). Recall\nthat the symbolic constants only appear as leaves of expression trees.\nThis means that the expression tree stored in a %mdk-data-line={662}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicType}}%mdk-data-line={662}\n{} will\nhave instances of a %mdk-data-line={663}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicType}}%mdk-data-line={663}\n{} as some of its leaves, namely\nthose leaves representing the symbolic constants.\n%mdk-data-line={665}\n{}%mdk-data-line={665}\n{}\nThe abstract class provides an %mdk-data-line={666}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{unwrap}}%mdk-data-line={666}\n{} method which returns\nthe pair of concrete value and expression tree associated with\nthe %mdk-data-line={668}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicType}}%mdk-data-line={668}\n{}, as well as a %mdk-data-line={668}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{wrap}}%mdk-data-line={668}\n{} method that takes a \npair of concrete value and expression tree and creates a %mdk-data-line={669}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicType}}%mdk-data-line={669}\n{}\nencapsulating them.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={672}]%\n%mdk-data-line={672}\n{}The class %mdk-data-line={672}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicObject}}%mdk-data-line={672}\n{} inherits from both %mdk-data-line={672}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{object}}%mdk-data-line={672}\n{} and %mdk-data-line={672}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicType}}%mdk-data-line={672}\n{} and\noverrides the basic comparison operations (%mdk-data-line={673}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_eq\\_\\_}}%mdk-data-line={673}\n{}, %mdk-data-line={673}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_neq\\_\\_}}%mdk-data-line={673}\n{}, %mdk-data-line={673}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_lt\\_\\_}}%mdk-data-line={673}\n{}, %mdk-data-line={673}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_le\\_\\_}}%mdk-data-line={673}\n{},\n%mdk-data-line={674}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_gt\\_\\_}}%mdk-data-line={674}\n{}, and %mdk-data-line={674}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_ge\\_\\_}}%mdk-data-line={674}\n{}).\nThe class %mdk-data-line={675}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicInteger}}%mdk-data-line={675}\n{} inherits from both %mdk-data-line={675}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{int}}%mdk-data-line={675}\n{} and %mdk-data-line={675}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicObject}}%mdk-data-line={675}\n{}\nand overrides a number of %mdk-data-line={676}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{int}}%mdk-data-line={676}\n{}{'}%mdk-data-line={676}\n{}s arithmetic methods\n(%mdk-data-line={677}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_add\\_\\_}}%mdk-data-line={677}\n{}, %mdk-data-line={677}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_sub\\_\\_}}%mdk-data-line={677}\n{}, %mdk-data-line={677}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_mul\\_\\_}}%mdk-data-line={677}\n{}, %mdk-data-line={677}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_mod\\_\\_}}%mdk-data-line={677}\n{}, %mdk-data-line={677}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_floordiv\\_}}%mdk-data-line={677}\n{})\nand bitwise methods\n(%mdk-data-line={679}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_and\\_\\_}}%mdk-data-line={679}\n{}, %mdk-data-line={679}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_or\\_\\_}}%mdk-data-line={679}\n{}, %mdk-data-line={679}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_xor\\_\\_}}%mdk-data-line={679}\n{}, %mdk-data-line={679}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_lshift\\_\\_}}%mdk-data-line={679}\n{}, %mdk-data-line={679}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_rshift\\_\\_}}%mdk-data-line={679}\n{}).%\n\\end{mdP}%\n\\mdHxxx[id=sec-tracing-control-flow,label={[4.3]\\{.heading-label\\}},toc={},data-line={682},caption={[[4.3]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Tracing control-flow},bookmark={4.3.{\\hspace{0.5em}}Tracing control-flow}]{%mdk-data-line={682}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{4.3}.{\\hspace{0.5em}}}%mdk-data-line={682}\n{}Tracing control-flow}\\begin{mdP}[data-line={684}]%\n%mdk-data-line={684}\n{}As Python interprets a program, it will evaluate expressions, \nsubstituting the value of a variable in its place in an \nexpression, applying operators (methods) to \nparameter values and assigning the return values of methods\nto variables. Value of type %mdk-data-line={688}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicInteger}}%mdk-data-line={688}\n{} will simply flow\nthrough this interpretation, without necessitating any change \nto the program or the interpreter. This takes care\nof the case of the strongest-postcondition rule for assignment,\nas elaborated in Section%mdk-data-line={692}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h2}]{sp-refined}{}{\\mdSpan[class={heading-label}]{3.2}}%mdk-data-line={692}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={694}]%\n%mdk-data-line={694}\n{}The strong-postcondition rule for a conditional test requires\na little more work. In Python, any object can be tested in\nan %mdk-data-line={696}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Keyword,Python}{if}}%mdk-data-line={696}\n{} or %mdk-data-line={696}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Keyword,Python}{while}}%mdk-data-line={696}\n{} condition or as the operand of a Boolean operation\n(%mdk-data-line={697}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Keyword,Python}{and}}%mdk-data-line={697}\n{}, %mdk-data-line={697}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Keyword,Python}{or}}%mdk-data-line={697}\n{}, %mdk-data-line={697}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Keyword,Python}{not}}%mdk-data-line={697}\n{})\nThe Python base class %mdk-data-line={698}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{object}}%mdk-data-line={698}\n{} provides a method named %mdk-data-line={698}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_bool\\_\\_}}%mdk-data-line={698}\n{} that\nthe Python runtime calls whenever it needs to perform such a conditional test.\nThis hook provides us what we need to trace the conditional\ncontrol-flow of a Python execution.  We override this method in the class\n%mdk-data-line={702}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicObject}}%mdk-data-line={702}\n{} in order to inform the %mdk-data-line={702}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{PathToConstraint}}%mdk-data-line={702}\n{} object (defined\nlater) of the symbolic expression for the conditional (as captured by\nthe %mdk-data-line={704}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicInteger}}%mdk-data-line={704}\n{} subclass).%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={706}]%\n%mdk-data-line={706}\n{}Note that the use of this hook in combination with the\ntainted types will only trace those conditionals\nin a Python execution whose values inherit from %mdk-data-line={708}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicObject}}%mdk-data-line={708}\n{}; \nby definition, %mdk-data-line={709}\n{}{\\textquotedblleft}untainted{\\textquotedblright}%mdk-data-line={709}\n{} conditionals do not depend on symbolic \ninputs so there is no value in adding them to the path-condition.%\n\\end{mdP}%\n\\mdHxxx[id=sec-recording-path-conditions,label={[4.4]\\{.heading-label\\}},toc={},data-line={712},caption={[[4.4]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Recording path-conditions},bookmark={4.4.{\\hspace{0.5em}}Recording path-conditions}]{%mdk-data-line={712}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{4.4}.{\\hspace{0.5em}}}%mdk-data-line={712}\n{}Recording path-conditions}\\begin{mdP}[data-line={714}]%\n%mdk-data-line={714}\n{}A %mdk-data-line={714}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Predicate}}%mdk-data-line={714}\n{} records a conditional (more precisely the symbolic expression\nfound in %mdk-data-line={715}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicInteger}}%mdk-data-line={715}\n{}) and\nwhich way it evaluated in an execution.  A %mdk-data-line={716}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={716}\n{}\nhas a %mdk-data-line={717}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Predicate}}%mdk-data-line={717}\n{}, a parent %mdk-data-line={717}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={717}\n{} and a set\nof %mdk-data-line={718}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={718}\n{} children. %mdk-data-line={718}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraints}}%mdk-data-line={718}\n{} form a tree, where\neach path starting from the root of the tree represents\na path-condition. The tree represents all path-conditions that have\nbeen explored so far.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={723}]%\n%mdk-data-line={723}\n{}The class %mdk-data-line={723}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{PathToConstraint}}%mdk-data-line={723}\n{} has a reference to the root of \nthe tree of %mdk-data-line={724}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={724}\n{}s\nand is responsible for installing a new %mdk-data-line={725}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={725}\n{} in the tree\nwhen notified by the overridden %mdk-data-line={726}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_bool\\_\\_}}%mdk-data-line={726}\n{} method of %mdk-data-line={726}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicObject}}%mdk-data-line={726}\n{}.\n%mdk-data-line={727}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{PathToConstraint}}%mdk-data-line={727}\n{} also tracks whether or not the current execution\nis following an existing path in the tree and grows the\ntree as needed. In fact, it actually\ntracks whether or not the current execution follows a particular\n%mdk-data-line={731}\n{}\\mdEm{expected path}%mdk-data-line={731}\n{} in the tree.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={733}]%\n%mdk-data-line={733}\n{}The expected path is the result\nof the %mdk-data-line={734}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{ExplorationEngine}}%mdk-data-line={734}\n{} picking a constraint %mdk-data-line={734}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={734}\n{} in the tree,\nand asking the ATP if the path-condition consisting of the prefix\nof predicates up to but not including %mdk-data-line={736}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={736}\n{} in the tree, \nfollowed by the negation of %mdk-data-line={737}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={737}\n{}{'}%mdk-data-line={737}\n{}s predicate is satisfiable. If the \nATP returns %mdk-data-line={738}\n{}{\\textquotedblleft}satisfiable{\\textquotedblright}%mdk-data-line={738}\n{} (with a new input %mdk-data-line={738}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$i$}%mdk-data-line={738}\n{}), then the assumption\nis that path-condition prefix is sound (that is, the execution of\nthe program on input %mdk-data-line={740}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$i$}%mdk-data-line={740}\n{} will follow the prefix).%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={742}]%\n%mdk-data-line={742}\n{}However, it is possible \nfor the path-condition to be unsound and for \nthe executed path to diverge early \nfrom the expected path, due to the fact that not every operation\nhas a symbolic encoding.  The tool simply reports the divergence\nand continues to process the execution as usual (as a diverging\npath may lead to some other interesting part of the code).%\n\\end{mdP}%\n\\mdHxxx[id=sec-from-symbolic-types-to-z3,label={[4.5]\\{.heading-label\\}},toc={},data-line={750},caption={[[4.5]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}From symbolic types to Z3},bookmark={4.5.{\\hspace{0.5em}}From symbolic types to Z3}]{%mdk-data-line={750}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{4.5}.{\\hspace{0.5em}}}%mdk-data-line={750}\n{}From symbolic types to Z3}\\begin{mdP}[data-line={752}]%\n%mdk-data-line={752}\n{}As we have explained DSE, the symbolic expressions are \nrepresented at the level of the source language. As detailed later\nin Section%mdk-data-line={754}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-int2z3}{}{\\mdSpan[class={heading-label}]{5}}%mdk-data-line={754}\n{}, we must translate \nfrom the source language to the input language of an\nautomated theorem prover (ATP), in this case%mdk-data-line={756}\n{}{\\mdNbsp}\\mdA[data-linkid={z3}]{http://z3.codeplex.org/}{}{Z3}%mdk-data-line={756}\n{}.  This\nseparation of languages is quite useful, as we may have\nthe need to translate a given symbolic expression\nto the ATP%mdk-data-line={759}\n{}{'}%mdk-data-line={759}\n{}s language multiple times, to make use of different\nfeatures of the underlying ATP. \nFurthermore, this separation\nof concerns allows us to easily retarget the DSE tool to a\ndifferent ATP.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={765}]%\n%mdk-data-line={765}\n{}The base class %mdk-data-line={765}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3Expression}}%mdk-data-line={765}\n{} represents a Z3 formula. The two \nsubclasses %mdk-data-line={766}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3Integer}}%mdk-data-line={766}\n{} and %mdk-data-line={766}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3BitVector}}%mdk-data-line={766}\n{} represent different ways \nto model arithmetic reasoning about integers in Z3. We will describe\nthe details of these encodings in Section%mdk-data-line={768}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-int2z3}{}{\\mdSpan[class={heading-label}]{5}}%mdk-data-line={768}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={770}]%\n%mdk-data-line={770}\n{}The class %mdk-data-line={770}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3Wrapper}}%mdk-data-line={770}\n{} is responsible for performing the\ntranslation from the source language (Python) to Z3%mdk-data-line={771}\n{}{'}%mdk-data-line={771}\n{}s input language, \ninvoking Z3, and lifting a Z3 answer back to the level of Python. \nThe %mdk-data-line={773}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{findCounterexample}}%mdk-data-line={773}\n{} method does all the work, taking as\ninput a list of %mdk-data-line={774}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Predicate}}%mdk-data-line={774}\n{}s (called %mdk-data-line={774}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{assertions}}%mdk-data-line={774}\n{}) \nas well as a single %mdk-data-line={775}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Predicate}}%mdk-data-line={775}\n{} (called\nthe %mdk-data-line={776}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{query}}%mdk-data-line={776}\n{}). The %mdk-data-line={776}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{assertions}}%mdk-data-line={776}\n{} represent a path-condition\nprefix derived from the %mdk-data-line={777}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={777}\n{} tree that we wish the next\nexecution to follow, while %mdk-data-line={778}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{query}}%mdk-data-line={778}\n{} represents the predicate\nfollowing the prefix in the tree that we will negate.%\n\\end{mdP}%\n\\begin{mdP}[class={indent,para-continue},data-line={781}]%\n%mdk-data-line={781}\n{}The method constructs the formula%\n\\end{mdP}%\n\\begin{mdDiv}[class={equation,para-block},label={[(4)]\\{.equation-label\\}},elem={equation},line-adjust={0},data-line={783}]%\n%mdk-data-line={783}\n{}\\mdSpan[class={equation-before}]{\\mdSpan[class={equation-label}]{(4)}}%mdk-data-line={783}\n{}\n\\begin{mdDiv}[class={mathdisplay,para-block,input-math},elem={mathdisplay},color={},math-needpdf={},line-adjust={0},data-line={784}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={784}\n(\\bigwedge_{a \\in asserts} a) \\wedge \\neg query\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[data-line={787}]%\n%mdk-data-line={787}\n{}and asks Z3 if it is satisfiable. The method performs\na standard syntactic %mdk-data-line={788}\n{}{\\textquotedblleft}cone of influence{\\textquotedblright}%mdk-data-line={788}\n{} (CIF)\nreduction on the %mdk-data-line={789}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{asserts}}%mdk-data-line={789}\n{} with respect to the \n%mdk-data-line={790}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{query}}%mdk-data-line={790}\n{} to shrink the size of the formula. For example,\nif %mdk-data-line={791}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{asserts}}%mdk-data-line={791}\n{} is the set of predicates %mdk-data-line={791}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\{ (x<a), (a<0), (y>0) \\}$}%mdk-data-line={791}\n{} and\nthe query is %mdk-data-line={792}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$(x=0)$}%mdk-data-line={792}\n{}, then the CIF yields the\nset  %mdk-data-line={793}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\{ (x<a), (a<0) \\}$}%mdk-data-line={793}\n{}, which does not include the predicate %mdk-data-line={793}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$(y>0)$}%mdk-data-line={793}\n{},\nas the variable %mdk-data-line={794}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$y$}%mdk-data-line={794}\n{} is not in the set of variables (transitively)\nrelated to variable %mdk-data-line={795}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x$}%mdk-data-line={795}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={797}]%\n%mdk-data-line={797}\n{}If the formula is satisfiable a model is requested\nfrom Z3 and lifted back to Python%mdk-data-line={798}\n{}{'}%mdk-data-line={798}\n{}s type universe.  Note that\nbecause of the CIF reduction, the model may not mention certain\ninput variables, in which case we simply keep their values from\nthe execution from which the %mdk-data-line={801}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{asserts}}%mdk-data-line={801}\n{} and %mdk-data-line={801}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{query}}%mdk-data-line={801}\n{} were derived.%\n\\end{mdP}%\n\\mdHxxx[id=sec-putting-it-all-together,label={[4.6]\\{.heading-label\\}},toc={},data-line={803},caption={[[4.6]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Putting it all together},bookmark={4.6.{\\hspace{0.5em}}Putting it all together}]{%mdk-data-line={803}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{4.6}.{\\hspace{0.5em}}}%mdk-data-line={803}\n{}Putting it all together}\\begin{mdP}[data-line={805}]%\n%mdk-data-line={805}\n{}The class %mdk-data-line={805}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{ExplorationEngine}}%mdk-data-line={805}\n{} ties everything together. It kicks off\nan execution of the Python code under test using %mdk-data-line={806}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{FunctionInvocation}}%mdk-data-line={806}\n{}.\nAs the Python code executes, building symbolic expressions via %mdk-data-line={807}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicType}}%mdk-data-line={807}\n{}\nand its subclasses, callbacks to %mdk-data-line={808}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{PathToConstraint}}%mdk-data-line={808}\n{} create\na path-condition, represented by %mdk-data-line={809}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={809}\n{} and %mdk-data-line={809}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Predicate}}%mdk-data-line={809}\n{}. \nNewly discovered %mdk-data-line={810}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraints}}%mdk-data-line={810}\n{} are added to the end of a deque maintained by\n%mdk-data-line={811}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{ExplorationEngine}}%mdk-data-line={811}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={813}]%\n%mdk-data-line={813}\n{}Given the first seed execution, %mdk-data-line={813}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{ExplorationEngine}}%mdk-data-line={813}\n{} starts the work of \nexploring paths in a breadth-first fashion. It removes a %mdk-data-line={814}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={814}\n{} %mdk-data-line={814}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={814}\n{}\nfrom the front of its deque and, if %mdk-data-line={815}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={815}\n{} has not been already %mdk-data-line={815}\n{}{\\textquotedblleft}processed{\\textquotedblright}%mdk-data-line={815}\n{}, \nuses %mdk-data-line={816}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3Wrapper}}%mdk-data-line={816}\n{} to find a new input (as discussed in the previous section)\nwhere %mdk-data-line={817}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={817}\n{} is the query (to be negated) and the path to %mdk-data-line={817}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={817}\n{} in the \n%mdk-data-line={818}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={818}\n{} tree forms the assertions.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={820}]%\n%mdk-data-line={820}\n{}A %mdk-data-line={820}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={820}\n{} %mdk-data-line={820}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={820}\n{} in the tree is considered %mdk-data-line={820}\n{}{\\textquotedblleft}processed{\\textquotedblright}%mdk-data-line={820}\n{} if an execution \nhas covered %mdk-data-line={821}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c'$}%mdk-data-line={821}\n{}, a sibling of %mdk-data-line={821}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={821}\n{} in the tree that represents\nthe negation of the predicate associated with %mdk-data-line={822}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={822}\n{}, or if constraint %mdk-data-line={822}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={822}\n{}\nhas been removed from the deque.%\n\\end{mdP}%\n\\mdHxx[id=sec-int2z3,label={[5]\\{.heading-label\\}},toc={},data-line={825},caption={[[5]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}From Python Integers to Z3 Arithmetic},bookmark={5.{\\hspace{0.5em}}From Python Integers to Z3 Arithmetic}]{%mdk-data-line={825}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{5}.{\\hspace{0.5em}}}%mdk-data-line={825}\n{}From Python Integers to Z3 Arithmetic}\\begin{mdP}[data-line={827}]%\n%mdk-data-line={827}\n{}In languages such as C and Java, integers are finite-precision,\ngenerally limited to the size of a machine word (32 or 64 bits, for example).\nFor such languages, satisfiability of finite-precision integer arithmetic\nis decidable and can be reduced to Z3%mdk-data-line={830}\n{}{'}%mdk-data-line={830}\n{}s theory of bit-vectors, where\neach arithmetic operation is encoded by a circuit. This translation permits \nreasoning about non-linear arithmetic problems, such as \n%mdk-data-line={833}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\exists x,y,z : x*z + y \\leq (z/y)+5$}%mdk-data-line={833}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={835}]%\n%mdk-data-line={835}\n{}Python (3.0) integers, however, are not finite-precision. They are only\nlimited by the size of machine memory. This means, for example, that\nPython integers don%mdk-data-line={837}\n{}{'}%mdk-data-line={837}\n{}t overflow or underflow. It also means that\nwe can%mdk-data-line={838}\n{}{'}%mdk-data-line={838}\n{}t hope to decide algorithmically whether or not a given\nequation over integer variables has a solution in general. Hilbert%mdk-data-line={839}\n{}{'}%mdk-data-line={839}\n{}s famous\n10th problem and its solution by Matiyasevich tells us that it is\nundecidable whether or not a polynomial\nequation of the form %mdk-data-line={842}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p(x_1, \\ldots, x_n) = 0$}%mdk-data-line={842}\n{} with integer coefficients\nhas an solution in the integers.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={845}]%\n%mdk-data-line={845}\n{}This means that we will resort to heuristic approaches in our use\nof the%mdk-data-line={846}\n{}{\\mdNbsp}\\mdA[data-linkid={z3}]{http://z3.codeplex.org/}{}{Z3}%mdk-data-line={846}\n{} ATP.  The special case of linear integer arithmetic (LIA)\nis decidable and supported by Z3. In order to deal with non-linear operations,\nwe use uninterpreted functions (UF). Thus, if Z3 returns\n%mdk-data-line={849}\n{}{\\textquotedblleft}unsatisfiable{\\textquotedblright}%mdk-data-line={849}\n{} we know that there is no solution, but if the\nZ3 %mdk-data-line={850}\n{}{\\textquotedblleft}satisfiable{\\textquotedblright}%mdk-data-line={850}\n{}, we must treat the answer as a %mdk-data-line={850}\n{}{\\textquotedblleft}don{'}t know{\\textquotedblright}%mdk-data-line={850}\n{}. \nThe class %mdk-data-line={851}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3Integer}}%mdk-data-line={851}\n{} is used to translate a symbolic expression\ninto the theory LIA+UF and check for unsatisfiability. We leave it as an\nimplementation exercise to check if a symbolic expression can\nbe converted to LIA (without the use of UF) in order to make\nuse of %mdk-data-line={855}\n{}{\\textquotedblleft}satisfiable{\\textquotedblright}%mdk-data-line={855}\n{} answers from the LIA solver.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={857}]%\n%mdk-data-line={857}\n{}If the translation to %mdk-data-line={857}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3Integer}}%mdk-data-line={857}\n{} does not return %mdk-data-line={857}\n{}{\\textquotedblleft}unsatisfiable{\\textquotedblright}%mdk-data-line={857}\n{}, we\nuse Z3%mdk-data-line={858}\n{}{'}%mdk-data-line={858}\n{}s bit-vector decision procedure (via the class\n%mdk-data-line={859}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3BitVector}}%mdk-data-line={859}\n{}) to heuristically\ntry to find satisfiable answers, even in the presence of non-linear \narithmetic. We start with bit-vectors of size %mdk-data-line={861}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$N=32$}%mdk-data-line={861}\n{} and %mdk-data-line={861}\n{}\\mdEm{bound}%mdk-data-line={861}\n{} the values\nof the symbolic constants to fit within 8 bits in order to find \nsatisfiable solutions\nwith small values. Also, because Python integers do not overflow/underflow, \nthe bound helps us reserve space in the bit-vector to allow the\nresults of operations to exceed the bound while not overflowing\nthe bit-vector. As long as Z3 returns %mdk-data-line={867}\n{}{\\textquotedblleft}unsatisfiable{\\textquotedblright}%mdk-data-line={867}\n{} we increase\nthe bound. If the bound reaches %mdk-data-line={868}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$N$}%mdk-data-line={868}\n{}, we increase %mdk-data-line={868}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$N$}%mdk-data-line={868}\n{} by 8 bits,\nleaving the bound where it is and continue.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={871}]%\n%mdk-data-line={871}\n{}If Z3 returns\n%mdk-data-line={872}\n{}{\\textquotedblleft}satisfiable{\\textquotedblright}%mdk-data-line={872}\n{}, it may be the case that Z3 found a solution\nthat involved overflow in the bit-vector world of arithmetic\n(modulo %mdk-data-line={874}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$2^N-1$}%mdk-data-line={874}\n{}). Therefore,\nthe solution is validated back in the\nPython world by evaluating the formula under that solution\nusing Python semantics. \nIf the formula does not evaluate to the same\nvalue in both worlds, then we increase %mdk-data-line={879}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$N$}%mdk-data-line={879}\n{} by 8 bits (to \ncreate a gap between the bound and %mdk-data-line={880}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$N$}%mdk-data-line={880}\n{}) and continue to search\nfor a solution.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={883}]%\n%mdk-data-line={883}\n{}The process terminates when we find a valid satisfying solution \nor %mdk-data-line={884}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$N=64$}%mdk-data-line={884}\n{} and the bound reaches 64 (in which case, we return %mdk-data-line={884}\n{}{\\textquotedblleft}don{'}t know{\\textquotedblright}%mdk-data-line={884}\n{}).%\n\\end{mdP}%\n\\mdHxx[id=sec-extensions,label={[6]\\{.heading-label\\}},toc={},data-line={886},caption={[[6]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Extensions},bookmark={6.{\\hspace{0.5em}}Extensions}]{%mdk-data-line={886}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{6}.{\\hspace{0.5em}}}%mdk-data-line={886}\n{}Extensions}\\begin{mdP}[data-line={888}]%\n%mdk-data-line={888}\n{}We have presented the basics of dynamic symbolic execution \n(for Python).\nA more thorough treatment would deal with other data types besides\nintegers, such as Python dictionaries, strings and lists, each\nof which presents their own challenges for symbolic reasoning. \nThere are many other interesting challenges in DSE, such\nas dealing with user-defined classes (rather than built-in types\nas done here) and multi-threaded execution.%\n\\end{mdP}%\n\\mdHxx[id=sec-acknowledgements,label={[7]\\{.heading-label\\}},toc={},data-line={897},caption={[[7]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Acknowledgements},bookmark={7.{\\hspace{0.5em}}Acknowledgements}]{%mdk-data-line={897}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{7}.{\\hspace{0.5em}}}%mdk-data-line={897}\n{}Acknowledgements}\\begin{mdP}[data-line={899}]%\n%mdk-data-line={899}\n{}Many thanks to the students of the 2014 Marktoberdorf Summer School\non Dependable Software Systems Engineering\nfor their questions and feedback about the first author%mdk-data-line={901}\n{}{'}%mdk-data-line={901}\n{}s lectures on dynamic\nsymbolic execution. The following students of the summer school\nhelpfully provided tests for the%mdk-data-line={903}\n{}{\\mdNbsp}\\mdA[data-linkid={pyexz3}]{https://github.com/thomasjball/PyExZ3/}{}{PyExZ3}%mdk-data-line={903}\n{} tool: Daniel Darvas,\nDamien Rusinek, Christian Dehnert and Thomas Pani. Thanks also to Peter\nChapman for his contributions.%\n\\end{mdP}%\n\n\\mdHxx[id=sec-references,label={8},toc={},data-line={963},caption={References},bookmark={References}]{%mdk-data-line={963}\n{}References}\\begin{mdBibliography}[class={bibliography,bib-numeric},elem={bibliography},bibstyle={plainnat},bibdata={dse},caption={14},data-line={964;out{\\textbackslash}DSE-bib.bbl:2}]%\n\\begin{mdBibitem}[class={bibitem},id=cadare05,label={[1]\\{.bibitem-label\\}},elem={bibitem},cite-label={Cadar and Engler(2005)},caption={Cristian Cadar and Dawson{\\textbackslash} R. 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Engler.\n%mdk-data-line={964;out\\DSE-bib.bbl:7}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:7}\n{} Execution generated test cases: How to make systems code crash\n  itself.\n%mdk-data-line={964;out\\DSE-bib.bbl:9}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:9}\n{} In %mdk-data-line={964;out\\DSE-bib.bbl:9}\n{}\\mdEm{Proceedings of 12th International SPIN Workshop}%mdk-data-line={964;out\\DSE-bib.bbl:9}\n{}%mdk-data-line={964;out\\DSE-bib.bbl:9}\n{}, pages\n  2%mdk-data-line={964;out\\DSE-bib.bbl:10}\n{}{\\textendash}%mdk-data-line={964;out\\DSE-bib.bbl:10}\n{}23, 2005.%\n\\end{mdBibitem}%\n\\begin{mdBibitem}[class={bibitem},id=cadars13,label={[2]\\{.bibitem-label\\}},elem={bibitem},cite-label={Cadar and Sen(2013)},caption={Cristian Cadar and Koushik Sen. \\\\Symbolic execution for software testing: three decades later.},searchterm={Symbolic+execution+software+testing+three+decades+later++Cristian+Cadar+Koushik+},data-line={964;out{\\textbackslash}DSE-bib.bbl:13}]%\n%mdk-data-line={964;out\\DSE-bib.bbl:14}\n{}\\mdSpan[class={bibitem-before}]{[\\mdSpan[class={bibitem-label}]{2}]{\\mdNbsp}{\\mdNbsp}}%mdk-data-line={964;out\\DSE-bib.bbl:14}\n{}Cristian Cadar and Koushik Sen.\n%mdk-data-line={964;out\\DSE-bib.bbl:15}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:15}\n{} Symbolic execution for software testing: three decades later.\n%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{} %mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}\\mdEm{Communications of the ACM}%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}, 56%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}\\mdSpan[penalty={0}]{}%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{} (2):%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}\\mdSpan[penalty={0}]{}%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{} 82%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}{\\textendash}%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}90,\n  2013.%\n\\end{mdBibitem}%\n\\begin{mdBibitem}[class={bibitem},id=cadargpde06,label={[3]\\{.bibitem-label\\}},elem={bibitem},cite-label={Cadar et{\\textbackslash} al.(2006)Cadar, Ganesh, Pawlowski, Dill, and\\\\  Engler},caption={Cristian Cadar, Vijay Ganesh, Peter{\\textbackslash} M. 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Dill, and Dawson%mdk-data-line={964;out\\DSE-bib.bbl:21}\n{}{\\mdNbsp}%mdk-data-line={964;out\\DSE-bib.bbl:21}\n{}R.\n  Engler.\n%mdk-data-line={964;out\\DSE-bib.bbl:23}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:23}\n{} EXE: automatically generating inputs of death.\n%mdk-data-line={964;out\\DSE-bib.bbl:24}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:24}\n{} In %mdk-data-line={964;out\\DSE-bib.bbl:24}\n{}\\mdEm{Proceedings of the 13th ACM Conference on Computer and\n  Communications Security}%mdk-data-line={964;out\\DSE-bib.bbl:25}\n{}%mdk-data-line={964;out\\DSE-bib.bbl:25}\n{}, pages 322%mdk-data-line={964;out\\DSE-bib.bbl:25}\n{}{\\textendash}%mdk-data-line={964;out\\DSE-bib.bbl:25}\n{}335, 2006.%\n\\end{mdBibitem}%\n\\begin{mdBibitem}[class={bibitem},id=clarke76,label={[4]\\{.bibitem-label\\}},elem={bibitem},cite-label={Clarke(1976)},caption={Lori{\\textbackslash} A. Clarke. \\\\A system to generate test data and symbolically execute programs.},searchterm={+system+generate+test+data+symbolically+execute+programs++Lori+Clarke+},data-line={964;out{\\textbackslash}DSE-bib.bbl:28}]%\n%mdk-data-line={964;out\\DSE-bib.bbl:29}\n{}\\mdSpan[class={bibitem-before}]{[\\mdSpan[class={bibitem-label}]{4}]{\\mdNbsp}{\\mdNbsp}}%mdk-data-line={964;out\\DSE-bib.bbl:29}\n{}Lori%mdk-data-line={964;out\\DSE-bib.bbl:29}\n{}{\\mdNbsp}%mdk-data-line={964;out\\DSE-bib.bbl:29}\n{}A. Clarke.\n%mdk-data-line={964;out\\DSE-bib.bbl:30}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:30}\n{} A system to generate test data and symbolically execute programs.\n%mdk-data-line={964;out\\DSE-bib.bbl:31}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:31}\n{} %mdk-data-line={964;out\\DSE-bib.bbl:31}\n{}\\mdEm{IEEE Transactions on Software Engineering}%mdk-data-line={964;out\\DSE-bib.bbl:31}\n{}%mdk-data-line={964;out\\DSE-bib.bbl:31}\n{}, 2%mdk-data-line={964;out\\DSE-bib.bbl:31}\n{}\\mdSpan[penalty={0}]{}%mdk-data-line={964;out\\DSE-bib.bbl:31}\n{}\n  (3):%mdk-data-line={964;out\\DSE-bib.bbl:32}\n{}\\mdSpan[penalty={0}]{}%mdk-data-line={964;out\\DSE-bib.bbl:32}\n{} 215%mdk-data-line={964;out\\DSE-bib.bbl:32}\n{}{\\textendash}%mdk-data-line={964;out\\DSE-bib.bbl:32}\n{}222, 1976.%\n\\end{mdBibitem}%\n\\begin{mdBibitem}[class={bibitem},id=demourab08,label={[5]\\{.bibitem-label\\}},elem={bibitem},cite-label={de{\\textbackslash} Moura and Bj{\\o}rner(2008)},caption={Leonardo{\\textbackslash} Mendon{\\c{c}}a de{\\textbackslash} Moura and Nikolaj Bj{\\o}rner. \\\\Z3: an efficient SMT solver.},searchterm={+efficient+solver++Leonardo+Mendon+Moura+Nikolaj+rner+},data-line={964;out{\\textbackslash}DSE-bib.bbl:35}]%\n%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}\\mdSpan[class={bibitem-before}]{[\\mdSpan[class={bibitem-label}]{5}]{\\mdNbsp}{\\mdNbsp}}%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}Leonardo%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}{\\mdNbsp}%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}Mendon%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}{\\c{c}}%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}a de%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}{\\mdNbsp}%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}Moura and Nikolaj Bj%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}{\\o}%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}rner.\n%mdk-data-line={964;out\\DSE-bib.bbl:37}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:37}\n{} Z3: an efficient SMT solver.\n%mdk-data-line={964;out\\DSE-bib.bbl:38}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:38}\n{} In %mdk-data-line={964;out\\DSE-bib.bbl:38}\n{}\\mdEm{Proceedings of the 14th International Conference of Tools\n  and Algorithms for the Construction and Analysis of Systems}%mdk-data-line={964;out\\DSE-bib.bbl:39}\n{}%mdk-data-line={964;out\\DSE-bib.bbl:39}\n{}, pages 337%mdk-data-line={964;out\\DSE-bib.bbl:39}\n{}{\\textendash}%mdk-data-line={964;out\\DSE-bib.bbl:39}\n{}340,\n  2008.%\n\\end{mdBibitem}%\n\\begin{mdBibitem}[class={bibitem},id=dijkstra76,label={[6]\\{.bibitem-label\\}},elem={bibitem},cite-label={Dijkstra(1976)},caption={Edsger{\\textbackslash} W. 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Mathur, and Mary%mdk-data-line={964;out\\DSE-bib.bbl:71}\n{}{\\mdNbsp}%mdk-data-line={964;out\\DSE-bib.bbl:71}\n{}Lou Soffa.\n%mdk-data-line={964;out\\DSE-bib.bbl:72}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:72}\n{} Generating test data for branch coverage.\n%mdk-data-line={964;out\\DSE-bib.bbl:73}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:73}\n{} In %mdk-data-line={964;out\\DSE-bib.bbl:73}\n{}\\mdEm{Proceedings of the Automate Software Engineering\n  Conference}%mdk-data-line={964;out\\DSE-bib.bbl:74}\n{}%mdk-data-line={964;out\\DSE-bib.bbl:74}\n{}, pages 219%mdk-data-line={964;out\\DSE-bib.bbl:74}\n{}{\\textendash}%mdk-data-line={964;out\\DSE-bib.bbl:74}\n{}228, 2000.%\n\\end{mdBibitem}%\n\\begin{mdBibitem}[class={bibitem},id=king76,label={[11]\\{.bibitem-label\\}},elem={bibitem},cite-label={King(1976)},caption={James{\\textbackslash} C. 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Conference\\_{\\textbackslash}/, pages 419--423, 2006.},searchterm={Koushik+Agha+CUTE+jcute+Concolic+unit+testing+explicit+path+model+checking+tools+\\_Proceedings+Computer+Aided+Verification+Conference\\_+pages++},data-line={964;out{\\textbackslash}DSE-bib.bbl:98}]%\n%mdk-data-line={964;out\\DSE-bib.bbl:99}\n{}\\mdSpan[class={bibitem-before}]{[\\mdSpan[class={bibitem-label}]{14}]{\\mdNbsp}{\\mdNbsp}}%mdk-data-line={964;out\\DSE-bib.bbl:99}\n{}Koushik Sen and Gul Agha.\n%mdk-data-line={964;out\\DSE-bib.bbl:100}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:100}\n{} CUTE and jcute: Concolic unit testing and explicit path\n  model-checking tools.\n%mdk-data-line={964;out\\DSE-bib.bbl:102}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:102}\n{} In %mdk-data-line={964;out\\DSE-bib.bbl:102}\n{}\\mdEm{Proceedings of 18th Computer Aided Verification Conference}%mdk-data-line={964;out\\DSE-bib.bbl:102}\n{}%mdk-data-line={964;out\\DSE-bib.bbl:102}\n{},\n  pages 419%mdk-data-line={964;out\\DSE-bib.bbl:103}\n{}{\\textendash}%mdk-data-line={964;out\\DSE-bib.bbl:103}\n{}423, 2006.%\n\\end{mdBibitem}%%\n\\end{mdBibliography}%\n\\begin{mdDiv}[class={logomadoko,block},elem={logomadoko},text-align={right},font-size={xx-small},margin-top={4em},data-line={967}]%\n%mdk-data-line={968}\n{}Created with{\\mdNbsp}\\mdA{https://www.madoko.net}{}{Madoko.net}.%\n\\end{mdDiv}%\n\n\n\\end{document}\n"
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/* so math displays well on mobile devices */\n}\n\nbody.madoko, .madoko-body, body.madoko.preview[data-view=full] {\n  padding: 1em 16% 1em 12%;\n}\n\nbody.madoko.preview {\n  padding: 0em 2em 0em 2em;\n}\n\n.madoko p,\n.madoko li {\n  text-align: justify;\n}\n\n/* style h5 and h6 headers a bit better */\n.madoko h5, .madoko h6 {\n  margin-top: 0.5em;\n  margin-bottom: 0pt;\n  font-size: 100%;\n}\n.madoko h5+p {\n  margin-top: 0.5em;\n}\n.madoko h6+p {\n  margin-top: 0pt;\n}\n\n/* Fix monospace display (see http://code.stephenmorley.org/html-and-css/fixing-browsers-broken-monospace-font-handling/) */\n.madoko pre, .madoko code, .madoko kbd, .madoko samp, .madoko tt, .madoko .monospace, .madoko .token.indent, .madoko .reveal pre, .madoko .reveal code, .madoko .email {\n  font-family: Consolas,\"Andale Mono WT\",\"Andale Mono\",Lucida Console,Monaco,monospace,monospace;\n  font-size: 0.85em;\n}\n.madoko pre code, .madoko .token.indent {\n  font-size: 0.95em;\n}\n\n.madoko pre code {\n  font-family: inherit !important;\n}\n\n/* Code prettify */\n.madoko ol.linenums li {\n  background-color: white;\n  list-style-type: decimal;\n}\n\n/* Merging */\n.madoko .remote {\n  background-color: #F0FFF0;\n}\n\n/* ---------------------------------------------------\n   Print settings\n----------------------------------------------------*/\n\n@media print {\n  body.madoko, .madoko-body {\n    font-size: 10pt;\n  }\n  @page {\n    margin: 1in 1.5in;\n  }\n}\n\n/* ---------------------------------------------------\n   Mobile device settings\n----------------------------------------------------*/\n\n@media only screen and (max-device-width:1024px) {\n  body.madoko, .madoko-body {\n    padding-left: 1em;\n    padding-right: 1em;\n  }\n}\n\n  </style>\n\n</head>\n<body class=\"madoko\">\n\n\n<div class=\"htmlonly\"><section class='titleblock align-center'><h1 class='title' data-line='1'>Deconstructing Dynamic Symbolic Execution</h1>\n\n\n<table><tr><td class='authorblock align-center'><div class='author'>Thomas Ball, Jakub Daniel</div><div class='affiliation address'>Microsoft Research, Charles University</div><div class='email'>tball@microsoft.com, jakub.daniel@d3s.mff.cuni.cz</div></td></tr></table></section></div>\n<div class=\"abstract\">\n<p data-line=\"1\"><span data-line=\"1\"></span>Dynamic symbolic execution (DSE) is a well-known technique\nfor automatically generating tests to achieve higher levels\nof coverage in a program. Two keys ideas of DSE are\nto: (1) seed symbolic execution by executing a program on an\ninitial input; (2) using concrete values from the program\nexecution in place of symbolic expressions whenever symbolic\nreasoning is hard or not desired. In this paper, we describe\nDSE for a simple core language and then present\na minimalist implementation of DSE for Python (in Python) \nthat follows this basic recipe. The code is available \nat http://www.github.com/thomasjball/pyexz3/ (tagged <span data-line=\"11\"></span>&#8220;v1.0&#8221;<span data-line=\"11\"></span>) \nand has been designed to make it easy to experiment with and\nextend. </p></div><h2 id=\"sec-intro\"   style=\"bookmark:1.&#8194;Introduction\"><span class=\"heading-before\"><span class=\"heading-label\">1</span>.&#8194;</span>Introduction</h2>\n<div class=\"mathdefs hidden input-mathdefs\"></div><!-- should we motivate more? -- this dives in deep very quickly -->\n\n\n<p>Static, path-based symbolic execution explores one control-flow path\nat a time through a (sequential) program <img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">, using an automated theorem\nprover (ATP) to determine if the current path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> is feasible&nbsp;<span class=\"citations\" style=\"target-element:bibitem\">[<a href=\"#clarke76\" title=\"Lori&#160;A. Clarke. \nA system to generate test data and symbolically execute programs.\" class=\"bibref localref\" style=\"target-element:bibitem\"><span class=\"bibitem-label\">4</span></a>, <a href=\"#king76\" title=\"James&#160;C. King. \nSymbolic execution and program testing.\" class=\"bibref localref\" style=\"target-element:bibitem\"><span class=\"bibitem-label\">11</span></a>]</span>. \nIdeally, symbolic execution a path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> through program\n<img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> yields a logic formula <img src=\"data:image/png;base64,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\" alt=\"$\\phi_p$\" class=\"math-inline math\" style=\"vertical-align:-0.304em;height:1.041em\"> that describes the set of inputs <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAcCAYAAABh2p9gAAAA8UlEQVRIx62W4Q2DIBBGH6QDmI7gCJ2hjtB0g47gEN2kcQSZxRFaN7j+KD+IoXgHkpAIOZ8fT4wgIvzrwASIoY8u3phtzrkO6OOwjw9I2xOYgTewishCKWEm8SdJM+dqPMoW03bJ1JSrUwOB62YcWoFDcv3zdWDC8K/IG/z1ydTcBNT6swDvGn+1CUOp0Cv89Zv9NzcBLf60wEHrryZh2Cv2Rn+vJqDVnwaY+ltEZD0yYdBsWH/U/tMkNPvbA5r9WRIG5TefB9b6KyWs8lcC3jbf76oFnuISL8A5LnPYJOyccw9giT/03Rc0Go8bU+kw8AVaaLebPZP9FwAAAABJRU5ErkJggg==\" alt=\"$I$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> (possibly empty)\nto program <img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> such that for any <img src=\"data:image/png;base64,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\" alt=\"$i \\in I$\" class=\"math-inline math\" style=\"vertical-align:-0.042em;height:0.781em\">, execution of <img src=\"data:image/png;base64,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\" alt=\"$P(i)$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\"> follows path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\">. \n</p>\n<p class=\"indent\">If the formula <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACgAAAAoCAYAAACM/rhtAAACp0lEQVRYw81Y0ZGbMBB978YFkOsgXAc4JXAdkEkHdx1wkxJIByQVJJSA00HsDrgrAXew+VlldggSBASxZhgbdiWedlfvIVFEsLSRbAE8i8grNmpcCpBkAqAXEWLDdreibw7ghI3bGoCPANpbBrhLBBfVoNbfm4i8u9UI7hK9NQA/7VF/a1LcAzhuyX+LI0gyBYA9wM0CSDIjWZE8a+Q6AAlJIdmRrElmmyEUkdELQKFgRH9LXRwtgFrtFYBefRoAiW+8pdcYsATA2QDLB/YeQDbwb4x/shlAAJmNyAj4VPV3bGKdr18UgApOQi/RNDeBknD9s6gANU0ucl2gLlsATx5bYgCWsQE2c2av9nTCHjXNrq7cwG3AMfPV3wjANhbAOwDPhnXqCf39MUXg2l5jEnVu7k8r9Lcw/6Pq9OTiMOlLAnbHnRKTZqzUXQPpywFcROQasDu5e/H41CqNZ5KleV6QbPW5k87E2Q9aL1Na+jiRfle7FxH5MgKuAnAVkQeSNYCKJAA8ALgH8FFErgrsJ4A3ku9dQMqp1IxJnrHV2r/3lYBd/fZ9AKoA4TeOZixJFz4C9ry4MhqcenyeLC+aCbUBOhM3KfcwN1FIR2bUjoBu55CyAsxGNNunSLmJcOL7UChdunTGpfFxETj70h4Aa+UwDej93wAHEWsUbG9S2GnUypDcTQAsbPo8Po1dE1MDpjF5zdRsM1fPpz75s8jbyzykNCStGr3M2ZPEPt5wfPvLY/+sv1//bMomUtItrbeJ1VkGFkcb3JMMVly/Qf2ddeLZiK0e9jvseLzh6u+7jvtNpe1e748ichl2OvyH+jspkOPajfuHWBHUrx1X85d/6XsIbOiPEaP36L529jzAXJTemwOo6c0Hh5/rz2YiUUtvuM9exdwxfgOHZY7f662mjQAAAABJRU5ErkJggg==\" alt=\"$\\phi_p$\" class=\"math-inline math\" style=\"vertical-align:-0.304em;height:1.041em\"> is unsatisfiable then <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAcCAYAAABh2p9gAAAA8UlEQVRIx62W4Q2DIBBGH6QDmI7gCJ2hjtB0g47gEN2kcQSZxRFaN7j+KD+IoXgHkpAIOZ8fT4wgIvzrwASIoY8u3phtzrkO6OOwjw9I2xOYgTewishCKWEm8SdJM+dqPMoW03bJ1JSrUwOB62YcWoFDcv3zdWDC8K/IG/z1ydTcBNT6swDvGn+1CUOp0Cv89Zv9NzcBLf60wEHrryZh2Cv2Rn+vJqDVnwaY+ltEZD0yYdBsWH/U/tMkNPvbA5r9WRIG5TefB9b6KyWs8lcC3jbf76oFnuISL8A5LnPYJOyccw9giT/03Rc0Go8bU+kw8AVaaLebPZP9FwAAAABJRU5ErkJggg==\" alt=\"$I$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> is empty and so path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> is not feasible; \nif the formula is satisfiable then <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAcCAYAAABh2p9gAAAA8UlEQVRIx62W4Q2DIBBGH6QDmI7gCJ2hjtB0g47gEN2kcQSZxRFaN7j+KD+IoXgHkpAIOZ8fT4wgIvzrwASIoY8u3phtzrkO6OOwjw9I2xOYgTewishCKWEm8SdJM+dqPMoW03bJ1JSrUwOB62YcWoFDcv3zdWDC8K/IG/z1ydTcBNT6swDvGn+1CUOp0Cv89Zv9NzcBLf60wEHrryZh2Cv2Rn+vJqDVnwaY+ltEZD0yYdBsWH/U/tMkNPvbA5r9WRIG5TefB9b6KyWs8lcC3jbf76oFnuISL8A5LnPYJOyccw9giT/03Rc0Go8bU+kw8AVaaLebPZP9FwAAAABJRU5ErkJggg==\" alt=\"$I$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> is not empty and so path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> is feasible.\nIn this case, a model of <img src=\"data:image/png;base64,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\" alt=\"$\\phi_p$\" class=\"math-inline math\" style=\"vertical-align:-0.304em;height:1.041em\"> provides a witness <img src=\"data:image/png;base64,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\" alt=\"$i \\in I$\" class=\"math-inline math\" style=\"vertical-align:-0.042em;height:0.781em\">.  Thus, an ATP\nthat can provide such models can be used in conjunction with\nsymbolic execution to automatically generate tests to cover paths\nin a program. Combined with a search strategy, one gets, in the limit,\nan exhaustive white-box testing procedure, for which there are many\napplications&nbsp;<span class=\"citations\" style=\"target-element:bibitem\">[<a href=\"#cadars13\" title=\"Cristian Cadar and Koushik Sen. \nSymbolic execution for software testing: three decades later.\" class=\"bibref localref\" style=\"target-element:bibitem\"><span class=\"bibitem-label\">2</span></a>, <a href=\"#cadargpde06\" title=\"Cristian Cadar, Vijay Ganesh, Peter&#160;M. Pawlowski, David&#160;L. Dill, and Dawson&#160;R. Engler. \nEXE: automatically generating inputs of death. \nIn Proceedings of the 13th ACM Conference on Computer and Communications Security, pages 322&#8211;335, 2006.\" class=\"bibref localref\" style=\"target-element:bibitem\"><span class=\"bibitem-label\">3</span></a>, <a href=\"#godefroidlm12\" title=\"Patrice Godefroid, Michael&#160;Y. Levin, and David&#160;A. Molnar. \nSAGE: whitebox fuzzing for security testing.\" class=\"bibref localref\" style=\"target-element:bibitem\"><span class=\"bibitem-label\">9</span></a>]</span>. \n</p>\n<p class=\"indent\">The formula <img src=\"data:image/png;base64,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\" alt=\"$\\phi_p$\" class=\"math-inline math\" style=\"vertical-align:-0.304em;height:1.041em\"> is called a <em>path-condition</em> of the path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\">. \nWe will see that a given path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> can induce many different path-conditions.\nA path-condition <img src=\"data:image/png;base64,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\" alt=\"$\\psi_p$\" class=\"math-inline math\" style=\"vertical-align:-0.304em;height:1.041em\"> for path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> is <em>sound</em> if \nevery input assignment satisfying <img src=\"data:image/png;base64,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\" alt=\"$\\psi_p$\" class=\"math-inline math\" style=\"vertical-align:-0.304em;height:1.041em\"> defines an\nexecution of program <img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> that follows path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\">&nbsp;<span class=\"citations\" style=\"target-element:bibitem\">[<a href=\"#godefroid11\" title=\"Patrice Godefroid. \nHigher-order test generation.\" class=\"bibref localref\" style=\"target-element:bibitem\"><span class=\"bibitem-label\">7</span></a>]</span>. \nBy its definition, the formula  <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACgAAAAoCAYAAACM/rhtAAACp0lEQVRYw81Y0ZGbMBB978YFkOsgXAc4JXAdkEkHdx1wkxJIByQVJJSA00HsDrgrAXew+VlldggSBASxZhgbdiWedlfvIVFEsLSRbAE8i8grNmpcCpBkAqAXEWLDdreibw7ghI3bGoCPANpbBrhLBBfVoNbfm4i8u9UI7hK9NQA/7VF/a1LcAzhuyX+LI0gyBYA9wM0CSDIjWZE8a+Q6AAlJIdmRrElmmyEUkdELQKFgRH9LXRwtgFrtFYBefRoAiW+8pdcYsATA2QDLB/YeQDbwb4x/shlAAJmNyAj4VPV3bGKdr18UgApOQi/RNDeBknD9s6gANU0ucl2gLlsATx5bYgCWsQE2c2av9nTCHjXNrq7cwG3AMfPV3wjANhbAOwDPhnXqCf39MUXg2l5jEnVu7k8r9Lcw/6Pq9OTiMOlLAnbHnRKTZqzUXQPpywFcROQasDu5e/H41CqNZ5KleV6QbPW5k87E2Q9aL1Na+jiRfle7FxH5MgKuAnAVkQeSNYCKJAA8ALgH8FFErgrsJ4A3ku9dQMqp1IxJnrHV2r/3lYBd/fZ9AKoA4TeOZixJFz4C9ry4MhqcenyeLC+aCbUBOhM3KfcwN1FIR2bUjoBu55CyAsxGNNunSLmJcOL7UChdunTGpfFxETj70h4Aa+UwDej93wAHEWsUbG9S2GnUypDcTQAsbPo8Po1dE1MDpjF5zdRsM1fPpz75s8jbyzykNCStGr3M2ZPEPt5wfPvLY/+sv1//bMomUtItrbeJ1VkGFkcb3JMMVly/Qf2ddeLZiK0e9jvseLzh6u+7jvtNpe1e748ichl2OvyH+jspkOPajfuHWBHUrx1X85d/6XsIbOiPEaP36L529jzAXJTemwOo6c0Hh5/rz2YiUUtvuM9exdwxfgOHZY7f662mjQAAAABJRU5ErkJggg==\" alt=\"$\\phi_p$\" class=\"math-inline math\" style=\"vertical-align:-0.304em;height:1.041em\"> is sound and the best representation of <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\">\n(as for all sound path-conditions <img src=\"data:image/png;base64,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\" alt=\"$\\psi_p$\" class=\"math-inline math\" style=\"vertical-align:-0.304em;height:1.041em\">, we have that <img src=\"data:image/png;base64,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\" alt=\"$\\psi_p \\implies \\phi_p$\" class=\"math-inline math\" style=\"vertical-align:-0.304em;height:1.041em\">).\nIn practice, we attempt to compute sound under-approximations of <img src=\"data:image/png;base64,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\" alt=\"$\\phi_p$\" class=\"math-inline math\" style=\"vertical-align:-0.304em;height:1.041em\"> \nsuch as <img src=\"data:image/png;base64,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\" alt=\"$\\psi_p$\" class=\"math-inline math\" style=\"vertical-align:-0.304em;height:1.041em\">. However, we also find it necessary (and useful) to \ncompute unsound path-conditions.\n</p>\n<p class=\"indent\">A path-condition can be translated into the input\nlanguage of an ATP, such as&nbsp;<a href=\"http://z3.codeplex.org/\">Z3</a><span class=\"citations\" style=\"target-element:bibitem\">[<a href=\"#demourab08\" title=\"Leonardo&#160;Mendon&#231;a de&#160;Moura and Nikolaj Bj&#248;rner. \nZ3: an efficient SMT solver.\" class=\"bibref localref\" style=\"target-element:bibitem\"><span class=\"bibitem-label\">5</span></a>]</span>, which provides an answer\nof &#8220;unsatisfiable&#8221;, &#8220;satisfiable&#8221; or &#8220;unknown&#8221;, due to theoretical or practical\nlimitations in automatically deciding satisfiability of various logics.\nIn the case that the ATP is able to prove &#8220;satisfiable&#8221; we can query it for \nsatisfying model in order to generate test inputs. A path-condition \nfor <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> can be thought of as function from a\nprogram&#39;s primary inputs to a boolean output representing whether\nor not <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> is executed under a given input. Thus, we are asking\nthe ATP to invert a function when we ask it to decide\nthe satisfiability/unsatisfiability of a path-condition\n</p>\n<p class=\"indent\">The static translation of a path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> through a program <img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> into \nthe most precise path-condition <img src=\"data:image/png;base64,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\" alt=\"$\\phi_p$\" class=\"math-inline math\" style=\"vertical-align:-0.304em;height:1.041em\"> is not a simple task, as \nprogramming languages and their semantics are very complex.\nCompletely characterizing the set of inputs <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAcCAYAAABh2p9gAAAA8UlEQVRIx62W4Q2DIBBGH6QDmI7gCJ2hjtB0g47gEN2kcQSZxRFaN7j+KD+IoXgHkpAIOZ8fT4wgIvzrwASIoY8u3phtzrkO6OOwjw9I2xOYgTewishCKWEm8SdJM+dqPMoW03bJ1JSrUwOB62YcWoFDcv3zdWDC8K/IG/z1ydTcBNT6swDvGn+1CUOp0Cv89Zv9NzcBLf60wEHrryZh2Cv2Rn+vJqDVnwaY+ltEZD0yYdBsWH/U/tMkNPvbA5r9WRIG5TefB9b6KyWs8lcC3jbf76oFnuISL8A5LnPYJOyccw9giT/03Rc0Go8bU+kw8AVaaLebPZP9FwAAAABJRU5ErkJggg==\" alt=\"$I$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> that follow\npath <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> means providing a symbolic interpretation of \nevery operation in the language so that the\nATP can reason about it. For example, consider a method call in Python. \nPython&#39;s algorithm for method resolution order (see&nbsp;<a href=\"https://www.python.org/download/releases/2.3/mro/\">MRO</a>)\ndepends on the inheritance hierarchy of the program, a directed, \nacyclic graph that can evolve during program execution. Symbolically\nencoding Python&#39;s method resolution order is possible but non-trivial.\nThere are other reasons it is hard or undesirable to symbolically \nexecute various operations, as will detail later.\n</p><h3 id=\"sec-dynamic-symbolic-execution\"   style=\"bookmark:1.1.&#8194;Dynamic symbolic execution\"><span class=\"heading-before\"><span class=\"heading-label\">1.1</span>.&#8194;</span>Dynamic symbolic execution</h3>\n<p><em>Dynamic</em> symbolic execution (DSE) is a form of path-based\nsymbolic execution based on two insights. First, the approach \nstarts by executing program <img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> on\nsome input <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAwAAAAcCAYAAABVo158AAABDUlEQVQ4y92U23GDQAxFj7aCrYES4hYowZMSSAduIeMSNiXEJVAD6cCkg5AOlB+tLQSG5Dea0cCuXlfSBVSVLQU6oACNqrLnXAA1vagqiW15du9fwG6Fs2XvgayqiBl+LYk/yr8LEJFWRAYRmUTkKiKnRUSgwHCjwH3L59luzPgETHU5dneygGktYADasOVL5VG4pwWuK7SopJvZEvBieH3znTuW2PfRY3cQa4W86CE4N865X9g3KK1xEI8CprVx+qZ9s0cg2/FtlRshe+/gNPFnMINkmavzEJxzXWB68MG/BiDdDeIeFdwgmlhhDM86iAK8q+oYK/iFZdMS+0luWiNwAD6AT6PHt6oefMUfyjONtCS7Ic0AAAAASUVORK5CYII=\" alt=\"$i$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">, seeding the symbolic execution process\nwith a feasible path&nbsp;<span class=\"citations\" style=\"target-element:bibitem\">[<a href=\"#gupta00\" title=\"Neelam Gupta, Aditya&#160;P. Mathur, and Mary&#160;Lou Soffa. \nGenerating test data for branch coverage.\" class=\"bibref localref\" style=\"target-element:bibitem\"><span class=\"bibitem-label\">10</span></a>, <a href=\"#korel90\" title=\"Bogdan Korel. \nAutomated software test data generation.\" class=\"bibref localref\" style=\"target-element:bibitem\"><span class=\"bibitem-label\">12</span></a>, <a href=\"#korel92\" title=\"Bogdan Korel. \nDynamic method of software test data generation.\" class=\"bibref localref\" style=\"target-element:bibitem\"><span class=\"bibitem-label\">13</span></a>]</span>. \nSecond,  DSE\nuses concrete values from the execution <img src=\"data:image/png;base64,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\" alt=\"$P(i)$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\"> in place of symbolic expressions \nwhenever symbolic reasoning is not possible or desired&nbsp;<span class=\"citations\" style=\"target-element:bibitem\">[<a href=\"#cadare05\" title=\"Cristian Cadar and Dawson&#160;R. Engler. \nExecution generated test cases: How to make systems code crash itself. \nIn Proceedings of 12th International SPIN Workshop, pages 2&#8211;23, 2005.\" class=\"bibref localref\" style=\"target-element:bibitem\"><span class=\"bibitem-label\">1</span></a>, <a href=\"#godefroidks05\" title=\"Patrice Godefroid, Nils Klarlund, and Koushik Sen. \nDART: directed automated random testing.\" class=\"bibref localref\" style=\"target-element:bibitem\"><span class=\"bibitem-label\">8</span></a>]</span>.\nThe major benefit of DSE is to\nsimplify the construction of a symbolic execution tool by\nleveraging the concrete execution behavior (given by\nactually running the program).\nAs DSE combines both \nconcrete and symbolic reasoning, it also has been called &#8220;concolic&#8221; \nexecution&nbsp;<span class=\"citations\" style=\"target-element:bibitem\">[<a href=\"#senacav06\" title=\"Koushik Sen and Gul Agha. \nCUTE and jcute: Concolic unit testing and explicit path model-checking tools. \nIn Proceedings of 18th Computer Aided Verification Conference, pages 419&#8211;423, 2006.\" class=\"bibref localref\" style=\"target-element:bibitem\"><span class=\"bibitem-label\">14</span></a>]</span>.\n</p>\n<figure id=\"fig-dse\" class=\"figure align-center\"   >\n<pre class=\"para-block pre-fenced pre-fenced3 language-python lang-python python highlighted\"  data-line=\"1\" data-line-code=\"2\"><code data-line=\"2\"><span class='token white python'>  </span><span class='token identifier python'>i</span><span class='token white python'> </span><span class='token keyword python'>=</span><span class='token white python'> </span><span class='token identifier python'>an</span><span class='token white python'> </span><span class='token identifier python'>input</span><span class='token white python'> </span><span class='token identifier python'>to</span><span class='token white python'> </span><span class='token identifier python'>program</span><span class='token white python'> </span><span class='token constructor identifier python'>P</span><br><span class='token white python'>  </span><span class='token keyword python'>while</span><span class='token white python'> </span><span class='token identifier python'>defined</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>i</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token keyword python bracket-open'>:</span><br><span class='token white python'>     </span><span class='token identifier python'>p</span><span class='token white python'> </span><span class='token keyword python'>=</span><span class='token white python'> </span><span class='token identifier python'>path</span><span class='token white python'> </span><span class='token identifier python'>covered</span><span class='token white python'> </span><span class='token identifier python'>by</span><span class='token white python'> </span><span class='token identifier python'>execution</span><span class='token white python'> </span><span class='token constructor identifier python'>P</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>i</span><span class='token delimiter parenthesis python bracket-close'>)</span><br><span class='token white python'>     </span><span class='token identifier python'>cond</span><span class='token white python'> </span><span class='token keyword python'>=</span><span class='token white python'> </span><span class='token identifier python'>pathCondition</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>p</span><span class='token delimiter parenthesis python bracket-close'>)</span><br><span class='token white python'>     </span><span class='token identifier python'>s</span><span class='token white python'> </span><span class='token keyword python'>=</span><span class='token white python'> </span><span class='token constructor identifier python'>ATP</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token namespace identifier python'>Not</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>cond</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token delimiter parenthesis python bracket-close'>)</span><br><span class='token white python'>     </span><span class='token identifier python'>i</span><span class='token white python'> </span><span class='token keyword python'>=</span><span class='token white python'> </span><span class='token identifier python'>s</span><span class='token delimiter python'>.</span><span class='token identifier python'>model</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token delimiter parenthesis python bracket-close'>)</span></code></pre>\n<hr  class=\"figureline madoko\" data-line=\"9\">\n\n<div data-line=\"10\"><span data-line=\"10\"></span><fig-caption class=\"figure-caption\"><span class=\"caption-before\"><strong>Figure&#160;<span class=\"figure-label\">1</span>.</strong> </span>Pseudo-code for dynamic symbolic execution</fig-caption><span data-line=\"10\"></span>\n</div></figure>\n<p class=\"indent\">The pseudo-code of Figure&nbsp;<a href=\"#fig-dse\" title=\"Pseudo-code for dynamic symbolic execution\" class=\"localref\" style=\"target-element:figure\"><span class=\"figure-label\">1</span></a> shows the high level process\nof DSE. The variable <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>i</span></code> represents an input\nto program <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token constructor identifier python'>P</span></code>. Execution of program <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token constructor identifier python'>P</span></code> on the input <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>i</span></code>\ntraces  a path <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>p</span></code>, from which\na logical formula <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>pathCondition</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>p</span><span class='token delimiter parenthesis python bracket-close'>)</span></code> is constructed.\nFinally, the ATP is called with the negation of the path-condition\nto find a new input (that hopefully will cover a new path).  This\npseudo-code ellides a number of details that we will deal with later. \n</p>\n<figure id=\"fig-easy-dse\" class=\"figure align-center\"   >\n<pre class=\"para-block pre-fenced pre-fenced3 language-python lang-python python highlighted\"  data-line=\"1\" data-line-code=\"2\"><code data-line=\"2\"><span class='token keyword python'>def</span><span class='token white python'> </span><span class='token identifier python'>max2</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>s</span><span class='token delimiter python'>,</span><span class='token identifier python'>t</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token keyword python bracket-open'>:</span><br><span class='token white python'>    </span><span class='token keyword python'>if</span><span class='token white python'> </span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>s</span><span class='token white python'> </span><span class='token operator python'>&lt;</span><span class='token white python'> </span><span class='token identifier python'>t</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token keyword python bracket-open'>:</span><br><span class='token white python'>        </span><span class='token keyword python'>return</span><span class='token white python'> </span><span class='token identifier python'>t</span><br><span class='token white python'>    </span><span class='token keyword python'>else</span><span class='token keyword python bracket-open'>:</span><br><span class='token white python'>        </span><span class='token keyword python'>return</span><span class='token white python'> </span><span class='token identifier python'>s</span><br><br><span class='token keyword python'>def</span><span class='token white python'> </span><span class='token identifier python'>max4</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token delimiter python'>,</span><span class='token identifier python'>b</span><span class='token delimiter python'>,</span><span class='token identifier python'>c</span><span class='token delimiter python'>,</span><span class='token identifier python'>d</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token keyword python bracket-open'>:</span><br><span class='token white python'>    </span><span class='token keyword python'>return</span><span class='token white python'> </span><span class='token identifier python'>max2</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>max2</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token delimiter python'>,</span><span class='token identifier python'>b</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token delimiter python'>,</span><span class='token identifier python'>max2</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>c</span><span class='token delimiter python'>,</span><span class='token identifier python'>d</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token delimiter parenthesis python bracket-close'>)</span></code></pre>\n<hr  class=\"figureline madoko\" data-line=\"11\">\n\n<div data-line=\"12\"><span data-line=\"12\"></span><fig-caption class=\"figure-caption\"><span class=\"caption-before\"><strong>Figure&#160;<span class=\"figure-label\">2</span>.</strong> </span>Easy example: computing the maximum of four numbers in Python.</fig-caption><span data-line=\"12\"></span>\n</div></figure>\n<p class=\"indent para-continue\">Consider the  Python function <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>max4</span></code> in Figure&nbsp;<a href=\"#fig-easy-dse\" title=\"Easy example: computing the maximum of four numbers in Python.\" class=\"localref\" style=\"target-element:figure\"><span class=\"figure-label\">2</span></a>,\nwhich computes the maximum of four numbers via three calls to\nthe function <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>max2</span></code>. Suppose we execute <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>max4</span></code> with values\nof zero for all four arguments.  In this case, the \nexecution path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> contains three comparisons (in the order <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token white python'> </span><span class='token operator python'>&lt;</span><span class='token white python'> </span><span class='token identifier python'>b</span><span class='token delimiter parenthesis python bracket-close'>)</span></code>, \n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>c</span><span class='token white python'> </span><span class='token operator python'>&lt;</span><span class='token white python'> </span><span class='token identifier python'>d</span><span class='token delimiter parenthesis python bracket-close'>)</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token white python'> </span><span class='token operator python'>&lt;</span><span class='token white python'> </span><span class='token identifier python'>c</span><span class='token delimiter parenthesis python bracket-close'>)</span></code>), all of which evaluate false.\nThus, the path-condition for path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> is <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token keyword python'>not</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token operator python'>&lt;</span><span class='token identifier python'>b</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token keyword python'>not</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>c</span><span class='token operator python'>&lt;</span><span class='token identifier python'>d</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token keyword python'>not</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token white python'> </span><span class='token operator python'>&lt;</span><span class='token white python'> </span><span class='token identifier python'>c</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token delimiter parenthesis python bracket-close'>)</span></code>.\nNegating this condition yields <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token operator python'>&lt;</span><span class='token identifier python'>b</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token white python'> </span><span class='token keyword python'>or</span><span class='token white python'> </span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>c</span><span class='token operator python'>&lt;</span><span class='token identifier python'>d</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token white python'> </span><span class='token keyword python'>or</span><span class='token white python'> </span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token operator python'>&lt;</span><span class='token identifier python'>c</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token delimiter parenthesis python bracket-close'>)</span></code>.\nTaking the execution ordering of the three comparisons into account, we\nderive three expressions from the negated path-condition to generate\nnew inputs that will explore execution prefixes of path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> of increasing length:\n</p>\n<ul class=\"list-star compact\">\n<li><em>length 0</em>: <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token operator python'>&lt;</span><span class='token identifier python'>b</span><span class='token delimiter parenthesis python bracket-close'>)</span></code>\n</li>\n<li><em>length 1</em>: <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token keyword python'>not</span><span class='token white python'> </span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token operator python'>&lt;</span><span class='token identifier python'>b</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>c</span><span class='token operator python'>&lt;</span><span class='token identifier python'>d</span><span class='token delimiter parenthesis python bracket-close'>)</span></code>\n</li>\n<li><em>length 2</em>: <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token keyword python'>not</span><span class='token white python'> </span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token operator python'>&lt;</span><span class='token identifier python'>b</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token keyword python'>not</span><span class='token white python'> </span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>c</span><span class='token operator python'>&lt;</span><span class='token identifier python'>d</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token operator python'>&lt;</span><span class='token identifier python'>c</span><span class='token delimiter parenthesis python bracket-close'>)</span></code>\n</li></ul>\n\n<p>The purpose of taking execution order into account should be clear, as the\ncomparison <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token operator python'>&lt;</span><span class='token identifier python'>c</span><span class='token delimiter parenthesis python bracket-close'>)</span></code> only executes in the case where <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token keyword python'>not</span><span class='token white python'> </span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token operator python'>&lt;</span><span class='token identifier python'>b</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token keyword python'>not</span><span class='token white python'> </span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>c</span><span class='token operator python'>&lt;</span><span class='token identifier python'>d</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token delimiter parenthesis python bracket-close'>)</span></code>\nholds.\n</p><!--\n\nassertions, query\n\n-->\n\n\n<p class=\"para-continue\">Integer solutions to the above three systems of constraints are:\n</p>\n<ul class=\"list-star compact\">\n<li><code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>a</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token number python'>0</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token identifier python'>b</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token number python'>2</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token identifier python'>c</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token number python'>0</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token identifier python'>d</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token number python'>0</span></code>\n</li>\n<li><code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>a</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token number python'>0</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token identifier python'>b</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token number python'>0</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token identifier python'>c</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token number python'>0</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token identifier python'>d</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token number python'>3</span></code>\n</li>\n<li><code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>a</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token number python'>0</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token identifier python'>b</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token number python'>0</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token identifier python'>c</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token number python'>2</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token identifier python'>d</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token number python'>0</span></code>\n</li></ul>\n\n<p>In the three cases above, we sought solutions that kept as many of\nthe variables as possible equal to the original input (in which \nall variables are equal to 0). Execution the <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>max4</span></code> function on the input\ncorresponding to the first solution produces the path-condition\n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>a</span><span class='token operator python'>&lt;</span><span class='token identifier python'>b</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token keyword python'>not</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>c</span><span class='token operator python'>&lt;</span><span class='token identifier python'>d</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token keyword python'>not</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>b</span><span class='token white python'> </span><span class='token operator python'>&lt;</span><span class='token white python'> </span><span class='token identifier python'>c</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token delimiter parenthesis python bracket-close'>)</span></code>, from which we can produce more\ninputs.   For this (loop-free function), there are a finite number\nof path-conditions. We leave it as an exercise to the reader to \nenumerate them all. \n</p><h3 id=\"sec-leveraging-concrete-values-in-dse\"   style=\"bookmark:1.2.&#8194;Leveraging concrete values in DSE\"><span class=\"heading-before\"><span class=\"heading-label\">1.2</span>.&#8194;</span>Leveraging concrete values in DSE</h3>\n<p>We now consider several situations where we can make use of concrete\nvalues in DSE. In the realm of (unbounded-precision) integer arithmetic \n(e.g., bignum integer arithmetic, as in Python 3.0 onwards),\nit is easy  to come up with  tiny programs that will be <em>very difficult</em>,\nif not <em>impossible</em>, \nfor any symbolic execution tool to deal with, such as the function <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>fermat3</span></code>\nin Figure&nbsp;<a href=\"#fig-fermat3\" title=\"Hard example for symbolic execution\" class=\"localref\" style=\"target-element:figure\"><span class=\"figure-label\">3</span></a>.  \n</p>\n<figure id=\"fig-fermat3\" class=\"figure align-center\"   >\n<pre class=\"para-block pre-fenced pre-fenced3 language-python lang-python python highlighted\"  data-line=\"1\" data-line-code=\"2\"><code data-line=\"2\"><span class='token keyword python'>def</span><span class='token white python'> </span><span class='token identifier python'>fermat3</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>x</span><span class='token delimiter python'>,</span><span class='token identifier python'>y</span><span class='token delimiter python'>,</span><span class='token identifier python'>z</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token keyword python bracket-open'>:</span><br><span class='token white python'>   </span><span class='token keyword python'>if</span><span class='token white python'> </span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>x</span><span class='token white python'> </span><span class='token operator python'>&gt;</span><span class='token white python'> </span><span class='token number python'>0</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token identifier python'>y</span><span class='token white python'> </span><span class='token operator python'>&gt;</span><span class='token white python'> </span><span class='token number python'>0</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token identifier python'>z</span><span class='token white python'> </span><span class='token operator python'>&gt;</span><span class='token white python'> </span><span class='token number python'>0</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token keyword python bracket-open'>:</span><br><span class='token white python'>      </span><span class='token keyword python'>if</span><span class='token white python'> </span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>x</span><span class='token operator python'>*</span><span class='token identifier python'>x</span><span class='token operator python'>*</span><span class='token identifier python'>x</span><span class='token white python'> </span><span class='token operator python'>+</span><span class='token white python'> </span><span class='token identifier python'>y</span><span class='token operator python'>*</span><span class='token identifier python'>y</span><span class='token operator python'>*</span><span class='token identifier python'>y</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token identifier python'>z</span><span class='token operator python'>*</span><span class='token identifier python'>z</span><span class='token operator python'>*</span><span class='token identifier python'>z</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token keyword python bracket-open'>:</span><br><span class='token white python'>          </span><span class='token keyword python'>return</span><span class='token white python'> </span><span class='token string delim python bracket-open'>&quot;</span><span class='token string python'>Fermat and Wiles were wrong!?!</span><span class='token string delim python bracket-close'>&quot;</span><br><span class='token white python'>   </span><span class='token keyword python'>return</span><span class='token white python'> </span><span class='token number python'>0</span></code></pre>\n<hr  class=\"figureline madoko\" data-line=\"8\">\n\n<div data-line=\"9\"><span data-line=\"9\"></span><fig-caption class=\"figure-caption\"><span class=\"caption-before\"><strong>Figure&#160;<span class=\"figure-label\">3</span>.</strong> </span>Hard example for symbolic execution</fig-caption><span data-line=\"9\"></span>\n</div></figure>\n<p class=\"indent\">Fermat&#39;s Last Theorem, proved\nby Andrew Wiles in the late 20th century, states that no \nthree positive integers <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABUAAAASCAYAAAC0EpUuAAABHklEQVQ4y6WUYY3DMAyFP08HoBh6DIohY7CjUAjlMAjDUAg5Cj0GHYVC8P04Z/KsrOl0liLV8fPTSxo/VJW4gARkYLO1AGPA9MAMrBFTI7wZILm9CVBgtnwwohT6FJhqhPmF+tWaRiMcKoQKLPHIa43Q6tk13kJtc7U5FtIO6eoa+1CbrD8DXdm87Kk0TCHcxakqJ/7iDnzxIkRkcOk3jfgAUNWfBi6579wiPXEszu8oFbuvfZBIAd1V9fPfSkUkvaPy6PHPR+5TRPrHD209DxvZ8py6Bu5SrrMzY9iAa8U0mu+zeIHLn+ZWA/jqZ/qIykLqZ3oOKjczEPVKAuHs+wrpGE2imEvxAneaa8AskfDhpzb7izPlXDGNGqZqQL/kpe0Z3roFswAAAABJRU5ErkJggg==\" alt=\"$x$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">, <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAaCAYAAAC3g3x9AAABcElEQVRIx5WV4W2EMAyFv9wEdAVGQB0h3QB1hOsG7MAI1xU6At2gohvQFdIN3D/OyWdCQiNFMth5+L28BESEPIEIrEACNmCy+dIEOl2zAZ1NTPqy1+cbIMDcAJy1ToDBdpaAzn1AgNQAXDOgiGBfRlf4YQsrgLm7VZ+JwFYoTFq4VcCiAZxFhAvwpnrdRwhhVLHxOTdeTLzkYLTaebo+V9PvrmHBBg+6VOyyq7sUaLyauEb32cSf96hGo0H3ZupikbKjsTTsspVs5SlHE38fcQ0hdEC/ows7wN7EXxX9YskuJUA7fk7676FDr8vVaNgfaNebmt2xrG3K9cB7W23jSh2M+ZYBBgN0VbDFAE5NQEPrQwGSzpsC22M5+LUhXzlnRwghL/gVkSefv/wTzNrlvVhkKK6tK9/ol46OpS88vFDNZgkwHn7Un8tSh7oZ6dRPy3S4nvDe3Pyt6qJBOxgd0GRsE1tgDz503svCL6Wua/MPz3MNm1mBZ2QAAAAASUVORK5CYII=\" alt=\"$y$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\">, and <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAASCAYAAAC5DOVpAAAA6klEQVQ4y52UURWDMAxFLzsTgAYkdBaQwCwgYR6QgAYkoKESmAUkZD+Bk7GmhfWc/JD08prmFRHBC2AEVkCAKVM3A8FL1sCiEBvR+eEsIniwaACLUSeaa7Tupfk6CQN63dQl1PZ6JAtv9hrn/F2hl40qCl/fc5uugC7DgOCBLsGAzjb/b5jeWtxuzYs7hVVV1QQgIo9S7a0AmoG3iDw5szIOiKUR2VqQm7PNSu0JUA8sSZhR1J6ctxXoPdjmyVXVjSmwsdySHA21kTVxmzB4PLwiv3YCBud5CQegGOUhUb83vM70JprnaPBqPzmXkP+U5dLxAAAAAElFTkSuQmCC\" alt=\"$z$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> can satisfy the equation\n<img src=\"data:image/png;base64,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\" alt=\"$x^n + y^n = z^n$\" class=\"math-inline math\" style=\"vertical-align:-0.212em;height:0.937em\"> for any integer value of <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAASCAYAAACw50UTAAABWUlEQVQ4y6WV4VHDMAyFP+UygI8RygYBNigb+GCDskE5RsgIYQOubJCO0IyQbsCFDcQf+1Bcu2nBvyTr6Z30IjmoKqoK0AADMAEjsI0xg1kBXcCNATsAPsWqKjHJBeA6+B2gQGeIu0DozZ0L5DNsSt7ZSgEfEjRU2+eSTccR2+TIJ8AVyCegzRGb6iN2JmUlIh44qOo3v+fR2AdVfaV8box9awMV8AzskoS1sV84fxpjDzZQAx/APl6IiAs6A+xV9bhA/mDsOTajoS9pWNB8jPg0VmUqsXrvz5UsIivT5ecJ4JpKMtit6dJn57wwVv1/JFFV6qSRJ2PvFiRpjCTvScwB9/Vf9U5GNC3kDfhK25ziVl4gScSOhdiqSr68u2JKXG5KRGQT96MubGV/jlxVjyIS3THRugXuZqMYdIuT4i6QJT7LbfDXJ0+yAW+CVotbaXJa89Po0yf3ByaK/TzOgVZHAAAAAElFTkSuQmCC\" alt=\"$n$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> greater than two.\nThe function <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>fermat3</span></code> encodes this statement for <img src=\"data:image/png;base64,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\" alt=\"$n=3$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\">.   It\nis not reasonable to have a computer waste time trying to find\na solution that would cause <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>fermat3</span></code> to print the string \n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token string delim python bracket-open'>&quot;</span><span class='token string python'>Fermat and Wiles were wrong!?!</span><span class='token string delim python bracket-close'>&quot;</span></code>.  In cases of complex (non-linear)\narithmetic operations,\nsuch as <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>x</span><span class='token operator python'>*</span><span class='token identifier python'>x</span><span class='token operator python'>*</span><span class='token identifier python'>x</span></code>, we might choose to treat the operation concretely.\n</p>\n<p class=\"indent\">There are a number of ways to deal with the above issue: one is \nto recognize all non-linear terms in a symbolic expression and replace\nthem with their concrete counterparts during execution. For the <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>fermat3</span></code> \nexample, this would mean that during DSE the symbolic expression \n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>x</span><span class='token operator python'>*</span><span class='token identifier python'>x</span><span class='token operator python'>*</span><span class='token identifier python'>x</span><span class='token white python'> </span><span class='token operator python'>+</span><span class='token white python'> </span><span class='token identifier python'>y</span><span class='token operator python'>*</span><span class='token identifier python'>y</span><span class='token operator python'>*</span><span class='token identifier python'>y</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token identifier python'>z</span><span class='token operator python'>*</span><span class='token identifier python'>z</span><span class='token operator python'>*</span><span class='token identifier python'>z</span><span class='token delimiter parenthesis python bracket-close'>)</span></code> would be reduced to the constant <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>False</span></code>\nby evaluation on the concrete values of variables <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>x</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>y</span></code> and <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>z</span></code>.\n</p>\n<p class=\"indent\">Besides difficult operations (such as non-linear arithmetic),\nother examples of code that we might treat\nconcretely instead of symbolically include\nfunctions that are hard to invert, such as cryptographic hash functions,\nor low-level functions that we do not wish to test (such as \noperating system functions).  Consider\nthe code in Figure&nbsp;<a href=\"#fig-hash\" title=\"Another hard example for symbolic execution\" class=\"localref\" style=\"target-element:figure\"><span class=\"figure-label\">4</span></a>, which applies the function\n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>unknown</span></code> to argument <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>x</span></code> and compares it to argument <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>y</span></code>.\nBy using the name <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>unknown</span></code> we simply mean to say that we \nwish treat this function as a black box, with no knowledge\nof how it operates internally. \n</p>\n<figure id=\"fig-hash\" class=\"figure align-center\"   >\n<pre class=\"para-block pre-fenced pre-fenced3 language-python lang-python python highlighted\"  data-line=\"1\" data-line-code=\"2\"><code data-line=\"2\"><span class='token keyword python'>def</span><span class='token white python'> </span><span class='token identifier python'>dart</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>x</span><span class='token delimiter python'>,</span><span class='token identifier python'>y</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token keyword python bracket-open'>:</span><br><span class='token white python'>  </span><span class='token keyword python'>if</span><span class='token white python'> </span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>unknown</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>x</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token identifier python'>y</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token keyword python bracket-open'>:</span><br><span class='token white python'>     </span><span class='token keyword python'>return</span><span class='token white python'> </span><span class='token number python'>1</span><br><span class='token white python'>  </span><span class='token keyword python'>return</span><span class='token white python'> </span><span class='token number python'>0</span></code></pre>\n<hr  class=\"figureline madoko\" data-line=\"7\">\n\n<div data-line=\"8\"><span data-line=\"8\"></span><fig-caption class=\"figure-caption\"><span class=\"caption-before\"><strong>Figure&#160;<span class=\"figure-label\">4</span>.</strong> </span>Another hard example for symbolic execution</fig-caption><span data-line=\"8\"></span>\n</div></figure>\n<p class=\"indent\">In such a case, we can use DSE to execute the function <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>unknown</span></code>\nfunction on a specific input (say <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token number python'>5013</span></code>) and observe its output\n(say 42). That is, rather than try to execute <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>unknown</span></code> symbolically\nand invoke an ATP to invert the function&#39;s path-condition, we\nsimply treat the call to <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>unknown</span></code> concretely, substituting\nits return value (in this case <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token number python'>42</span></code>) for the specialized expression \n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>unknown</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token number python'>5013</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token identifier python'>y</span></code> to get the predicate <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token number python'>42</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token identifier python'>y</span><span class='token delimiter parenthesis python bracket-close'>)</span></code>. \n</p>\n<p class=\"indent\">Adding the constraint <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>x</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token number python'>5013</span><span class='token delimiter parenthesis python bracket-close'>)</span></code> yields the sound but\nrather specific path-condition \n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>x</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token number python'>5013</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token white python'> </span><span class='token keyword python'>and</span><span class='token white python'> </span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token number python'>42</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token identifier python'>y</span><span class='token delimiter parenthesis python bracket-close'>)</span></code>.\nNote that the path-condition <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token number python'>42</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token identifier python'>y</span><span class='token delimiter parenthesis python bracket-close'>)</span></code> is not sound, as it admits\nany value for the variable <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>x</span></code>, which likely includes many values\nfor which <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>unknown</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>x</span><span class='token delimiter parenthesis python bracket-close'>)</span><span class='token white python'> </span><span class='token operator python'>==</span><span class='token white python'> </span><span class='token identifier python'>y</span><span class='token delimiter parenthesis python bracket-close'>)</span></code> is false.\n</p><h3 id=\"sec-overview\"   style=\"bookmark:1.3.&#8194;Overview\"><span class=\"heading-before\"><span class=\"heading-label\">1.3</span>.&#8194;</span>Overview</h3>\n<p class=\"para-continue\">This introduction elides many important \nissues that arise in implementing DSE for a real language, which we will \nfocus on in the remainder of the paper. These include how to:\n</p>\n<ul class=\"list-star compact\">\n<li>Identify the code under test <img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> and the symbolic inputs to <img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">;\n</li>\n<li>Trace the control flow path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> taken by execution <img src=\"data:image/png;base64,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\" alt=\"$P(i)$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\">;\n</li>\n<li>Reinterpret program operations to compute symbolic expressions;\n</li>\n<li>Generate a path-condition from <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> and the symbolic expressions;\n</li>\n<li>Generate a new input <img src=\"data:image/png;base64,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\" alt=\"$i&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"> by negating (part of) the path-condition, translating\nthe path-condition to the input language of an ATP, invoking the ATP, and\nlifting a satisfying model (if any) back up to the source level;\n</li>\n<li>Guide the search to expose new paths.\n</li></ul>\n\n<p>The rest of this paper is organized as follows. Section&nbsp;<a href=\"#sec-semantics\" title=\"2.&#8194;Instrumented Types\" class=\"localref\" style=\"target-element:h1\"><span class=\"heading-label\">2</span></a> \ndescribes an instrumented typing discipline where we lift each type (representing \na set of concrete values) to a symbolic type (representing\na set of pairs of concrete and symbolic values).\nSection&nbsp;<a href=\"#sec-sp2dse\" title=\"3.&#8194;From Strongest Postconditions to DSE\" class=\"localref\" style=\"target-element:h1\"><span class=\"heading-label\">3</span></a> shows how strongest postconditions defines a symbolic\nsemantics for a small programming language and how strongest postconditions\ncan be refined to model DSE.\nSection&nbsp;<a href=\"#sec-impl\" title=\"4.&#8194;Architecture of PyExZ3\" class=\"localref\" style=\"target-element:h1\"><span class=\"heading-label\">4</span></a> describes an implementation of DSE for the Python language\nin the Python language that follows the instrumented semantics pattern closely\n(full implementation and tests available at&nbsp;<a href=\"http://www.github.com/thomasjball/pyexz3\">PyExZ3</a>, tagged &#8220;v1.0&#8221;).\nSection&nbsp;<a href=\"#sec-int2z3\" title=\"5.&#8194;From Python Integers to Z3 Arithmetic\" class=\"localref\" style=\"target-element:h1\"><span class=\"heading-label\">5</span></a> describes the symbolic encoding of Python integer \noperations using two decision procedures of Z3: linear arithmetic with\nuninterpreted functions in place of non-linear operations;\nfixed-width bitvectors with precise encodings of most operations.\nSection&nbsp;<a href=\"#sec-extensions\" title=\"6.&#8194;Extensions\" class=\"localref\" style=\"target-element:h1\"><span class=\"heading-label\">6</span></a> offers a number of ideas for projects\nto extend the capabilities of&nbsp;<a href=\"http://www.github.com/thomasjball/pyexz3\">PyExZ3</a>.\n</p><h2 id=\"sec-semantics\"   style=\"bookmark:2.&#8194;Instrumented Types\"><span class=\"heading-before\"><span class=\"heading-label\">2</span>.&#8194;</span>Instrumented Types</h2>\n<p>We are given a universe of classes/types <img src=\"data:image/png;base64,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\" alt=\"$U$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">; a type <img src=\"data:image/png;base64,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\" alt=\"$T \\in U$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.807em\"> carries\nalong a  set of operations that apply to values of type <img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\">,\nwhere an operation <img src=\"data:image/png;base64,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\" alt=\"$o \\in T$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.807em\">  takes an argument list of typed \nvalues as input (the first being of type <img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\">) and produces a single \ntyped value as output. Nullary (static) operations of type <img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> can be \nused to create values of type <img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> (such as constants, objects, etc.)\n</p>\n<p class=\"indent para-continue\">A program <img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> has typed input variables\n<img src=\"data:image/png;base64,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\" alt=\"$v_1 : T_1 \\ldots v_k : T_k$\" class=\"math-inline math\" style=\"vertical-align:-0.164em;height:0.911em\"> and a body from the language of statements <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABkAAAAcCAYAAACUJBTQAAABnElEQVRIx62W7XHDIAyGXzyBL93A3cB33cAjcB3B2SA7eAS3E/TcDcgGbbqBvUHP2UD9EeFTKF+m1Z0OEgMPEkIIRIRcBVADOAEwAFYAxO2F/6+d8RrAuGfxkRedecEOQMNtzyACoHlOx79POYBW7LpPjNXbwreWALQ5gG1wptVazFmJCCkXrdIFO85u5nkTEaFCWF4Z9EVE79gnZ24NAD9EKVWz2QDwhv1ylbCQJc+ivxRAvgEsRLTEII+i/1QAeRAui56Jlb4A8sH36iYZYUh80Zo9ESZV8aK+g189O1zYDQbAmYiuWXZFYr13rPHpBcCQuqipS5UDsjoWQTyZNwWaiiCBfBaD6j9DIk+AVfOvEAGbBGR2v1cibFulVIcyGUX/4H6UN/7IL12JfMZyXeUkxaUQcvCk+XuIUqrhQyyVLuC6O0vs21HqriO3Lza9/0orIuZNQWT1ooqpg5fRifN+B0CL2quJjNvqo1XsagjtSkwchAVNYuw2uBVpw1YbhlOI5s1oPlhbxQyZFsMA6AKumATQWmt8JWlMfwDWk6gpP8EGnAAAAABJRU5ErkJggg==\" alt=\"$S$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">:\n</p>\n<div class=\"mathpre para-block input-mathpre\"><img src=\"data:image/png;base64,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\" alt=\"\\[\\begin{mdMathprearray}%\n\\mathid{S}\\mdMathspace{1}\\rightarrow   &amp;\\mdMathspace{1}\\mathid{v}\\mdMathspace{1}:=\\mdMathspace{1}\\mathid{E}\\mdMathbr{}\n\\mdMathindent{3}|\\mdMathspace{1}&amp;\\mdMathspace{1}\\mathkw{skip}\\mdMathspace{1}\\mdMathbr{}\n\\mdMathindent{3}|\\mdMathspace{1}&amp;\\mdMathspace{1}\\mathid{S}_1\\mdMathspace{1};\\mdMathspace{1}\\mathid{S}_2\\mdMathspace{1}\\mdMathbr{}\n\\mdMathindent{3}|\\mdMathspace{1}&amp;\\mdMathspace{1}\\mathkw{if}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{S}_1\\mdMathspace{1}\\mathkw{else}\\mdMathspace{1}\\mathid{S}_2\\mdMathspace{1}\\mathkw{end}\\mdMathbr{}\n\\mdMathindent{3}|\\mdMathspace{1}&amp;\\mdMathspace{1}\\mathkw{while}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{do}\\mdMathspace{1}\\mathid{S}\\mdMathspace{1}\\mathkw{end}\n\\end{mdMathprearray}\\]\" class=\"math-display math\" style=\"vertical-align:-0.000em;height:6.218em\"></div>\n<p>The language of expressions (<img src=\"data:image/png;base64,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\" alt=\"$E$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\">) is defined by the application of operations\nto values, where constants (nullary operations) and \nprogram variables form the leaves \nof the expression tree and non-nullary operators \nform the interior nodes of the tree.\nFor now, we will consider all values to be immutable.\nThat is, the only source of mutation in the language is the \nassignment statement.\n</p>\n<p class=\"indent para-continue\">To introduce symbolic execution into the picture,\nwe can imagine that a type <img src=\"data:image/png;base64,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\" alt=\"$T \\in U$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.807em\"> has\n(one or more) counterparts in a symbolic universe <img src=\"data:image/png;base64,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\" alt=\"$U&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\">. A type <img src=\"data:image/png;base64,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\" alt=\"$T&#39; \\in U&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.885em\">\nis a subtype of <img src=\"data:image/png;base64,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\" alt=\"$T \\in U$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.807em\"> with two purposes:\n</p>\n<ul class=\"list-star loose\">\n<li>\n<p>First, a value of type <img src=\"data:image/png;base64,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\" alt=\"$T&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"> represents a pair of values: \na concrete value <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA30lEQVQ4y52UURHCMAyGv3AI6CFhEtAwLZMwD5UwDZUwDUgoEigOwksHIUdhJXd52d98f65NhqryLYERSEAGSs0LMANBVflVnAEFIjAYLQBThYUWYKnFxRY7g/I0+HBgNYDQMFGTyYvJiGMDEBxktOJkhHXHZcfNyNLLry6a0AqZDaD0ACwkG0jshRxEJAADr1jpjIMDAFz/gZzsB1Xthvh3z92XCuejqt5F5F5hXSEiCbhttGi6CTs7SNtQ2mHLe57YbPfyNidVHNzqB6dNVc9+oj85TXVWivsJxdY6PAB/ne4jt7G24QAAAABJRU5ErkJggg==\" alt=\"$c$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> of (super)type <img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> and a symbolic expression <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA/0lEQVQ4y6WU/W2DQAzFfxdlALoCIzADIzAD2YAdOkJmYIRkBbIBHaFkg9c/+ogceiWgnGTpZPve2c8fSGJNgBbogRGYLBegfvisPO78YPS9CLbGti4LApTAAAjoVz5pAP0BMcD0CsC+lf3KqCwCwLSBq8/ZLyrPBhDQbiBbM7kxDQUpMmnWJni0PKpz5PeceD5DSmmh4gu4ASdJ1yeLfxpDFN0rPpZyMFYZcG/sPEkSKSUF3Yek+1sgktLeSA6BNID77lRSqo6Bh9INtwegB77n6tShOvWWiniyL8uO7TfOTOOWOGdXgfeEPALFomPbXLf+twoag01BBg9clYvuB0wmtUhHaPolAAAAAElFTkSuQmCC\" alt=\"$e$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">. \nA symbolic expression is a tree\nwhose leaves are either nullary operators (i.e., constants) of a type in <img src=\"data:image/png;base64,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\" alt=\"$U$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">\nor are Skolem constants representing the (symbolic) inputs (<img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIkAAAAYCAYAAADOHt4vAAADM0lEQVRo3u2a4VHbQBCFv2UoQKEEpwMDFdh0YEIHpgOnhIxLEFQQ7A4wFQS7A6kErA42P9ibXMTZEnAyGriduZEsyavVvKfdvXdCVXEDGANrYAsUwMw/Xx9ADiiQ77sujbePPmDiO59aMIPazeY7gpnYeTeGCdToBOkFJkcAIjIE5sBIVUuerbDtlLCVtd8DkkWzPmFyZNtb4FJVK+/cuW2z0B9VdQP89A5VCdqo1itMxsA2kLq2lrKKhpS4tusGLdLnEBinUtKqD+kEEytJa8tKzl/WVG6ugV+1VDfx2Jo3kGxlLC5bpM91jenJwtYlJivzXZq/qpatguXmN3BTO37l7S8bAiqAzT5yiEhuBEnWzjrDRFUrVV16/cuqsSdR1aXPJBHJLCUBbJoyBHAaupGIrEVEgQfgZB+Rkr0AshNMAm0GwH3bxtW3H95+3uKZxjtuNFJVUdVvqnoZ6LyTtbdYmDjSDbyZz+otJLn2GH3T0GcMgBNVXYXSWsI2ao/ybkwCWaRqkZWCJBm2ZZgFf5cw7NxiY3LxCn//k8RmILStVSbq5AnDzkW12JgE+xERyURkLiKFjTyUSc68/U1D8BPg0QScZN1ZVEysHGX1TGJkfAD+qOp3m0FNRWRyXPPhK3lPLRTBUY/euDFQtqmxh/IVyU9sTF70IyIytTI1UtXK4p7t6kl8Bp7sefgcuOlLFhGRwlJnYQ/84b4ixhQbE9eP3Nn/ZsCFqp56k41Hu+9SVZfskX7zHZLvDFi/QWpemN/7yBJ2feVz+9G+YsYUGxPP18xIPGu1ClzXN1wD5L8BppwugHNVPe1Rza5Ptcse+IoZUzRMav3I3MVmYt1edS/EtsyYVrg3gWdZffqOt6GTTGK+5+a74J3fUMTyFTOmWJjYzMfFNLYYXWZZsGOh75Arm52RJI1XY5DvWDVetC03yT6v7VqvyWvnXXmai0iWSPJ1RLmgPlKbZpe+sGalrEok+XpZpAysq1WBBnuKfa5wfEAWO3n5TEQGMUSvZK+yfes17tjAU1+v3IzpqENi5CKi9k1Jwb+l6cwEJrVxn/A7iD1ZppgHZrilI5GJgLfApTv/F4urg/voq4NjAAAAAElFTkSuQmCC\" alt=\"$v_1 \\ldots v_k$\" class=\"math-inline math\" style=\"vertical-align:-0.161em;height:0.624em\">)\nto the program <img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">, and whose interior nodes represent operations \nfrom types in <img src=\"data:image/png;base64,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\" alt=\"$U$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">. We refer to Skolem constants as &#8220;symbolic constants&#8221;\nfrom this point on. Note that symbolic expressions do not contain\nreferences to program variables.\n</p></li>\n<li>\n<p>Second, the type <img src=\"data:image/png;base64,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\" alt=\"$T&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"> redefines some of the operations <img src=\"data:image/png;base64,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\" alt=\"$o \\in T$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.807em\">,\nnamely those for which we wish to compute symbolic expressions.\nAn operation <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAG0AAAAiCAYAAABV9lfvAAADyElEQVRo3u1b0XHbMAx98HkAJZ2g6gbKdYIwGzjJBs4GznWCnrNBMkLsDexsEHsDeYNa3gD9KJiijGRRlCxLPeGOF50EiTTBBzwCDDEzBum+EFEMYM/Mh9EwHb0wmAGQAsiIaDIYrR/yqK5vaHCPnUdZBCBTt27Hw7QETdyp5IKZD849o653zLwcjOYnP/TEAVhKjNkDOMhfALgUV2Yc17Z1dAAgBnADYGIVcwwGAPfqem4Vh1bSxEAMYOqhuxJdBpB66CeimxU8Z/f5QETKXWMiqHhk5hePVzTKlmXKzLwVtL4XsEYrH30PRiuXewBbZn7yNLCWlWcf7+J2XblV1z/txbjHKz8G8OWIauqJjDKZOpTbF2Vg5rXne5cAFjn37yzK/ol3PYkpifyoTPl4nxY10G9WQV/Hs03FmGmce7H6VqyfjTuOqkiMZVfwFsBaJscyt6JYsWvINb5W0NdIW1d4L86JaZZVrj/9lg6jyyhkLdzV1tIYMt9+FQu0zXi+F+WhGcBGvpN8etZhg7FMmjnjOOIKujNttDr9iCELXezIw0UZIloQUUpEmbQNEc1OmM1eieu7qhDMG5eKLvZGXW9r9nPnMkYv9yir3W4q53o1CKOySEgaXt2FbqHjZEm7xnkDnmZR+LzgpWe7oy9yERIoc9lNjcFOmvjRZ2K3leNZcH9HaGtpEFZIXDWMsqhnRguOZyFt5MQTTa+vPXy69d1G6Hldep8AWBYkTrsswfEsREZq0qbO3qBq56bmWEzF1E+XJHR/FiRjtcrnBZXSsk0hnJJDqNhv7YVBtsn86qbVQvKN9YwmbDBShTZflOkB72uOxeYRFw1NZlFR8ZQoQxtbFGu0hyrlBLWfglMcrCO/VAmiiYlOW4qNrcYzABiLa4wD4G1y6kJ1xBp9xcxL9EdajWeWiMQ5tR0fua2KTk8m+r1nZaJW45k1miYQBx+XIug0AcSljDjsJL72EWVoK+U2cmKRb1zSB11eGmRqzwAi2X4M+7Mi0DAziCgT9nhg5gsPlNnjZFtmvmrY5aSC/q9d32QTkT40+sTMj20hzTI2KNp/TN4UKq9PMKYHGcdb3SzL/xjPrB+2xvrI6B8p1tnc4AonzA/ibxVhgzMUPz3HaJPqjRxtCEoYC4vUpZhIGWsmLjGDx9m/hg3HAGbnLIRKEsFImm8qCQD3PMpCnk1E15zKkEWTtVFGSgVZkzOVPOxCymR1T2Qio5YPqoa0kyy2XvwDBhFNhLEmFV479IHMBLPHHm1mI/wpxX8Tt30sSb1ri821Lb8BSDTsGhm4KYgAAAAASUVORK5CYII=\" alt=\"$o \\in T&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.885em\"> has the same parameter list as <img src=\"data:image/png;base64,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\" alt=\"$o \\in T$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.807em\">, allowing it\nto take inputs with types from both <img src=\"data:image/png;base64,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\" alt=\"$U$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> and <img src=\"data:image/png;base64,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\" alt=\"$U&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\">. The return type\nof <img src=\"data:image/png;base64,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\" alt=\"$o \\in T&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.885em\"> generally is from <img src=\"data:image/png;base64,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\" alt=\"$U&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"> (though it can be from <img src=\"data:image/png;base64,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\" alt=\"$U$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">). \nThus, <img src=\"data:image/png;base64,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\" alt=\"$o \\in T&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.885em\"> is a proper function subtype of <img src=\"data:image/png;base64,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\" alt=\"$o \\in T$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.807em\">. \nThe purpose of <img src=\"data:image/png;base64,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\" alt=\"$o \\in T&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.885em\"> is to:\n(1) perform operation <img src=\"data:image/png;base64,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\" alt=\"$o \\in T$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.807em\"> on the concrete \nvalues associated with its inputs; \n(2) build a symbolic expression tree rooted at operation <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAASCAYAAAC5DOVpAAABBElEQVQ4y5WU4W2EMAyFP24CZsgIzMAIzMBtcFJH6AjtCtcNmIHbIDdCGcH9cQ56Chy4kSJF5vn5+ZEYM+NoAz1wBzKw+J6B2wZ7QpIBAz6BJN9Gjy9Ad0gGfDk4K0mFGRxjBbMHmqRqOrGgKJ82ZO5NqdYF/FR8y44Pa6V/kg0l2HpbYVWeN0vOeOG1RicEeJrZg9jq5PxbyK4S/ImwNE2TqtCztGiy+2CL6rF5jK4ia4Nkk+TcC1kvwSVIVHeTzIxL6VX7DqwPOX+b2SvPKy1RZZWqefPQ/SGvRgbvVq791WrrhDhQVIimvR+l4FSNnFZIbjLLxreq39yfWZKzKxnOLPgD9K5IIvtC36IAAAAASUVORK5CYII=\" alt=\"$o$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> \nwhose children are the trees associated with the inputs to <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAASCAYAAAC5DOVpAAABBElEQVQ4y5WU4W2EMAyFP24CZsgIzMAIzMBtcFJH6AjtCtcNmIHbIDdCGcH9cQ56Chy4kSJF5vn5+ZEYM+NoAz1wBzKw+J6B2wZ7QpIBAz6BJN9Gjy9Ad0gGfDk4K0mFGRxjBbMHmqRqOrGgKJ82ZO5NqdYF/FR8y44Pa6V/kg0l2HpbYVWeN0vOeOG1RicEeJrZg9jq5PxbyK4S/ImwNE2TqtCztGiy+2CL6rF5jK4ia4Nkk+TcC1kvwSVIVHeTzIxL6VX7DqwPOX+b2SvPKy1RZZWqefPQ/SGvRgbvVq791WrrhDhQVIimvR+l4FSNnFZIbjLLxreq39yfWZKzKxnOLPgD9K5IIvtC36IAAAAASUVORK5CYII=\" alt=\"$o$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">. \n</p></li></ul>\n\n<p>Figure&nbsp;<a href=\"#fig-subtype\" title=\"Type instrumentation to carry both concrete values and symbolic expressions.\" class=\"localref\" style=\"target-element:figure\"><span class=\"figure-label\">5</span></a> presents pseudo code for the instrumention\nof a type <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB0AAAAdCAYAAABWk2cPAAABYklEQVRIx72W4W3CMBSEP0cMQBkhI2QG2KASG9ANgjpCu0E7QsUGyQiUDWCEwgbXPzYyUSueHYcnWXasd7mcz3aek8SjYxYGzrk5cH4A59MseniNxidgBxyBH+Die4AFsAWWUf4WOAxyAGpgBTyHREkXJOGX+AgI2IS5/xrQ+VwBR0N+43PPkhhOtvde4PMVtTcj5gx0kqi86jVwkPR+zxDnXDOY6oxe7r1tV9IN8GEEx14iqTfiFsA3QBW+XNKnEbyKxoeEXTuPla6BrwRwrLRPwNV+iQkG18bN0Aw20dKIm4edGy4jE6FPbmNSK85jrzyVpFPCEuX6ScxTJV5huX7ehJl0xPnMJx1xPkeRZvtZSmk/OWlJP1OUFvMzhbSYn7lK+8lJS/tpVVrUTytpUT9zlPaTk07hp0Xpyx/F1ehwknDO1f7vvoj6myLZx86rDQU4wF7SJYl1UGjntDalgpDEL7QAl68Ddnz/AAAAAElFTkSuQmCC\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> via a type <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACYAAAAgCAYAAAB+ZAqzAAABvUlEQVRYw82Y4W3CMBCFP0cdIOoIjJAZwga03aDdgKoTVIzAClU2SEYANoARCBtc/5wl9+QUJ5AQS5aN7fM9v3tOcjgRYS7FObcAziJyyWYEqgSOQOucW80GGPAZ9JduDqF0zuVAGwy9ICIouByQCWrufQa+V8H8UUR4ClB+Bf0TUGnMz8BFW4Bnpb00YTiYNQALYKmOARCRS4S0t6C/8Qs96qMifrcnipywtie8sr7QtW3HvNh5a7i+5sRsJMAm0aYF6sh4GdvLT26AfaKDwgArE+1qYBsZ38b0F57mPdHBOgSWYqN2+5gP9S0WtGeg7eEg1Ne+h93RsquXw++1COcyvRE/PR474W1setgtgJ0Z87e1EZGTNWgt2hH0lceiouEVoIjMpYG6RV8+bBGwnXLIYhT+U5ZB/9DntRPx86rtd2x935f4UH3FygmoRKTqOglj6mtozQayhYg0Y35xZFPoa2xg99TXfYA55wozVM+FsUn11QfYpPoaylgzC2CP0FcqY5PrKxXY5PoawlgzC2CP0lcKYx/m926y7FxE/L8suSaqvv2TqGqplDWfBAPsOpLY24pJdofU9RifPb/7eZ3zKDyarQAAAABJRU5ErkJggg==\" alt=\"$T&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\">.\nThe class <code class=\"code code2 language-cpp2 lang-cpp2 cpp2 highlighted\" ><span class='token type identifier cpp2'>Symbolic</span></code> is used to hold an expression tree (<code class=\"code code2 language-cpp2 lang-cpp2 cpp2 highlighted\" ><span class='token type identifier cpp2'>Expr</span></code>).\nGiven a class <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHAAAAAfCAYAAAA/QknEAAADy0lEQVRo3u2b0XHiMBCGPzEU4JAKzteBb66CkA7IpYIjHcCkgpukAygB6ICkgkvoADq4QAd7D6zmFA8G2ZaDILczHnuwJFa72v1Xq7UREf7T6VLLPhhjEmOMfMCVnKswjTGDkrLoFoyTevbvt51+987zCpgBS+AN2OgdoAMMAffPh8Ai1wYgBa6Bnm0oIpszNoiZyq4DfAUGO97PrUxF5GnXICKyMsbcqPxugcx5/aRjbIAJIoK60SUgQN/+VnTpAKLX0qN9pm3Xh9qe0wWsHTnNA4yT5d+11WQz1fZQRMYeK62bW1V7SUQWxpgN8PJZsMkYkwIuXExrDPcGTERkUYSBt8BCRB49GMtyP809mXhR9/JZKI9vTzXGSoHRrhcWA/uKY6UZK/LjO6hTcxWWtQDrVVLgck/TpafXKUvXbkwhIquK8+haL7ZTgdaiSkzCZWxRgpekaQvUudzrIktK9Js0EFx1A1nfzb7+bXWfkw9gLG0KA3VrMnV4WzjRmo2ii/B51QA/efyb11wIo30N1kDqGQ1lTlQlQNezX9JUBKoTtFHa1HcuDUefg5yckorjJEXRp9PGf8J5xkoykzakPFEFdiPaPpTaZu0Zp3do4bdLupCq+BfcVambshvab024wgjw7/ZQ/9aRGAtBNqK9ikl5O7ZZ05oLYR5EgTX2f00Iqad4/FgUXsey/yuxzSoKhIJZYBDGApHN2/6KcANfGWZ2yHtzyLu0jsBYiC1DBswiTYyHgplrn/7tE8S/7rFd+KGsSSCY6ekmvr4CY8I/TQgAvClO1KLAAdB1IPzLfA2lXcH6jo1/lwGiO1dYABeB3HEoL9Vlmz/dhFJgFPin9Efv430pshK0DIilWSAvdYvHMV1VCzz2/s+6vLmIzIiEdpSKLCqOk+pC+OnTvuUxYEz45wrme2QxTB6Pqybuh2zPZr0WgDlUlWaMGQAPDv6ZCFb7EuiIyEVkFriuIyc1lle2qcFFEAuMDP8sjYDEGNOPRYGKo5ucKyy7AKbAuEx2yUeBMeGfFdajYuFDZGWKk4KAxoeeNfNyV6ZT68Twz6U7trnC54iU6JalPPgGLQoJAFdVVvO+86gRAQ4mGzx36ytfr0RwkJs7oxRg5Hm+Oq36f0ZE3Mx3x7m/K8hVeleYaqOtY+YkFQdtycHQp7LuA3jKFM9SldMY+K3PbrHzSnme1cET+FfUW+UaRLDqM2cOa1VoT4WVHJGvnirSLfBdKn9BKgjMOX3coueE9yUDiA3w5VRL/s05fp2kQc0Ptt8nWEgozOyIyPBU5/oXJlUEAeXOaG8AAAAASUVORK5CYII=\" alt=\"$T \\in U$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.807em\">, a symbolic type <img src=\"data:image/png;base64,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\" alt=\"$T&#39; \\in U&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.885em\"> is defined by inheriting \nfrom both <img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> and <code class=\"code code2 language-cpp2 lang-cpp2 cpp2 highlighted\" ><span class='token type identifier cpp2'>Symbolic</span></code>. This ensures that a <img src=\"data:image/png;base64,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\" alt=\"$T&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"> can be used\nwhereever a <img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> is expected. \n</p>\n<figure id=\"fig-subtype\" class=\"figure align-center\"   >\n<pre class=\"para-block pre-fenced pre-fenced3 language-cpp2 lang-cpp2 cpp2 highlighted\" data-line=\"1\" data-line-code=\"2\" ><code data-line=\"2\"><span class='token white cpp2'>  </span><span class='token keyword cpp2'>class</span><span class='token white cpp2'> </span><span class='token type identifier cpp2'><span class=\"code-escaped\"><img src=\"data:image/png;base64,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\" alt=\"$T&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"></span></span><span class='token white cpp2'> </span><span class='token operator cpp2'>:</span><span class='token white cpp2'> </span><span class='token type identifier cpp2'><span class=\"code-escaped\"><img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"></span></span><span class='token delimiter cpp2'>,</span><span class='token white cpp2'> </span><span class='token type identifier cpp2'>Symbolic</span><span class='token white cpp2'> </span><span class='token delimiter curly cpp2 bracket-open'>{</span><br><span class='token white cpp2'>    </span><span class='token type identifier cpp2'><span class=\"code-escaped\"><img src=\"data:image/png;base64,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\" alt=\"$T&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"></span></span><span class='token delimiter parenthesis cpp2 bracket-open'>(</span><span class='token identifier cpp2'>c</span><span class='token operator cpp2'>:</span><span class='token type identifier cpp2'><span class=\"code-escaped\"><img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"></span></span><span class='token delimiter cpp2'>,</span><span class='token white cpp2'> </span><span class='token identifier cpp2'>e</span><span class='token operator cpp2'>:</span><span class='token type identifier cpp2'>Expr</span><span class='token delimiter parenthesis cpp2 bracket-close'>)</span><span class='token white cpp2'> </span><span class='token operator cpp2'>:</span><span class='token white cpp2'> </span><span class='token type identifier cpp2'><span class=\"code-escaped\"><img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"></span></span><span class='token delimiter parenthesis cpp2 bracket-open'>(</span><span class='token identifier cpp2'>c</span><span class='token delimiter parenthesis cpp2 bracket-close'>)</span><span class='token delimiter cpp2'>,</span><span class='token white cpp2'> </span><span class='token source cpp2'>S</span><span class='token identifier cpp2'>ymbolic</span><span class='token delimiter parenthesis cpp2 bracket-open'>(</span><span class='token identifier cpp2'>e</span><span class='token delimiter parenthesis cpp2 bracket-close'>)</span><span class='token white cpp2'> </span><span class='token delimiter curly cpp2 bracket-open'>{</span><span class='token delimiter curly cpp2 bracket-close'>}</span><br><br><span class='token white cpp2'>    </span><span class='token keyword extra cpp2'>override</span><span class='token white cpp2'> </span><span class='token identifier cpp2'>o</span><span class='token delimiter parenthesis cpp2 bracket-open'>(</span><span class='token keyword cpp2'>this</span><span class='token operator cpp2'>:</span><span class='token type identifier cpp2'><span class=\"code-escaped\"><img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"></span></span><span class='token delimiter cpp2'>,</span><span class='token white cpp2'> </span><span class='token identifier cpp2'>f1</span><span class='token operator cpp2'>:</span><span class='token type identifier cpp2'><span class=\"code-escaped\"><img src=\"data:image/png;base64,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\" alt=\"$T_1$\" class=\"math-inline math\" style=\"vertical-align:-0.164em;height:0.911em\"></span></span><span class='token delimiter cpp2'>,</span><span class='token white cpp2'> </span><span class='token delimiter cpp2'>.</span><span class='token delimiter cpp2'>.</span><span class='token delimiter cpp2'>.</span><span class='token white cpp2'> </span><span class='token delimiter cpp2'>,</span><span class='token white cpp2'> </span><span class='token identifier cpp2'>fk</span><span class='token operator cpp2'>:</span><span class='token type identifier cpp2'><span class=\"code-escaped\"><img src=\"data:image/png;base64,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\" alt=\"$T_k$\" class=\"math-inline math\" style=\"vertical-align:-0.164em;height:0.911em\"></span></span><span class='token delimiter parenthesis cpp2 bracket-close'>)</span><span class='token white cpp2'> </span><span class='token operator cpp2'>:</span><span class='token white cpp2'> </span><span class='token type identifier cpp2'><span class=\"code-escaped\"><img src=\"data:image/png;base64,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\" alt=\"$R&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"></span></span><span class='token white cpp2'> </span><span class='token delimiter curly cpp2 bracket-open'>{</span><br><span class='token white cpp2'>      </span><span class='token keyword extra cpp2'>var</span><span class='token white cpp2'> </span><span class='token identifier cpp2'>c</span><span class='token white cpp2'> </span>:=<span class='token white cpp2'> </span><span class='token type identifier cpp2'><span class=\"code-escaped\"><img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"></span></span><span class='token delimiter cpp2'>.</span><span class='token identifier cpp2'>o</span><span class='token delimiter parenthesis cpp2 bracket-open'>(</span><span class='token keyword cpp2'>this</span><span class='token delimiter cpp2'>,</span><span class='token white cpp2'> </span><span class='token identifier cpp2'>f1</span><span class='token delimiter cpp2'>,</span><span class='token white cpp2'> </span><span class='token delimiter cpp2'>.</span><span class='token delimiter cpp2'>.</span><span class='token delimiter cpp2'>.</span><span class='token white cpp2'> </span><span class='token delimiter cpp2'>,</span><span class='token identifier cpp2'>fk</span><span class='token delimiter parenthesis cpp2 bracket-close'>)</span><br><span class='token white cpp2'>      </span><span class='token keyword extra cpp2'>var</span><span class='token white cpp2'> </span><span class='token identifier cpp2'>e</span><span class='token white cpp2'> </span>:=<span class='token white cpp2'> </span><span class='token keyword cpp2'>new</span><span class='token white cpp2'> </span><span class='token source cpp2'>E</span><span class='token identifier cpp2'>xpr</span><span class='token delimiter parenthesis cpp2 bracket-open'>(</span><span class='token type identifier cpp2'><span class=\"code-escaped\"><img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"></span></span><span class='token delimiter cpp2'>.</span><span class='token identifier cpp2'>o</span><span class='token delimiter cpp2'>,</span><span class='token white cpp2'> </span><span class='token identifier cpp2'>expr</span><span class='token delimiter parenthesis cpp2 bracket-open'>(</span><span class='token identifier cpp2'>self</span><span class='token delimiter parenthesis cpp2 bracket-close'>)</span><span class='token delimiter cpp2'>,</span><span class='token white cpp2'> </span><span class='token identifier cpp2'>expr</span><span class='token delimiter parenthesis cpp2 bracket-open'>(</span><span class='token identifier cpp2'>f1</span><span class='token delimiter parenthesis cpp2 bracket-close'>)</span><span class='token delimiter cpp2'>,</span><span class='token white cpp2'> </span><span class='token delimiter cpp2'>.</span><span class='token delimiter cpp2'>.</span><span class='token delimiter cpp2'>.</span><span class='token delimiter cpp2'>,</span><span class='token white cpp2'> </span><span class='token identifier cpp2'>expr</span><span class='token delimiter parenthesis cpp2 bracket-open'>(</span><span class='token identifier cpp2'>fk</span><span class='token delimiter parenthesis cpp2 bracket-close'>)</span><span class='token delimiter parenthesis cpp2 bracket-close'>)</span><br><span class='token white cpp2'>      </span><span class='token keyword cpp2'>return</span><span class='token white cpp2'> </span><span class='token keyword cpp2'>new</span><span class='token white cpp2'> </span><span class='token type identifier cpp2'><span class=\"code-escaped\"><img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACgAAAAgCAYAAABgrToAAAACHUlEQVRYw81Y7XHCMAx95jpAjhEygq9sEDZIu0HbDcIMGQFGaLoBdIIWNkhGINlA/eNQVTiOFaCH73znJJb9JD19gCEi3NswxqQAjkTUze4QXAagBtAaY/K7AwhgxdZLc08uNsYkAFr26glENDgBvAKgCbN1bqoAFACS0D3svpydURMRghZ0Gj0CSADMAZRuzd1xEGIJgBTAAkDG9m8ArIioC9xXOZAA8EZEG8RoxjTcMw2rSJlSWDYN7D3tO71TAuRuzBVylXSdZ0/G9pRqgOIAiuWVh1te5QCsfWdr0sySrZsQlzyjC5zVj+eeq/xsDcCMrXfKDJKGALvK0QdTyb9pAFq23ioBSot9iec+cndE1Pz58g/8S4RsHcgO9uzbhFSxV0Z+KQDaAQW8507Jf+XEyG+9FvqtVvklALkFskiZgsmsh2jhlBhM+mr+jXAtcy5t+2oTqhwx80EbgcaY1rMnESnkhYg+rtLiKPlXjFhvz/iWX2K5KBd7UoRV1t381gCj+Cdk0qk50zdnCv4dIinTBEqkesyuXX9dXeVjfhOArpueUn+teG5uZcFMuC62g1mI5+OtAKr5N8C57j8sqOn/7EjQXA7Q8S/V8s8TIEP7rLtjsgWlm74jFT6OBYgD9hkb3bNr/v5w+w4jKaYE8B7tepb9LetG5L8EuftmxyqDO6uV7ZkrmxWArbLhPStNY7OIbPML0TzUMbJy/gAZmUMwaJ60OwAAAABJRU5ErkJggg==\" alt=\"$R&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"></span></span><span class='token delimiter parenthesis cpp2 bracket-open'>(</span><span class='token identifier cpp2'>c</span><span class='token delimiter cpp2'>,</span><span class='token identifier cpp2'>e</span><span class='token delimiter parenthesis cpp2 bracket-close'>)</span><span class='token white cpp2'> </span><br><span class='token white cpp2'>    </span><span class='token delimiter curly cpp2 bracket-close'>}</span><br><span class='token white cpp2'>    </span><span class='token delimiter cpp2'>.</span><span class='token delimiter cpp2'>.</span><span class='token delimiter cpp2'>.</span><br><span class='token white cpp2'>  </span><span class='token delimiter curly cpp2 bracket-close'>}</span><br><span class='token white cpp2'>  </span><br><span class='token white cpp2'>  </span><span class='token keyword cpp2'>class</span><span class='token white cpp2'> </span><span class='token type identifier cpp2'><span class=\"code-escaped\"><img src=\"data:image/png;base64,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\" alt=\"$R&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"></span></span><span class='token white cpp2'> </span><span class='token operator cpp2'>:</span><span class='token white cpp2'> </span><span class='token type identifier cpp2'><span class=\"code-escaped\"><img src=\"data:image/png;base64,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\" alt=\"$R$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"></span></span><span class='token delimiter cpp2'>,</span><span class='token white cpp2'> </span><span class='token type identifier cpp2'>Symbolic</span><span class='token white cpp2'> </span><span class='token delimiter curly cpp2 bracket-open'>{</span><span class='token white cpp2'> </span><span class='token delimiter cpp2'>.</span><span class='token delimiter cpp2'>.</span><span class='token delimiter cpp2'>.</span><span class='token white cpp2'> </span><span class='token delimiter curly cpp2 bracket-close'>}</span><br><span class='token white cpp2'>  </span><br><span class='token white cpp2'>  </span><span class='token keyword extra cpp2'>function</span><span class='token white cpp2'> </span><span class='token identifier cpp2'>expr</span><span class='token delimiter parenthesis cpp2 bracket-open'>(</span><span class='token identifier cpp2'>v</span><span class='token delimiter parenthesis cpp2 bracket-close'>)</span><span class='token white cpp2'> </span><span class='token operator cpp2'>=</span><span class='token white cpp2'> </span><span class='token identifier cpp2'>v</span><span class='token white cpp2'> </span><span class='token keyword extra cpp2'>instanceof</span><span class='token white cpp2'> </span><span class='token type identifier cpp2'>Symbolic</span><span class='token white cpp2'> </span><span class='token operator cpp2'>?</span><span class='token white cpp2'> </span><span class='token identifier cpp2'>v</span><span class='token delimiter cpp2'>.</span><span class='token identifier cpp2'>getExpr</span><span class='token delimiter parenthesis cpp2 bracket-open'>(</span><span class='token delimiter parenthesis cpp2 bracket-close'>)</span><span class='token white cpp2'> </span><span class='token operator cpp2'>:</span><span class='token white cpp2'> </span><span class='token identifier cpp2'>v</span></code></pre>\n<hr  class=\"figureline madoko\" data-line=\"17\">\n\n<div data-line=\"18\"><span data-line=\"18\"></span><fig-caption class=\"figure-caption\"><span class=\"caption-before\"><strong>Figure&#160;<span class=\"figure-label\">5</span>.</strong> </span>Type instrumentation to carry both concrete values and symbolic expressions.</fig-caption><span data-line=\"18\"></span>\n</div></figure>\n<p class=\"indent para-continue\">A type such as <img src=\"data:image/png;base64,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\" alt=\"$T&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\">  only can be constructed by providing a concrete value <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA30lEQVQ4y52UURHCMAyGv3AI6CFhEtAwLZMwD5UwDZUwDUgoEigOwksHIUdhJXd52d98f65NhqryLYERSEAGSs0LMANBVflVnAEFIjAYLQBThYUWYKnFxRY7g/I0+HBgNYDQMFGTyYvJiGMDEBxktOJkhHXHZcfNyNLLry6a0AqZDaD0ACwkG0jshRxEJAADr1jpjIMDAFz/gZzsB1Xthvh3z92XCuejqt5F5F5hXSEiCbhttGi6CTs7SNtQ2mHLe57YbPfyNidVHNzqB6dNVc9+oj85TXVWivsJxdY6PAB/ne4jt7G24QAAAABJRU5ErkJggg==\" alt=\"$c$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> of \ntype <img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> and a symbolic expression <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA/0lEQVQ4y6WU/W2DQAzFfxdlALoCIzADIzAD2YAdOkJmYIRkBbIBHaFkg9c/+ogceiWgnGTpZPve2c8fSGJNgBbogRGYLBegfvisPO78YPS9CLbGti4LApTAAAjoVz5pAP0BMcD0CsC+lf3KqCwCwLSBq8/ZLyrPBhDQbiBbM7kxDQUpMmnWJni0PKpz5PeceD5DSmmh4gu4ASdJ1yeLfxpDFN0rPpZyMFYZcG/sPEkSKSUF3Yek+1sgktLeSA6BNID77lRSqo6Bh9INtwegB77n6tShOvWWiniyL8uO7TfOTOOWOGdXgfeEPALFomPbXLf+twoag01BBg9clYvuB0wmtUhHaPolAAAAAElFTkSuQmCC\" alt=\"$e$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> to the constructor for <img src=\"data:image/png;base64,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\" alt=\"$T&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\">.\nThis will be done in exactly two places:\n</p>\n<ul class=\"list-star loose\">\n<li>\n<p>by the creation of symbolic constants associated with the primary\ninputs (<img src=\"data:image/png;base64,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\" alt=\"$v_1 \\ldots v_k$\" class=\"math-inline math\" style=\"vertical-align:-0.161em;height:0.624em\">) to the program;\n</p></li>\n<li>\n<p>by the instrumented operations as shown  in Figure&nbsp;<a href=\"#fig-subtype\" title=\"Type instrumentation to carry both concrete values and symbolic expressions.\" class=\"localref\" style=\"target-element:figure\"><span class=\"figure-label\">5</span></a>. \n</p></li></ul>\n\n<p>An instrumented operation <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAASCAYAAAC5DOVpAAABBElEQVQ4y5WU4W2EMAyFP24CZsgIzMAIzMBtcFJH6AjtCtcNmIHbIDdCGcH9cQ56Chy4kSJF5vn5+ZEYM+NoAz1wBzKw+J6B2wZ7QpIBAz6BJN9Gjy9Ad0gGfDk4K0mFGRxjBbMHmqRqOrGgKJ82ZO5NqdYF/FR8y44Pa6V/kg0l2HpbYVWeN0vOeOG1RicEeJrZg9jq5PxbyK4S/ImwNE2TqtCztGiy+2CL6rF5jK4ia4Nkk+TcC1kvwSVIVHeTzIxL6VX7DqwPOX+b2SvPKy1RZZWqefPQ/SGvRgbvVq791WrrhDhQVIimvR+l4FSNnFZIbjLLxreq39yfWZKzKxnOLPgD9K5IIvtC36IAAAAASUVORK5CYII=\" alt=\"$o$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> on arguments (<code class=\"code code2 language-cpp2 lang-cpp2 cpp2 highlighted\" ><span class='token keyword cpp2'>this</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>f1</span></code>, &#8230;, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>fk</span></code>)\nfirst invokes its corresponding underlying\noperator <img src=\"data:image/png;base64,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\" alt=\"$T.o$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> on arguments (<code class=\"code code2 language-cpp2 lang-cpp2 cpp2 highlighted\" ><span class='token keyword cpp2'>this</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>f1</span></code>, &#8230;, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>fk</span></code>) to get concrete value <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>c</span></code>.\nIt then constructs a new expression tree <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>e</span></code>\nrooted at operator <img src=\"data:image/png;base64,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\" alt=\"$T.o$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\">, whose children are the result of\nmapping the function <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>expr</span></code> over (<code class=\"code code2 language-cpp2 lang-cpp2 cpp2 highlighted\" ><span class='token keyword cpp2'>this</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>f1</span></code>, &#8230;, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>fk</span></code>). \nThe helper function <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>expr</span><span class='token delimiter parenthesis python bracket-open'>(</span><span class='token identifier python'>v</span><span class='token delimiter parenthesis python bracket-close'>)</span></code>\nevaluates to an expression tree in the case that <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>v</span></code> is of <code class=\"code code2 language-cpp2 lang-cpp2 cpp2 highlighted\" ><span class='token type identifier cpp2'>Symbolic</span></code> type\n(representing a type in <img src=\"data:image/png;base64,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\" alt=\"$U&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\">) and evaluates to <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>v</span></code> itself, a concrete value\nof some type in <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAcCAYAAAB2+A+pAAABj0lEQVRIx7WW7XHCMAyGH+c6QK4jpBt4hrBBemxAN+gOHYEVSDeAjgAbhBEgG6g/qvRUY4gTp7rTXS6WrI9XkoWIALwDMoFrESFkoErU3zgRwTlXAR54Bl7UEUufwB64AL2IHLhDzrlGHVjrnQMd9I4e2HHH86vxbh+TSWFzjw/PniIeV0BpfrXMpwuwE5HTzUnEy02AR5URcTRaEYkabo3RLsNo/RNX/LyIpKcOCmIuvT7SL0bw3WcYrh/qB+kJ+7mcmebyEb6xVK/M91lE+oxo+2g1x1K9IL7rMf3C4OuDs/bf8LUYh/hmtFGV0v/FHXxPmdH2InJOSvWC+K6S9O2UGXv2JozJZlROhT8Wwten9n+xcJrr1P4fDPuFxuRal4YkKpfA17SRT5T/xSV3Pm+B4wT5vxFnFpVP1onsWNWMl6gDtlP0huLaGcz9xIL60kn1NkkreD+T1x0tpg44zqmLmx1JeTtidHhQ2tnDJlIknV561YnWqFMbswh2KWMx2bBxoFEjtug6bZk6x+DA32lYn1cpV7d9AAAAAElFTkSuQmCC\" alt=\"$U$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">, otherwise.\nFinally, having computed the values <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>c</span></code> and <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>e</span></code>, the instrumented operator \nreturns <code class=\"code code2 language-cpp2 lang-cpp2 cpp2 highlighted\" ><span class=\"code-escaped\"><img src=\"data:image/png;base64,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\" alt=\"$R&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"></span><span class='token delimiter parenthesis cpp2 bracket-open'>(</span><span class='token identifier cpp2'>c</span><span class='token delimiter cpp2'>,</span><span class='token identifier cpp2'>e</span><span class='token delimiter parenthesis cpp2 bracket-close'>)</span></code>, where <img src=\"data:image/png;base64,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\" alt=\"$R$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> is the return type of operator <img src=\"data:image/png;base64,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\" alt=\"$T.o$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\">,\nand <img src=\"data:image/png;base64,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\" alt=\"$R&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"> is a subtype of <img src=\"data:image/png;base64,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\" alt=\"$R$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> from universe <img src=\"data:image/png;base64,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\" alt=\"$U&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\">.\n</p>\n<p class=\"indent\">Looked at another way, the universe <img src=\"data:image/png;base64,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\" alt=\"$U&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"> represents the &#8220;tainting&#8221; of\ntypes from <img src=\"data:image/png;base64,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\" alt=\"$U$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">. Tainted values flow from program inputs to the \noperands of operators. If an operator has been redefined\n(as above) then taintedness propagates from its inputs to its outputs.\nOn the other hand, if the operator has not been redefined, then it will\nnot propagate taintedness. In the context of DSE, &#8220;taintedness&#8221; means\nthat the instrumented semantics carries along a symbolic expression tree <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA/0lEQVQ4y6WU/W2DQAzFfxdlALoCIzADIzAD2YAdOkJmYIRkBbIBHaFkg9c/+ogceiWgnGTpZPve2c8fSGJNgBbogRGYLBegfvisPO78YPS9CLbGti4LApTAAAjoVz5pAP0BMcD0CsC+lf3KqCwCwLSBq8/ZLyrPBhDQbiBbM7kxDQUpMmnWJni0PKpz5PeceD5DSmmh4gu4ASdJ1yeLfxpDFN0rPpZyMFYZcG/sPEkSKSUF3Yek+1sgktLeSA6BNID77lRSqo6Bh9INtwegB77n6tShOvWWiniyL8uO7TfOTOOWOGdXgfeEPALFomPbXLf+twoag01BBg9clYvuB0wmtUhHaPolAAAAAElFTkSuQmCC\" alt=\"$e$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">\nalong with a concrete value <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA30lEQVQ4y52UURHCMAyGv3AI6CFhEtAwLZMwD5UwDZUwDUgoEigOwksHIUdhJXd52d98f65NhqryLYERSEAGSs0LMANBVflVnAEFIjAYLQBThYUWYKnFxRY7g/I0+HBgNYDQMFGTyYvJiGMDEBxktOJkhHXHZcfNyNLLry6a0AqZDaD0ACwkG0jshRxEJAADr1jpjIMDAFz/gZzsB1Xthvh3z92XCuejqt5F5F5hXSEiCbhttGi6CTs7SNtQ2mHLe57YbPfyNidVHNzqB6dNVc9+oj85TXVWivsJxdY6PAB/ne4jt7G24QAAAABJRU5ErkJggg==\" alt=\"$c$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">.\n</p>\n<p class=\"indent\">The choice of types from the universe <img src=\"data:image/png;base64,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\" alt=\"$U&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"> determines how symbolic\nexpressions are constructed. For each <img src=\"data:image/png;base64,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\" alt=\"$T \\in U$\" class=\"math-inline math\" style=\"vertical-align:-0.044em;height:0.807em\">, the &#8220;most symbolic&#8221;\n(least concrete) choice is the <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACYAAAAgCAYAAAB+ZAqzAAABvUlEQVRYw82Y4W3CMBCFP0cdIOoIjJAZwga03aDdgKoTVIzAClU2SEYANoARCBtc/5wl9+QUJ5AQS5aN7fM9v3tOcjgRYS7FObcAziJyyWYEqgSOQOucW80GGPAZ9JduDqF0zuVAGwy9ICIouByQCWrufQa+V8H8UUR4ClB+Bf0TUGnMz8BFW4Bnpb00YTiYNQALYKmOARCRS4S0t6C/8Qs96qMifrcnipywtie8sr7QtW3HvNh5a7i+5sRsJMAm0aYF6sh4GdvLT26AfaKDwgArE+1qYBsZ38b0F57mPdHBOgSWYqN2+5gP9S0WtGeg7eEg1Ne+h93RsquXw++1COcyvRE/PR474W1setgtgJ0Z87e1EZGTNWgt2hH0lceiouEVoIjMpYG6RV8+bBGwnXLIYhT+U5ZB/9DntRPx86rtd2x935f4UH3FygmoRKTqOglj6mtozQayhYg0Y35xZFPoa2xg99TXfYA55wozVM+FsUn11QfYpPoaylgzC2CP0FcqY5PrKxXY5PoawlgzC2CP0lcKYx/m926y7FxE/L8suSaqvv2TqGqplDWfBAPsOpLY24pJdofU9RifPb/7eZ3zKDyarQAAAABJRU5ErkJggg==\" alt=\"$T&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"> that redefines every operator of <img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\">\n(as shown in Figure&nbsp;<a href=\"#fig-subtype\" title=\"Type instrumentation to carry both concrete values and symbolic expressions.\" class=\"localref\" style=\"target-element:figure\"><span class=\"figure-label\">5</span></a>).\nThe &#8220;least symbolic&#8221; (most concrete) choice is <img src=\"data:image/png;base64,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\" alt=\"$T&#39; = T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"> which\nredefines no operators.  Let <img src=\"data:image/png;base64,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\" alt=\"$symbolic(T)$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\"> be the\nset of types in <img src=\"data:image/png;base64,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\" alt=\"$U&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"> that are subtypes of <img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\">. The types\nin <img src=\"data:image/png;base64,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\" alt=\"$symbolic(T)$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\"> are partially ordered by subset inclusion on the \nset of operators from <img src=\"data:image/png;base64,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\" alt=\"$T$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> they redefine.\n</p><!-- Example of an instrumentation -->\n\n<h2 id=\"sec-sp2dse\"   style=\"bookmark:3.&#8194;From Strongest Postconditions to DSE\"><span class=\"heading-before\"><span class=\"heading-label\">3</span>.&#8194;</span>From Strongest Postconditions to DSE</h2>\n<p>The previous section showed how symbolic expressions can be computed via a\nset of instrumented types, where the expressions are computed as a side-effect \nof the execution of program operations.\nThis section shows how these symbolic expressions can be used to form a \n<em>path-condition</em> (which then can be compiled into a logic formula and \npassed to an automated theorem prover to find new inputs to drive a \nprogram&#39;s execution along new paths). We derive a <em>path-condition</em>\ndirectly from the <em>strongest postcondition</em> (symbolic) semantics of our\nprogramming language, refining it to model the basic operations\nof an interpreter. \n</p><h3 id=\"sec-strongest-postconditions\"   style=\"bookmark:3.1.&#8194;Strongest Postconditions\"><span class=\"heading-before\"><span class=\"heading-label\">3.1</span>.&#8194;</span>Strongest Postconditions</h3>\n<p class=\"para-continue\">The strongest postcondition transformer <img src=\"data:image/png;base64,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\" alt=\"$SP$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">&nbsp;<span class=\"citations\" style=\"target-element:bibitem\">[<a href=\"#dijkstra76\" title=\"Edsger&#160;W. Dijkstra. \nA Discipline of Programming.\" class=\"bibref localref\" style=\"target-element:bibitem\"><span class=\"bibitem-label\">6</span></a>]</span> is defined over\na predicate <img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> representing a set of \npre-states and a statement <img src=\"data:image/png;base64,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\" alt=\"$S$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> from our language. The transformer <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPkAAAArCAYAAAC6uS60AAAIDUlEQVR42u1d63HbOBD+VpMCmKSC0B3QuQrCdECng1AdyJMKbpQO6FSQyB3IV8GF6oBKBYnUAe6HFtaKIi0SAF8n7AxHkk0C2F3sEwuQlFLw4GEoIKJIKbXxlDCjHYCtUmr/0n0zTyoPA03QgIgKAKGnhjG8AZATUeiF3MPYBDwE8AvAo1Lq0VPEDJRSTwCWlwSdvLvuYQAhL9jN/Oip4YSeGYAYwG2V6z4zbDQkoiURrYkoJ6IdESn+zIloccmF8FDrwi6YrlU0DUr3J8zgJm0vuD3Ta8fjWtrwlohW7GbeGTw7ChzGxmul1Jy/rupMfuMLwAJAAUAByAGkACIAAV8Ruw+KrxWAsE0f13gx7TKmWcF0jjlejZnOOf8/4Wdi/r1o0UcMIOH2doJPivtM+B55Jfy/Ven+zADPVOJgSKdBcRgrr1n2Ku9pOrBEEPSi4DIia0HMxAvzi8zRtE0b8EFPZk3byLDfXCrjFhNUPrdu2ecOQO6Qdr3jMGZeCyUWNBbyEkF2bYW1RMzYC3Wt9m0srIL5CsDOom9p0dKWlkg+u2z43NL1POgbh7Hzmr2BM4XXVOvsTNxu0amq0jDeRX+mb1vlWbSxXhXPxyW+hBbKe9dCqAqH9OsVh6nwWuD1rEhmNUmBmG8OAOw5a7c1SPFvAcglkqVPrz3DN6bvxmAZ6Yk/14Z9y6z23oC38v6AizJegi/8mTmkX984TIXXf/PnvDa7zsjKBo0EXMB38T31sn3IrLIrVqZPU9iXJkBbiCsmURsIWt6v+f7gkIx94zAJXgslklYKOS8r/CP+NLcU8LPBsZdw7fCpxqI0hd+8zmzKG2m1TLyB9y9YxSqvMECD8sux4jBBXm+Y9mmVJV8JDbdVSllr3grG+gII4EZ8/8vg+bemVrxCyT61fD4qWcGnC8J7JyeeI+vYNw5T4/VPSfuZQDwtace7jpAOvIxXurJt4F+L+NY2lp2Xfl/Ks8SW+YMx4DA1Xq9PaF+R/XO6dojzDPvKZ9ZPlkZ0YVHYU9+5KS/QcumpdH80RRymyOsSjuFMZD+lhc06tNx/vPE+c78iAAURFUSUcQljVx6PTSwryyYflVL3TZNjjreT9onD5HjNocf+mQeiEqmTNcMKTbZw0Oai1GaX17ojbZs26DtnVzIacm2ZvbG8LQ8FnwZfHzfFYcK81rhmryqSED8ca7KPVZk/S9jgsP7edXy/dxxLSm37QESXvKaIrwURPYiNCJ3Hsmxd3nP8qpeAHgHct4iBbwQde4/HHeEwVV5rjznUHUuNkjjWYoVs31e7VcZPC5zW+tddK4exrK5krLpOPBken0nF47qDHE+vOEyY17qOvUBFh2GHSbe1F+xGNc4vTYTEQTJGsWsY1F2OBXLVQUKpFxymyGtuOxNK8MTSuo7Hy7Gz36Rivi3RSlFWxLJRD+PXcyvrKB6PPK8vbgjazXB6xpbrzLeMKzZ8XI2HFllSjs1kvXPoIJZ1ne2ugzeOY/IhcJgir4FDpRwABLMSA5wlI4goKQ3y/poFlogii5LerEJw2oJtrbdN8scVDIHDFHl9AjN0V7Mrq4gevBXH3EIz/7RRxJxltq31toHAgeAMjcMkeC3grfaiXnGjkQP3QDJkKdraOlgOKLcfMyH7KJFdK6W+OmjnE+rO4Gru9ppaMKtab8slSFcWaSgcpsbrsmL984o1YupKyHndfSGYfNsBET/iuO7ZB3y1pEloqZDiGndu7LHsFsczAK8iHh8Br8/yITPef7otxdKV1lmczJpXbbIvbVXVh004392jlLpXSlFPl4tdc4mlEp2LsGdrOXH6tIDbjiz5mK340LwuW/Ktrl2XSbEv5ViIz8n+DeCDUuo1gM8Avsljbfm7Pk1mA+Bdn1VFIwetKEyOIda7A7cwSF5yLBsOFMsWLjzEPnEgotiylnwwXtdY8m3V4vlJXS/HFmlNTfpKrF/qKqPMr3E7O3BQ1/0bnbFXs3cg6rnYw/p8v75wwGnNSDI1XteMI6mrknkWdNRsLsDxoEd5hnTkhbq2gGOH40aF5aVJj2MxQ2HD9BJP1YAKLh4zDhWKZDc1Xtcp17qBFqLTggesywVD/q1L8fKXGIgrP6FVMDASDND01TXV+oUACU9o7RUtDUuJI25vifNSyZT/F/XBGxxLW5djxgHn1XStFVPfvG6w6624dCRzzIPIcVrwX7ALv7pECGZWceVCvq6ikQ53Si7iTkyGwFKDN7mWPSq59dhxKHmlOdofn9wbrxvikSmlzF94yFn4FQ6HPT7U3JNxNnwOD1cJnJAtcNgW+npC485weNvLwwRprl8JfauU2ti8ulhn725fyIimcHvKjIeJAa+wPOFwtvmUTup9g+6qQbtWqiEORWgbwM37yT/V/D2XHXm4ashMl5UGhAinJaZTgXmJ5lZCrtcSAyJa6bVFPrNKuwv+jSkeIAqu0g7PrnNpDSMOL/YTJHfKY//qQsilK5MA2BHR81tP4ejcdg//G5iLSTh2+ILj64am5KrrVbDPJ3+3SLzphEod3HpX3UNpzqxxOHPt3VitJFvxlVLqZoL03bFxPcmTGVtyTqjUCfGdF3APNdY8GGsYx6HEamK5Az32ZZUVt3XXAeADjidZ7Pn7jcGbGz1cR2y+ZQFKR5ppX+JwkutmYgKud37Oq8Zu7K578GBpdVLYvzHX0/LgffwC8KOuHsULuYehJmeGQ3XYjaeGFR31UnVtiOGF3MOQE3QB4G0Pryq6avr9B9EGZJD+DjRIAAAAAElFTkSuQmCC\" alt=\"$Q = SP(P,S)$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\">\nyields a predicate <img src=\"data:image/png;base64,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\" alt=\"$Q$\" class=\"math-inline math\" style=\"vertical-align:-0.207em;height:0.937em\"> such that for any state <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAASCAYAAABSO15qAAAA7UlEQVQ4y42UYRGDMAxGX1FQDUhAAxKYBSSwmwQkoAEJzEIlYGHMQfYncKU0Zb3LwTXNo/3yFUSEXAAeGIAAbBorsADdsc4obrUg6LuPcjUwAnMWADSA7Asy+VrBArS5BasmvQFYNC9Al9u6AFtBm16Pt1yOoKLt9MaCxFFhj+Ccm5xzLaWREUiMCKr+SRurC1sBtAF10QeRWLMBW28BmaONCaQ9AKr+yaIGqIsAvc6dWid/7ObUZiJbChBuin1qtAr4AF/gDTwoj5c+n/Fkf/flxObTxQeq8Koi+UzhrMXDBZwYaFJQ+gMZrNv5A62t9IKDIoPaAAAAAElFTkSuQmCC\" alt=\"$s$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> satisfying\npredicate <img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">, the execution of statement <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABkAAAAcCAYAAACUJBTQAAABnElEQVRIx62W7XHDIAyGXzyBL93A3cB33cAjcB3B2SA7eAS3E/TcDcgGbbqBvUHP2UD9EeFTKF+m1Z0OEgMPEkIIRIRcBVADOAEwAFYAxO2F/6+d8RrAuGfxkRedecEOQMNtzyACoHlOx79POYBW7LpPjNXbwreWALQ5gG1wptVazFmJCCkXrdIFO85u5nkTEaFCWF4Z9EVE79gnZ24NAD9EKVWz2QDwhv1ylbCQJc+ivxRAvgEsRLTEII+i/1QAeRAui56Jlb4A8sH36iYZYUh80Zo9ESZV8aK+g189O1zYDQbAmYiuWXZFYr13rPHpBcCQuqipS5UDsjoWQTyZNwWaiiCBfBaD6j9DIk+AVfOvEAGbBGR2v1cibFulVIcyGUX/4H6UN/7IL12JfMZyXeUkxaUQcvCk+XuIUqrhQyyVLuC6O0vs21HqriO3Lza9/0orIuZNQWT1ooqpg5fRifN+B0CL2quJjNvqo1XsagjtSkwchAVNYuw2uBVpw1YbhlOI5s1oPlhbxQyZFsMA6AKumATQWmt8JWlMfwDWk6gpP8EGnAAAAABJRU5ErkJggg==\" alt=\"$S$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> from state <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAASCAYAAABSO15qAAAA7UlEQVQ4y42UYRGDMAxGX1FQDUhAAxKYBSSwmwQkoAEJzEIlYGHMQfYncKU0Zb3LwTXNo/3yFUSEXAAeGIAAbBorsADdsc4obrUg6LuPcjUwAnMWADSA7Asy+VrBArS5BasmvQFYNC9Al9u6AFtBm16Pt1yOoKLt9MaCxFFhj+Ccm5xzLaWREUiMCKr+SRurC1sBtAF10QeRWLMBW28BmaONCaQ9AKr+yaIGqIsAvc6dWid/7ObUZiJbChBuin1qtAr4AF/gDTwoj5c+n/Fkf/flxObTxQeq8Koi+UzhrMXDBZwYaFJQ+gMZrNv5A62t9IKDIoPaAAAAAElFTkSuQmCC\" alt=\"$s$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">,\nif it does not go wrong or diverge, yields a state <img src=\"data:image/png;base64,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\" alt=\"$s&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"> satisfying predicate <img src=\"data:image/png;base64,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\" alt=\"$Q$\" class=\"math-inline math\" style=\"vertical-align:-0.207em;height:0.937em\">. \nThe strongest postcondition for the statements in our language\nis defined by the following five rules:\n</p>\n<div class=\"mathpre para-block input-mathpre\"><img src=\"data:image/png;base64,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\" alt=\"\\[\\begin{mdMathprearray}%\n1.\\mdMathspace{2}&amp;\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mdMathspace{1}\\mathid{x}\\mdMathspace{1}:=\\mdMathspace{1}\\mathid{E})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{1}\\exists \\mathid{y}\\mdMathspace{1}.\\mdMathspace{1}(\\mathid{x}\\mdMathspace{1}=\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}[\\mdMathspace{1}\\mathid{x}\\mdMathspace{1}\\rightarrow \\mathid{y}\\mdMathspace{1}])\\mdMathspace{1}\\wedge \\mathid{P}\\mdMathspace{1}[\\mdMathspace{1}\\mathid{x}\\mdMathspace{1}\\rightarrow \\mathid{y}\\mdMathspace{1}]\\mdMathbr{}\n2.\\mdMathspace{2}&amp;\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mdMathspace{1}\\mathkw{skip})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{1}\\mathid{P}\\mdMathbr{}\n3.\\mdMathspace{2}&amp;\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mathid{S}_{1};\\mathid{S}_{2})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{1}\\mathid{SP}(\\mathid{SP}(\\mathid{P},\\mathid{S}_{1}),\\mdMathspace{1}\\mathid{S}_{2})\\mdMathbr{}\n4.\\mdMathspace{2}&amp;\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mathkw{if}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{S}_1\\mdMathspace{1}\\mathkw{else}\\mdMathspace{1}\\mathid{S}_2\\mdMathspace{1}\\mathkw{end})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathbr{}\n\\mdMathindent{4}&amp;\\mdMathspace{7}\\mathid{SP}(\\mathid{P}\\wedge \\mathid{E},\\mathid{S}_1)\\mdMathspace{1}\\vee  \\mathid{SP}(\\mathid{P}\\mdMathspace{1}\\wedge \\neg \\mathid{E},\\mathid{S}_2)\\mdMathbr{}\n5.\\mdMathspace{2}&amp;\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mathkw{while}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{do}\\mdMathspace{1}\\mathid{S}\\mdMathspace{1}\\mathkw{end})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathbr{}\n\\mdMathindent{4}&amp;\\mdMathspace{6}\\mathid{SP}(\\mathid{P},\\mathkw{if}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{S};\\mdMathspace{1}\\mathkw{while}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{do}\\mdMathspace{1}\\mathid{S}\\mdMathspace{1}\\mathkw{end}\\mdMathspace{1}\\mathkw{else}\\mdMathspace{1}\\mathkw{skip}\\mdMathspace{1}\\mathkw{end})\n\\end{mdMathprearray}\\]\" class=\"math-display math\" style=\"vertical-align:-0.000em;height:10.849em\"></div>\n<p>Rule (1) defines the strongest postcondition for \nthe assignment statement. The assignment is modelled\nlogically by the equality <img src=\"data:image/png;base64,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\" alt=\"$x = E$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> where any free occurrence of <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABUAAAASCAYAAAC0EpUuAAABHklEQVQ4y6WUYY3DMAyFP08HoBh6DIohY7CjUAjlMAjDUAg5Cj0GHYVC8P04Z/KsrOl0liLV8fPTSxo/VJW4gARkYLO1AGPA9MAMrBFTI7wZILm9CVBgtnwwohT6FJhqhPmF+tWaRiMcKoQKLPHIa43Q6tk13kJtc7U5FtIO6eoa+1CbrD8DXdm87Kk0TCHcxakqJ/7iDnzxIkRkcOk3jfgAUNWfBi6579wiPXEszu8oFbuvfZBIAd1V9fPfSkUkvaPy6PHPR+5TRPrHD209DxvZ8py6Bu5SrrMzY9iAa8U0mu+zeIHLn+ZWA/jqZ/qIykLqZ3oOKjczEPVKAuHs+wrpGE2imEvxAneaa8AskfDhpzb7izPlXDGNGqZqQL/kpe0Z3roFswAAAABJRU5ErkJggg==\" alt=\"$x$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> in <img src=\"data:image/png;base64,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\" alt=\"$E$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\">\nis replaced by the existentially quantified variable <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAaCAYAAAC3g3x9AAABcElEQVRIx5WV4W2EMAyFv9wEdAVGQB0h3QB1hOsG7MAI1xU6At2gohvQFdIN3D/OyWdCQiNFMth5+L28BESEPIEIrEACNmCy+dIEOl2zAZ1NTPqy1+cbIMDcAJy1ToDBdpaAzn1AgNQAXDOgiGBfRlf4YQsrgLm7VZ+JwFYoTFq4VcCiAZxFhAvwpnrdRwhhVLHxOTdeTLzkYLTaebo+V9PvrmHBBg+6VOyyq7sUaLyauEb32cSf96hGo0H3ZupikbKjsTTsspVs5SlHE38fcQ0hdEC/ows7wN7EXxX9YskuJUA7fk7676FDr8vVaNgfaNebmt2xrG3K9cB7W23jSh2M+ZYBBgN0VbDFAE5NQEPrQwGSzpsC22M5+LUhXzlnRwghL/gVkSefv/wTzNrlvVhkKK6tK9/ol46OpS88vFDNZgkwHn7Un8tSh7oZ6dRPy3S4nvDe3Pyt6qJBOxgd0GRsE1tgDz503svCL6Wua/MPz3MNm1mBZ2QAAAAASUVORK5CYII=\" alt=\"$y$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\">, which represents\nthe value of <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABUAAAASCAYAAAC0EpUuAAABHklEQVQ4y6WUYY3DMAyFP08HoBh6DIohY7CjUAjlMAjDUAg5Cj0GHYVC8P04Z/KsrOl0liLV8fPTSxo/VJW4gARkYLO1AGPA9MAMrBFTI7wZILm9CVBgtnwwohT6FJhqhPmF+tWaRiMcKoQKLPHIa43Q6tk13kJtc7U5FtIO6eoa+1CbrD8DXdm87Kk0TCHcxakqJ/7iDnzxIkRkcOk3jfgAUNWfBi6579wiPXEszu8oFbuvfZBIAd1V9fPfSkUkvaPy6PHPR+5TRPrHD209DxvZ8py6Bu5SrrMzY9iAa8U0mu+zeIHLn+ZWA/jqZ/qIykLqZ3oOKjczEPVKAuHs+wrpGE2imEvxAneaa8AskfDhpzb7izPlXDGNGqZqQL/kpe0Z3roFswAAAABJRU5ErkJggg==\" alt=\"$x$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> in the pre-state. The same substitution (<img src=\"data:image/png;base64,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\" alt=\"$[x \\rightarrow y ]$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\">)\nis applied to the pre-state predicate <img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">. \n</p>\n<p class=\"indent\">Rules (2)-(5) define the strongest postcondition for the four control-flow\nstatements. The rules for the <strong>skip</strong> statement and sequencing (;) are\nstraightforward. \nOf particular interest, note that the rule for the <strong>if-then-else</strong> \nstatement splits cases on the expression <img src=\"data:image/png;base64,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\" alt=\"$E$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\">. \nIt is here that DSE will choose one of the cases for us, as the concrete\nexecution will evaluate <img src=\"data:image/png;base64,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\" alt=\"$E$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> either to be true or false. This gives rise\nto the path-condition (either <img src=\"data:image/png;base64,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\" alt=\"$P \\wedge E$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.781em\"> or <img src=\"data:image/png;base64,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\" alt=\"$P \\wedge \\neg E$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.781em\">). The\nrecursive rule for the <strong>while</strong> loop unfolds as many times as the\nexpression <img src=\"data:image/png;base64,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\" alt=\"$E$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> evaluates true, adding to the path-condition.\n</p><h3 id=\"sp-refined\"   style=\"bookmark:3.2.&#8194;From $SP$ to DSE\"><span class=\"heading-before\"><span class=\"heading-label\">3.2</span>.&#8194;</span>From <img src=\"data:image/png;base64,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\" alt=\"$SP$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> to DSE</h3>\n<p>Assume that an execution begins with the assignment of initial values <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIUAAAAYCAYAAADUIj6hAAAC7ElEQVRo3u1a0W3bMBB9z8gASjaouoHaDaJsYKMbyBt4B42gZILC2UDxBKm6gTxCrQ2uP0eAZWibsSmpMHgAgYSUj5Te0/HuURQRnDKSJYA1gALAg3bvAfwE8CwiA5JNYlNhsTi1AJI9gFYnfhKRexG5B/AIYACwI5kluMYnw6RYiMiHBqABIAAOAHLPeKljAqD2+UgtTpsDizsPK1udaADw5UhIaq2/8/QujxYhZsFi4Sxiq4sAgJVvEZ4Q1QTcXKH7YbJwQkTBguSSZEeyJ3kgKWe3GSsMVRqCBEB7JqSVAGoAZUD4K0J8piajYAEgA7DUiCIADmfnt354sBZSRrixwtoPEynCn1t0LJzcZHvuWrN9VLoYABhE5O2KsNeRFAA7LZt+p83gUxYNC8dKTw5yMqdYW33PV07+KCLUkmmlJVSycIuJhXlRcysJPUuyhSYd+ZFs9pISN4lZl4MXFQtPlBhEZB8SKdwyJr3Z89lYWDyFRglDigfnTU+kmM/GwsKbT5DMSNZarvYkG0OKX9cwk2SRsIxm0bHQfCJzI4VeuwPwLiJfAbwCqEguF5oDXJQHqMCynnkfLvXG/ws/1/gaCYsP+QTJCsCLFgWvKixu/hGvVPwwdXEWWPduQ7QHvW4UnQJAb627mttPDF+xsbCef6P/b1ytQiNJZ/rtTnMzdYCC1ptJ5iKFqnRitcOcfmL5io2FJYRtNKfYBCma+uPcXozNUh2rdLz/jMo2IilKB4BuTj+R1xQFC73WXk+rxM2CSOHo7q0yzLQu9Kxj4u3DhNoeQDG3nxF8XYWFdYbSW2ckJnJsj5FjCi1/NFKkFvzsG6e/O3UOskhV4E3bsfOOxhk3FUxNMkukuF3J3KtPqGWuFqISeyUiQyLF7UeJvec8avAIZBX0AO5uArYale07yTzJ6JPZqfOON/usRdXNHyLyzcjcsYnQ6CdfJuvNrZDVmzH9/jDZePZHI0HtUU73hjT6lfgLgJUZ/wuKnTTsnTsflQAAAABJRU5ErkJggg==\" alt=\"$c_1 \\ldots c_k$\" class=\"math-inline math\" style=\"vertical-align:-0.161em;height:0.624em\"> to the program <img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">&#39;s inputs\n<img src=\"data:image/png;base64,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\" alt=\"$V = \\{ v_1 : T_1 \\ldots v_k : T_k \\}$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\">.  To seed symbolic execution, some of the types <img src=\"data:image/png;base64,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\" alt=\"$T_i$\" class=\"math-inline math\" style=\"vertical-align:-0.164em;height:0.911em\"> are\nreplaced by symbolic counterparts <img src=\"data:image/png;base64,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\" alt=\"$T&#39;_i$\" class=\"math-inline math\" style=\"vertical-align:-0.281em;height:1.093em\">, in which case\n<img src=\"data:image/png;base64,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\" alt=\"$v_i$\" class=\"math-inline math\" style=\"vertical-align:-0.161em;height:0.624em\"> is initialized to the value <img src=\"data:image/png;base64,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\" alt=\"$sc_i = T&#39;_i (c_i,SC(v_i))$\" class=\"math-inline math\" style=\"vertical-align:-0.288em;height:1.119em\"> instead of the value <img src=\"data:image/png;base64,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\" alt=\"$c_i$\" class=\"math-inline math\" style=\"vertical-align:-0.161em;height:0.624em\">,\nwhere <img src=\"data:image/png;base64,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\" alt=\"$SC(v_i)$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\"> is the symbolic constant representing the initial value of variable <img src=\"data:image/png;base64,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\" alt=\"$v_i$\" class=\"math-inline math\" style=\"vertical-align:-0.161em;height:0.624em\">.\nThe symbolic constant <img src=\"data:image/png;base64,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\" alt=\"$SC(v_i)$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\"> can be thought of as representing any value of\ntype <img src=\"data:image/png;base64,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\" alt=\"$T_i$\" class=\"math-inline math\" style=\"vertical-align:-0.164em;height:0.911em\">, which includes the value <img src=\"data:image/png;base64,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\" alt=\"$c_i$\" class=\"math-inline math\" style=\"vertical-align:-0.161em;height:0.624em\">.  \n</p>\n<p class=\"indent para-continue\">Let <img src=\"data:image/png;base64,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\" alt=\"$V_s$\" class=\"math-inline math\" style=\"vertical-align:-0.159em;height:0.885em\"> and <img src=\"data:image/png;base64,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\" alt=\"$V_c$\" class=\"math-inline math\" style=\"vertical-align:-0.159em;height:0.885em\"> partition the variables of <img src=\"data:image/png;base64,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\" alt=\"$V$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> into those variables\nthat are treated symbolically (<img src=\"data:image/png;base64,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\" alt=\"$V_s$\" class=\"math-inline math\" style=\"vertical-align:-0.159em;height:0.885em\">) and those that are treated concretely \n(<img src=\"data:image/png;base64,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\" alt=\"$V_c$\" class=\"math-inline math\" style=\"vertical-align:-0.159em;height:0.885em\">). The initial state of the program is characterized by the formula\n</p>\n<div class=\"equation align-center para-block\" style=\"line-adjust:0\"><span class=\"equation-before\"><span class=\"equation-label\">(1)</span></span>\n\n<div class=\"math para-block input-math\"   style=\"line-adjust:0\"><img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAt8AAABdCAYAAAB97Qo+AAAWFElEQVR42u2d7XEayRaG33ZtANg3gkUZoN0IFmWA7AgWZSCVI3DhDJAjsFEGcCNYiQxgI7iGDPr+mG6p1Z5PNAMzzPNUUZIQzEd/vH3m9OnTxlorgL5gjBlKWkr6IOnaWruiVAAA0GmAo7VxjG/omagvJE3cn1tr7QWlAgCATgNgfAM0I+pxg7+01q4pGQAAdBrgGLyjCKBHgj5JefsTJQMAgE4DHK2d4/mGHon6UtI4entvrX1P6QAAoNMAGN8A9Qn6QNIu499MaQIAoNMAR4GwE+gLH3P+d0PxAACg0wAY3wD1cR38vpe0Lin4AACATgNgfMPpMcaMOnKdA72OIbyXNA/+HhhjxtQo9K3/ur4BaCo6DXBEXcX4hoNE0hizkTTsyCXHHpPvkn5E711Ts9AzPkh6chuaAJqKTgMcSVcxvqHqIDGU9K+kB2vtQ0cuO4wV3Fpr19bavaRw17QptQt9wu0aOMMAR1PRaYDj6irGN1RlKenRWnvXoYEtnMoNB7d59NkJ1Qs9GyjulXgXl4SgoKnoNMBxdPVdQYdYGGNshddtyzr01Bgzx6tTW3kulEypdGnqLxbqedBBYi8TGzlAHwcK73FcUBpoKjoN0Lyu5ub5dha7N1yHKQf56p7afypJgr9tkajN9TJF9WCtJVbsjQ8yThCvuzQ1GsVRrq21lymD3yToMIbahh7275GkJ0l31tqvlAiaik4DNKerlTbZMcbsJHkX+spae9Ximw6v9T54CinjiRg7QVzRfF6V5zYWxZZf81DSJnjr1w6QrJ5fBm/1fiCE3vZxb+C8d7G2gKai0wAN6Oq7CgcYBMas1P4pynv/kCDpruQ9jl0hDfQ6/qzvjWfmyqRrMYk3GW3imZQHLDZygL7i+/e3Lg5wXUpD12FNRacBatDV0p5vt8ghNLgv2hRmUqMg+rh1trLVq+1+t9bai45de+5UZvC5MERJwvMH/e3vT87x0Bn9C8amvbX2PZrauTaHTkPvdLVKtpMwxGR/boa3Yxw8aa9pMpKkz+7nvGONfaTXOXPzrj+exWEnNegrX9zPLnkWvUYNOpIJo5Oaik4D1KerVTzf4dPpWS5gNMb4wmh1PPuRy8THznfKyxDNYqjo+ru0ngHgGDrYhUVtwYKmByUhg+u2x1B3VVPRaYD6dPVdyS8NoqfT5RkWzPic7+8NZTJQMj3atUEinJ5clbj+cCe1MTmPocesXf/vwoYm3ot8pyRWeNTmLdo7rqnoNEBNulo27CReyHKOWUCuzvz+DuE6bDQdfGjwlJnejT/DlCb0lceo/7e1nw+VeLsfXBjkLDLI0VR0GqCVulrW+A6T2hPv3R98mXRtJuCV0VAmJZWr89Drwmp66CvLWBNbis8i8MX14a0Sx8mkxRurdVVT0WmAGnX1EM93Ja+wMWZkjHkyxuyMMZu0XTCNMUO3E+WT+8zO/T6pcJ5x0XmCz27i3TkVpBZM2bmzdwtjolCjx45dfugNqZILNkxxNerqzqhV+oL7/Lyv7bzO/mKMuQ3K3Zf98pBFgK4OF4Eeek28PcJU+yrU5hbr01RJjHfoLJm11SjruKai02j0uWj1qbT1ta5aa3NfTixs8JoWfSf4rk+pNHZ/z90x5sFn5koS7E+i7z3Fn805z9R9fhidZ5ZzT+PgNQvubx79byRpUPaez+WlZDrXJs6GTl33OGqvkwrfjdv6bQfrrWpfmET3POpbW6+pze1cuY9DvXBtaiZpUeFYG19nvh4DXfT1O2j4nnZV9f7IZX6b1b99+aGp6DQazatN2hrqatmGElb8sMKJ5mHHiBrR0Lnh5xnfHZVpbO5zu2jA88K8K3mds6YadnAtx3gtG7juTcc62CIskwO+vwm+37V7r9wXon5WaRDsc/9IKb9FjqHgHQnjEnppXR0Oc4z8zIG6xvsq7fw4UVvf5bTpaRuNsq5qKjrdb40+E51uhbaGuvpbCVf5W/J7f5T0e8b/npRs+561w9c2CnvJisP+pmSr2TAO7E/3s+wUwiSKKauTtZtSa3qqeK964wgvguN2icmBU5meeTB1PTTGDDu0xqFyX7DWro0xd8E9H7u+u9o/PD738N857WlUpEfGmKXTub2k3zOyPoTX3/RU+88jneeQaWO/C/FdRpu+dynsPkv62qJL76qmotP91uhO63TLtPVFVyu4yW3ZqdPAy73MePoofLqJppbmedMIKe+XfiJ2jamxJ64OeyaWXSuTlOm58QHHiKc0Zx2aUtvl9N9NySfyYYmZsHkfQ7Fyps53BeW1y+tHkRdwXEKnynjR31RPwTVtWljuvq0OSsxmTtBUdBqNZoalDdoa6uq7gieGYfSk872Chf9Jv+5IFS7cLFoQM4q85GncKEo9FOWmLbM4gfze6XzooJcmzspTOWWk856sMzw0beZGL7toxR7CMn1hFdx/lh6M3XGmkv6gi7x4tLNyS1tr762177M2A3F65dvYKqvNOm/NlfPkXuW17Zrq6WekA23xeo9cud8X5IT2faFNaQe7qKnoNBrdSVqqrc+6WhR28pb83t/1enVnuNJ7VWKK6M/g922Zc6QY9fclrpP83ukMosbS9o42iAT4xxsO9z0wrIbGmFEH0k+m9YVwkCua2t2oOPfw1r8OGTDPnCdjzL2bHVxVaLOz4K1ZgcGxKqlRddRTWw3EWcmy2rv6mLao/3ZKU9FpNLrj7ayN2vqsq0WpBkPDtNKOXNbah+jzVT3Mk6hgCs/hPPWjwMAvc73k9z4PL0282cJbUjLd5whkK0npC+Egty7xsHtZJD7W2q219oItnTMHy6mkpUsJ9mSMmRWkrpoGBtm+rsGypnr6X2QwtmFA9VmqViXje9u26Q6e7x7rNBp9VNqqrc+6WmR8H5zfu8CQX5UQ2WHJp8GQm7JPOkHjH9Z0f+dG17wz4YYN+7c8SDmBXEcducuDXNnwK8KuKgqxGxDTjKmRkgwB/+bkIa46S9d37spqe1A/ftOdNjxE/KQK0Wk0+ii0XlvfFRjAoWC9tdKreJjDaakqcebTik864cPFgvaayqDtF+gG1nHNnS0Uw8yY3i6Ij7X2vsTD7gdCSQ4yANbW2veuvB9SDPFBmnZGD/516Gvd/MdraYv6uN9Up0o7beOW84M+9hV0Go0+Yjtrq7Y+6+pvJQ1T6Q2e4QM8zGGjfCh5jnDhQtlO3Xi8twvQvzmS4C6ttXWl1vKD7ocO9Ld4KvN7Dcf8EQn7jbq1lfGoYn/7cYqL7HD/iI3we687bqC8UeL5lpJ41HE0cMbe8LalSWtbfLJ3rHypWC8rY8zW1cXdie+hS5qKTqPRXdXpNmvri66WTNGyeWO6l3CjnmnJTSt+STHoLnxckMbpl1Q8WefUS6L+vFRh8zfee7iBT5c22VmoI2mx9JKCqdbUaFGb2nVsI4fSu78pSXU1yjnWk+srGx2QFuzc+ocz5JYqSGOn1ynVpnk7/NVU57XVU9D/n9q+qU6F8WfakrRny56mfUOnO6jRXdPpNmtrqKtl83vPa8y1WJSjMswFPk5pALcFeRqf0v5X8J1Fjmj3VShnbRp8S+ZprzXfq2rIR3vC7YvL5iydZLVxJzobvWyD7B9WBz3OG3tbRdiVsXNu1G43hwwITdZTYNAsWlDmE71xx0o3nm3QVHQaje5VW2uVtoa6+i4nvqiJeO8yO2T6qam0VC5TpS/ADKezvqR8Jy0MJczR+E/GtcxaMFV5KjYZUzhtI15kM6/rwCkhT12Zzgz7blHYwLecNr5QshvbNpou73P+2DCjwrpgqjZcbb+O2tZeB8ZTG2MWUVtsop4+tGjK1sdrvyVG+N6F/5wyH3RXNBWdRqM7S8u19VlX3xUYy6oQk6QShvyqwmcfov9NlZ1i6iqnI35W+ur4Qd4g6rYn/tHj9IOPvpxakimglDHUwBbDYXvqyoY765TOntbf5ko2K0lr/xP3ALx2f4+CPvPYY+P7pxP2lV5nbsgzGu9yDEI5o7BUH3ODw8Bae9NwPY0KHBNHocKmOkW0YdOdrmgqOo1Gd522auuzrmYZ33E6oLeIXun83lGH3ASF4hOmZw1i2zQvjWu4PzI6+jqnEqZK4jlv+tpyo87eyqfoKK+7VM8CnpjvKYLX9rpbBU/+1xlldyvpD2vtXc4TevjQ6vvC6o160HUWTpCvSuw0d+sGzvscg3Bbxig0xoyNMRtJP6Mcs7XXU5Qx4tTZFUptqlOiT+ydgTY6VUaMLmgqOo1Gnwmt09ZfdDWIaxm7J8Yw5vp5sZD7/0gVY130Ot57UOLz/vyzIHh+o5zFTUqm8Z7P4V5zFcTWBdc2D2KF5kqC63sbM5WyQGbW8rj00u3rwPOE51h0cEHPNHp/UfU+gmNN6BeaeU2K25zTK68rZRZSDfUSRzgLj+f+N1WFhT511JNe4lFPHSPtdX1Zc59YoKnoNBp99u2uVdoa66oULSAq8VoccMKdKiyWCQa3nfOWjyqsSN25785s+QVUm+B7tzTcX0Rz2dLr2yhjoW1DWQpqWT195IUnt0E57VwfmR4oGjv6xSu9mQfa4fVj6cp8cEAZL4Nj+bqaqeQCsrrqKXCAzFuiP+Maj7nUCRektV1T0Wk0+gzbXyu0NdZV4y1wgIzpwo0LPXrfsmvzD1vPU0JFGxW84VzxzmLXZfPPn0k7eHLG5tecKVA4k3pyU7BDSZenXPPipt3/tNZe16wbC0lXDcQdd1pT0Wn6PjRXvrGuvqNYIScuzW/PPHDC1iY+RX//aLAcwvi8MParD6LjF7z5J3f//ryHi8bOvp6ccThUsNDohPrztU7D2x1zba29OIXh3QFNRafRaGigfNN0FeMbivCN7bpl1xWmrjrGApNw0Bj3SNT84PmcocAJyZhFPWdZTzfxIAO90VR0Go2GZsr3F13F+IYib8KDklXD07YIWZTy51iGQnyOjz1pAmlbIM/0xuwT0Np6mioJifhKkfZHU9Fp+j40Wr6/6CrGN1R5apu27HrCwazpAXOtfoae+NCDjRtQZ6487ukW51VPLsXqQNLfFGfvNBWdRqOhgfLN0lUWXELZgXmpJDft76eeyjLG7BRsxlR3XGjOeWdKVqZ73vdhWs/ly/+oZIOZVZ/z359zPbl+tbXWXlKa/dJUdBqNhmbKN0tXMb6hbAPyq/TvT9mxT7miPWXl/h3T83Am/dsbLJe2v7v69lJT0WmA4+sqxjdUaUgTvaTpWp3oGhYKtg+21pojn9+nC5KSBRh4CaHr/dobKzdMVfdPU9FpgOPrKjHfUBrnufgqaeG8Nqcg3Db4FDlcwwU9oxOWA0AdA8RA0n+VeF8xvPupqeg0wJF1FeMbqg4Wd0rSOS1P0KAnOQJ7LB5yBhmArvFfESPaW01FpwFOo6uEncChAnsr6T/H3E0rmso82Q5xwW5XUrKQ4oIWAfRhoD7QaaAfl+nHGN/QpUYdNtaTLVJynSvM8XmyHfMAANBpdBq6BWEn0BVBb8NU5vOAEv3NlD0AoNPoNADGN5wVoXDuT5kOzeWMDc9PPCEAADoNgPEN54FbOTwO3mpDVobQozN0aYUAANBpdBoA4xs6z8fo7+8tuKYf0d+fqCYAQKfRaQCMbzgHwlXq2zbswOemNMNNMfCoAAA6jU4DYHzDWTCXtHe/t2nRzF1wXTOqCQDQaXQaoAhSDQIAAAAAHAk83wAAAAAAGN8AAAAAABjfAAAAAACA8Q0AAAAAgPENAAAAAIDxTREAAAAAAGB8AwAAAABgfAMAAAAAAMY3AAAAAADGNwAAAAAAxjcAZGKMmRtjbM5rVPD9Ucb3dpQuAAA6DRjfABBgrb2RdCHpa/D2VtKlpPfW2nXB99fu+1v31oOkS2vte0oXAACdhh4+MFprKQWAoo5izECS94KsrLVXFb+/kbS21l5TmgAA6DRgfAOcsyCPJH2SNJI0zPjYXtJf1tp9znF2kgaS9lU8IsaYW0lXVQcCAAB0Gp2G8+M3igDOXMy/OTGXpJV7pbHJE3THo6SxpIExZlDi8zLGDCV9lvQ7NQIAgE4DYHzDuQr6raSZkhi+a2vtQw2H3Qa//5EzQIQsJP1dZgAAAECn0WnA+AbosqB/tdbe1Xjop+D3Ycnr+FnTgAIAgE6j03AGkO0EmhDVkTHmyRizca9xymcGwWcGNZ573JCgS8l0puey4Dr8NCYLdwAAnUanATC+oTlBVxK/d22tvXBvL1OE+6NeFtZ8rPES5kpWq9ct6IrSVf1R8HGmMQEAnUanATC+oXG8mPm4uw8ZIvhDL7F5tXhUnDdlKOmuwfvz1zzKuY6ZpC3TmACATqPTABjf0BjGmIkTs7X7exQI9mPkndhLuomEMlMkjTG7EtOe10rSS60avM11cF3DlGsdSZpK+psWAQDoNDoNgPENTfJBSRyfx4v2Km1aLxDfdcFxx0oWxBRNDQ7jwaMB/gl+T/OqMI0JAOg0Og2QCdlOoDastffRWx8DoVOK98FvhLAtOO5lyUsYShoaYw7ZOWov6fcSYryNzhfez0xJHCPTmACATqPTABjfcDxcXN8gQ+w9YyUxhXWxV+Kd+X7Ad7clvSCh9+fP4H79NCabNAAAOo1OA2B8w9HxqZvy4vo+q96Yu62kobX2a1M3Za3dGmP8n+F05kJJ5gCmMQEAnUanATIh5huawq+aT40TdItgBlFaqPD/E2PMwuWXXZQ851LSqM58tBn4ax66a50piZestIDI5dldGGOW7vVkjJnTdAAAnW6NTg+MMXOnz8tAs0c0H8D4hrbyT8b7c2WkmnKr8T9Za6+VLAaaGGOmJc7lp0Y/N3xPj8G13kqaWGtvKgr6WEme3b+ttVfW2itJf0kau2MCAKDTp9XpW0n/Snqy1l46rb5Wsmj1G80GML6hbTyGXodI0KZKFvBkLXiZ6WWa88r93Bad0E0l3km6bdgr8RRd680Bx7iT9D2a/hw6UV/TfAAAnT6dTjvDe6bEQXIfvD9UEsqypfnAoRhrLaUA9TesRKC8+F36lfJO0D7lrYw3xoyCHLQ7JemrLiqce6lkOvWvrOnSN97bKLi3+6reFHeMmaRbJbGWa0nLhvPeAgCg0+W+O1YSHrOukMUFoDR4vqERnIhfKplifHIxgU/uf5cF3/WCPlWyEn9W8fTXzivx1EQIRzBQbA8xvN0x7tx17iVNlGztbEtO2wIAoNPN6bT/PGtwoJkHXzzf0NrGacxGyap44z0ZVTwkbvHi1Bm4P5R4QbaSfqZ8vDCPbXTsqZLFO9sD7mss6TEMOXGLj6ZuAHvPanwAQKdPo9POKz9WMhtAGCBgfENvBN1PGd5ba2+ccfpUZVrTHWfovBgTpcQ1RjRu9LrBYC7pLi3VljHG+kEMAACdPolO+7DAy7xML8rYFRSgCPJ8Q1vxArx0P7+p+rSmn1a9c6/n1FkpH90fSUQvlHh17iMhHyjJQXtD1QMAOn1Snf6ixBv/WS+50MMHhTtJH9glEw5+wMPzDa1tnEne2JHcbmjnInQuvvFKyTSrH2D2kr4wxQkA6HQr7svHsfsHjA/u56OkBQvk4S38Hwe45X0U8fkEAAAAAElFTkSuQmCC\" alt=\"\\[Init = (\\bigwedge_{v_i \\in V_s} v_i = sc_i) ~\\wedge~ (~\\bigwedge_{v_i \\in V_c} v_i = c_i)\\]\" class=\"math-display math\" style=\"vertical-align:-0.000em;height:2.420em\"></div></div>\n<p>Thus, we see that the initial value of every input variable is characterized by \na symbolic constant <img src=\"data:image/png;base64,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\" alt=\"$sc_i$\" class=\"math-inline math\" style=\"vertical-align:-0.161em;height:0.624em\"> or constant <img src=\"data:image/png;base64,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\" alt=\"$c_i$\" class=\"math-inline math\" style=\"vertical-align:-0.161em;height:0.624em\">.  We assume that every non-input\nvariable in the program is initialized before being used.\n</p>\n<p class=\"indent\">The strongest postcondition is formulated to deal with open programs,\nprograms in which some variables are used before being assigned to. \nThis surfaces in Rule (1) for assignment, which uses existential quantification\nto refer to the value of variable <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABUAAAASCAYAAAC0EpUuAAABHklEQVQ4y6WUYY3DMAyFP08HoBh6DIohY7CjUAjlMAjDUAg5Cj0GHYVC8P04Z/KsrOl0liLV8fPTSxo/VJW4gARkYLO1AGPA9MAMrBFTI7wZILm9CVBgtnwwohT6FJhqhPmF+tWaRiMcKoQKLPHIa43Q6tk13kJtc7U5FtIO6eoa+1CbrD8DXdm87Kk0TCHcxakqJ/7iDnzxIkRkcOk3jfgAUNWfBi6579wiPXEszu8oFbuvfZBIAd1V9fPfSkUkvaPy6PHPR+5TRPrHD209DxvZ8py6Bu5SrrMzY9iAa8U0mu+zeIHLn+ZWA/jqZ/qIykLqZ3oOKjczEPVKAuHs+wrpGE2imEvxAneaa8AskfDhpzb7izPlXDGNGqZqQL/kpe0Z3roFswAAAABJRU5ErkJggg==\" alt=\"$x$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> in the pre-state.\n</p>\n<p class=\"indent para-continue\">By construction,\nwe have that every variable is defined before being used.  This means\nthat the precondition <img src=\"data:image/png;base64,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\" alt=\"$P$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> can be reformulated as a pair <img src=\"data:image/png;base64,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\" alt=\"$&lt;\\sigma,P_c&gt;$\" class=\"math-inline math\" style=\"vertical-align:-0.207em;height:0.937em\">,\nwhere <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAAaCAYAAACHD21cAAAA8klEQVQ4y52UYRGDMAxGX1CABiRMA7OCBDxMApMwJICFzQGTMCRkf9pdKG3g1rsegTQkjy8BVcXbwAisQLt5fhDUAhp2b30V/roae954DjI+Y8ad7yAwljmlvmKpItKa2yn1V3/xeYwe348RWAzPmT2IqiIiDdAkZfbBvocmiOsDvEtl3szbL8VSM4GLx5cNBGpPP09HVz9PR1+/ko5n+HaMZ/lyjJZv9Oat+osvZTR8q9PDw6ZUEalN282FUet+Z878X8yZNbagZayN/cpkuwEPVX2lGRuTsU0ydcDi6TjGeTO6DmGo66Mm78PXXcM1y/sF71cMZTYuzH0AAAAASUVORK5CYII=\" alt=\"$t$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.676em\"> is a store mapping variables to values and <img src=\"data:image/png;base64,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\" alt=\"$P_c$\" class=\"math-inline math\" style=\"vertical-align:-0.159em;height:0.885em\"> is\nthe path-condition, a list of symbolic expressions (predicates) \ncorresponding\nto the expressions <img src=\"data:image/png;base64,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\" alt=\"$E$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> evaluated in the context of an <strong>if-then-else</strong> \nstatement. Initially, we have that :\n</p>\n<div class=\"equation align-center para-block\" style=\"line-adjust:0\"><span class=\"equation-before\"><span class=\"equation-label\">(2)</span></span>\n\n<div class=\"math para-block input-math\"   style=\"line-adjust:0\"><img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAt4AAAArCAYAAABPVI/cAAAP9ElEQVR42u2d/XHbOBPGn824AMZXwdEdSLkKTu7AylUQuQN5roIbugM5FVzkDqRUcJE7kN4KYqkDvH9wEcM0KZEiQIHyszOcOLFCkYvFDx/7ATHGgNKdiMgNgGsAKYBPAB6MMXct7zkwxjxRu72ygwGAjTFm58mmnowxG2qWQiGnyVVylRKvfKAKOgPCVES2AOYALgHMAPzZBuYikojIWgcHSr/kEsBKRHy03RzALVVKqeDESkTWIrIQkZmIjKgVcppcJVcpp+EqJ97dNNAcQKZ/HRpjxsaYxza7HwqW/wF4NMY8Usv9EmPMUm2i1SAhIon+mFCrlApbGxpjrgDc6cRkISJbEZlQO+Q0uUquUrrlKife4WE+BXCjf73z6GpcAPjR1v1JOWnHfQDwTTssAU8JbW9PxpgxgLFOKGYiklEz5DS5SqF0x1VOvMPL3/rnToHgY5CwbtAx1dv7TmtdmXNqo1cTtUxETM1rUPOeqwP3WXmyuUcAlkXTus9HTpPT5CqFXG3PVU68w4tdcW88GeYE+c7MFx8JJJQoZAxgpLtulH7IP8iT78YAliW/f0AeH3rdYPf0C4B7AG6/3um9xp4ncDPnZ8axktPkKoVc7YirF2znzuTZ030y5JnWjBc8EzHGPInII4BMRB44UPeizXZ2YBCRDQB31+T+mNACHUieROSn9vMlgHEge3AnmEz6I6fJVQq52hFXuePdI9G4oQR5MD/lvMS26Veqon8DfOGf2saVXiMviXYdarIQehIiIhMRmXm+Z6IVBKIOjSGnyVUKubpPOPHuz6Q7ATBVw1lSI2cHmQ2AJwA3jLntpbjAbVNNYQRgpINEn2UM4LPne6aqm0/kNIVcJVf7ylWGmvRHbPLPjKo4W/kHL7VjGXfbDKwDBXMK4Lc9H137Sp4ryAbAoO0AYdv/TA7tSHp2X3KaXKWQq5x4U36JrQ35QFWcpxhjHkXEtjUHiHqDwt/IdzKSBv/vWwBXY+sBQkMzfgQawCjkNLlKrpKrnHhTahqO7QAbJoicvTwBGIjIhBOwyv6QIN/BGDk6WyKvmfyM1+7J4iAcasdjU3jGtMl3aR+fALhiC5PTFHKVXD1frnLi3Q8ZOx2Bct7yQ1f4Y3DXrAqkc53gPCI/7CQG9+F/hb+naFaabobzCTEhpynkKrlKrlYIkyv7IXYFuqAqzl4WhTanvB4crH6u9UjvWIC6KRkg6r7XDB4PbqGQ0xRylVyNl6sfDjx4omWh7Bn0dU8U2orIDU3aS6dIHGP7cUyn0pObtiKyPnSYgIjMtA1nfdKRiEyd97TvujjGDlVnc72Hvd9KvyN0YtfSeY40gK56aQ+qiwVyd+cwwooRxQHiquZ7DZC7Qnm6ITl97pwmV8lVcnXfxFsNZ4t8q75RkD3izjrvGsa+dlHQ4KQm+/0T5AXjx8aYjwqfTOvMln3+Bi/JQZM+lF/S1fr/APyFvGbr78aYj8aYK+SJNH/o0c114blWEG109f9Rdfenwul7yEFCY0N3IXZnem4Ptg3/jDEcoySmN23wXncMMSGnz5XT5Cq5Sq4emHjr7uFKDcmuOO51EnOHt8d43uLl2E17Dduc2KUrYNPRFdIteOn8fGyyTVqx8quz4ssKHWptO3vN1WXUJ9rpOy4ALI0xQ2PMsqSjjpDXcB0d2oHQe10CuDLGFDvtJ9WnzfjuYpU/9KyrXtqDDlQD5CeXxRw/2+g0SB2Yd8aY+xM/9+U7n3uT0+H6LrlKrpKrBbkoWfl/18bYAfhSMoG+1xWedZFcBYiheUIe4B9653yHsPF4rpH8d+Q9ro4cEL7i7bGof+ifScXq8klE7pxFV+yZ+Xa1/qXi9zO8lCJK9nTUhU7Qd8h3zMvee9EhKJ8DfE+f7cEOyP9Ebo8bp80GNQbsKU6bbW+f970fGU9Oh5nYkavkKrlaYQS/Lu0ERq/U/V3x0lWdAbDd97n3fOnE0OozOfIetk0WDf7PqKxdkIcOGeTF7vf9/1VNG7ALsOQEuh0dsj99vu0+3RXaaFTxmcT5TOXnfOnFeaZ1jc/aZ5udoz047zfvQX+fFewkOcDP6Ymfd1LXpo/llud7DvRZp+R03JwmV8lVcrXapj84K4UpXmKfrmvExtid8KSDxIjeierTJvbdtqjrennEqvi2uIpVt1Li7ATvk6UuyjZ73m+k95ngNEc4Dxz7G1QsKh80lvC64h0mThstqxJLtO2ukYdcXe9LQPGkl+dDrqqG0md76FOliNWendSiKxSndoWqp9JyfB4i6Yycfn+cJlfJVXJ1P1cv9IETx+2xrJnZ+rMAHR4YgFcxXyN1Ody2zBROCtCoI//ibSz+XyWLpipZ43At2o29IsiEXonIg67elzXbybV5FH4u60zLEp2G0ovvvtRne7DQevYxMQycbFMWu/lU4QodxsArY8zYCR1ci8i97vKddbInOR1Mr+QquUquHuKqbo1P0dDliNfb/ynDSn65jew19XRf67bKPLiVDIBVzbbNItd3WtC3e9nk4H0uKdfmt5G9269n8+ES7bM9aDsaz1fSkU1OKziRRdifErwONWzLnChDTcjp7thFrpKr5Go5Vy8clwmQZ4IuG66YYFgKC8aYoa72P6k+MxG5RZ540SZj+Bntk0w/Oz/XqRs6cmwiVn1vRGSIPBm4qJ+BXhMRGVbYp/t+7+3gkj7Zw0+njXzsWK1DHeetNun+01VhV2aqg9NdZLu/mU5KgNztf7Y73uR0cCFXyVVy9QBXL9TNYCfRTVwZowo3wHuefO9Uh0uN9ZojD4MYtymvWHBltoLhoQo0ag+XERbSL9P3E4CP6tK5xtt683a1Weyo7mEXdncuJvnNLoRDD449sAfLl4WHPtSF7Bwb/FTQY4ZIXKHOc7mVJ4bvYROFnA5mS+QquUqu1uDqBY4opVSoiTwL8NB2JdhF0uYiRDC+MeZR444nyAPsr44c1Cwk2iSEDBosrG4BfOvZQPpgd1e0I946K81UREYFwKUVEIpFjokXPVd7sLuQf+BwzGQMsnH069rZHJHVyy0k1Ps+JjrE5Cb13V/J6SDtQ66Sq+TqAa42LimF1/HdSU/ij6quRcD4noHzPbOWJZAWHp5hWjNWcbDnXivkSSFreC5B1iBObwHg5sDnbpz3nlSUIqwV71dTx9704rT5yncsYh/tQe+97Umux7xoW2qzaw9xjjNH11vtB6OWeg0Si+swPA1wzwE5HR+nyVVytYdcTfQ+rr7XaJercZCrxc6SNkwgyPrQaCcOrDctgZzVhUXbWr06WV3sgcHa2ohjXMmJEnfqJMiYsoG60C7rYwaEkHrBSzLGPMAA0Tt7cNp90oM+Py0s7EdoWSfbWUROSibjRw/MbfpAg0lYFvtEgZwOokdylVyNnau2zVYlc4QFjqxxXqcPfADwY4+rqEy+2q3/2ILZIwyB8JK00KBt9rnX6rjYvgKoatN5wW1yWYy3qhNC1LLm+18lrrKq77Lfsyu6obRdjmobEZnjdUJMa72UyGVAV2009tCgH92rLrIenBmwKdHTw7GxnPq+X9Wd+lDi2k4BjD0/s482W+p9pz7aTPM4UgRI2COn/XCaXCVXe8TVmXoQH40xQ3eO4ISK3ITi6gcn0cSNUdoHvxvtXNecWncidmF07EFFTyXgqTLEh7JYKU1A2tjfac3MpPB8hwx9ravIrd7vGHlW21vWmGzY43CrAGcH8LSuXnVwSIwxt770UiG2H/4XwJ6isIcjxOZ8fI98kNiUDHBtKhd81vf+WfI7ezR1rIdg2D66alMr2DlEZIN4j7cmp8lVcrUHXNVqIxPV9XgPV8MlvhbqJG5rbMv/coPwauR2WHi4x7EuZVtjdrbHjbM60Pajkhj/RUNXeav6rvocqwZu7tkBV6J142U17rcu3q+tXmrECia+XaIx2IMn12Dag/Cy1kex47WLNUNJ2NQp2dSgzbY4kJdxIIRjG7LNyWk/nCZXydWYuVqwg9Ge7xqFZElZp3sV76KdYwHGdJ8S6Ks2+i8Y26Tw73M0jGVy7nXTMN7TR8fJFNY3RYDq99gkjDoJLqk7SLj3099N0CDBpaleDi1wGwJp1hd78DRIeDv8JGCfn3m4V9VBUQsPk3rT0cA+ciYla52ApAc+7ybwrxA4Rpmc9sdpcpVcjZWrzvxge0qW7Muc3+q1UlAmnEifDOiZh3vY7N21s4O0QsPECncHq+H/mzkD76olRAcldmpdpI1tVd9p4dzL6iZD/Uo/Ex8d2tHTLNQAEYM9eKh24D73TBdiaQycUlta+3oWZ2ewbAI+iX3iXeDYtriripeKAsXfrbuYdJDTwThNrpKrUXHV56aIt4k3r2iBniKSI3jb7uo4EJycWTuvfHiFHPANQg4Qsb13i2ofKzQrH7rt6yaCdYFq/9m2ZULXE+/CQiJzBtKtM9GbH9oRJ6ffD6fJVXI10NHzk1Oy5IK5i/GLHpm6BDAqOQymM9FkD5ugMiskjNw1qA5wiTM68dSXXpxTZDcxHQjQgT0c2y8eATxqUtBn5CeUJth/iMkm9HN50q09ke1fe7Jc4cRFG3KV9K2/KL+WODMhp8lVcjVqed6XrNmlcOLdH5npTtH4hIOWLef3ZMsdKdRGDTvdAOGytPusl9sibN/Je7cdKHYIUGbuxDLHS7nAx4pJ3u6cFrDkNDlNrpKrId9HmVm32k6G3OPhnbEfyMneGM2jDrKTE5b9KTsSN9OryWp+14ddxy71ojJR3dy/s/emvJVE+/tdRT8a6We+UFXk9JlymlwlV32LLUd6fcD+bfJ6kI0NTry7k0sP97h1QHIKsW66tbMiRMnBHvvkb8Rbi/dketEa+X2bSPmwB0q5ZMhr1j8X7CRRW5kDuO2D65ycJqfJVXI1koXxPXIP4rRYp15EUhHJtJb9kzFmGOo5xJ4bTwmkYJGtdvydMeajh/stkJ9i9fspdiM0zuyzTgiWTYrY6y7K3BhzdYbtfLReHDvZNO3suqu2RX4ow23f3puyV7cj5DvexcNnlsiTrjYt7m1LvgH56W3jd65rcjpCTpOr5Gog/d7o4jJ1Njc2yBO+H47ts3W5yol3+AbOkGfpA8B124Qbp2Ef+tQZFWQr5Mfhcpeu3EaGTXVz6gGCchZcGtsETuqDnD4XTpOrlFi5ylCTwGKMucOL62jW5uhkvd8GeeLORHfE+iIZ8oxsTrpfd9SBdlSGDVC6tjkg35V5fO86IafPi9PkKiVmrnLi3Q3Uh8izg1MAKxGZtgG7Nug9gHnbAaJDHdxygH/TURMA35HvqjwcqdcdNUmpOzCoC3ul/3T/3kNMyOnz4zS5Somdqww16R4In5Fn1KbIYwpnx2Zba2OPzjFm+p3Ywwp5/OG45X3WyGN+mXxDqbKPFHmy5gZ5WbVvnFyQ0+QquUrpnqucePe/8acAflNXKYXtRqG8mUhyks3+Tj1TKHFw9f/VCTN7xUYvCwAAAABJRU5ErkJggg==\" alt=\"\\[\\sigma = \\{ (v_i,sc_i) | v_i \\in V_s \\} \\cup \\{ (v_i,c_i) | v_i \\in V_c \\}\\]\" class=\"math-display math\" style=\"vertical-align:-0.000em;height:1.119em\"></div></div>\n<p class=\"para-continue\">representing the initial condition <img src=\"data:image/png;base64,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\" alt=\"$Init$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">, and <img src=\"data:image/png;base64,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\" alt=\"$P_c = []$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\">, the empty list.\nWe use <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACEAAAAgCAYAAACcuBHKAAABqklEQVRYw8WW4W3CQAyFv2OCtCOEDegKYYPQbpBuQEeoMkK7AmxAO0EFG4QNSrKB+8eHrJCSg0S5k06YcDk/7OdnOxEhxnLOpcBJRJpZJAAZUAG1cy6PAgJ4M/bSTZ0O51wC1ObRKkYkMmMfRWQbA8SLsUuAGOnwDhsReQCYRagKvz69MXU6VsZ+jwXi2UdBRJpzhKbihCpkpV/nInKMEYlcP78sAABEZJIN7AEBFu3fJkmHUcmDiDxdHOhBnwAFsNNLJHDXQG7uKfR53unnCoD1DU67tgWRAZv/fF2kQ0P3DSy8vgNbZXYCLFv6/wqcWgE+isghOF8d4d93hbR1rjD/uBxM2tblO3N52sOXyoMdDUSLA1lAyZXmfDIExMzwoDRi8hWQyV9jPw4pYa+YRbvHB6z5aEJyb34tf0ZQU1KT280NL/p3qqEgZgrCr587hpOPodlogwgVmFXXhDR0+g3SBiNoowmV50Ryoz5sxuLCmRM6ZnldWPRwodDhpNEeMmqJpn0lavpFFZK2e2U7Vyd7O/0oZ3ZjcqCvgaVacpV20VpBrYf2h2v7D7qTVLcY/XchAAAAAElFTkSuQmCC\" alt=\"$\\sigma&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\"> to refer to the formula that the store <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAASCAYAAACw50UTAAABEUlEQVQ4y5WU623DMAyEPwUZwDOoG2QGdwN3BXcD76AR2hWSDYxOUCQbKCPUI6h/6JQV9GAEEDCE45l3pEhKiVoAAzADK7AByRgbMLWIlyfISjEdyY5zbgC+gJNc3YELEEXJKzCqlHfgJ6O5p5RuJRuuWlpF1awqDFX1WdKqknynH3EvokueeTy2iAUfFH6okosdO3DtEReKKao8SANm1YyA7bx0EVb/CpU/+tPA4JW88xPke06sYQ5Cvp9vix/OOT3nHzVcTn4z+v2mvj9bwNE624XJCh3sP7Blvs89r/M53zu/dIhntRq8ldx3n/IfcbQQ589/kuQrcFL3o1IWrKNaWlxeRiuK9E1+ttT2Ryt+AVgpvMAK6aihAAAAAElFTkSuQmCC\" alt=\"$\\sigma$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">\ninduces: \n</p>\n<div class=\"equation align-center para-block\" style=\"line-adjust:0\"><span class=\"equation-before\"><span class=\"equation-label\">(3)</span></span>\n\n<div class=\"math para-block input-math\"   style=\"line-adjust:0\"><img src=\"data:image/png;base64,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\" alt=\"\\[\\sigma &#39; =  \\bigwedge_{(v,V) \\in \\sigma} (v = V)\\]\" class=\"math-display math\" style=\"vertical-align:-0.000em;height:2.550em\"></div></div>\n<p>Thus, the pair <img src=\"data:image/png;base64,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\" alt=\"$&lt;\\sigma,P_c&gt;$\" class=\"math-inline math\" style=\"vertical-align:-0.207em;height:0.937em\"> represents the predicate \n<img src=\"data:image/png;base64,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\" alt=\"$P = \\sigma&#39; \\wedge (\\bigwedge_{c \\in P_c} c)$\" class=\"math-inline math\" style=\"vertical-align:-0.434em;height:1.249em\">.\nA store <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAASCAYAAACw50UTAAABEUlEQVQ4y5WU623DMAyEPwUZwDOoG2QGdwN3BXcD76AR2hWSDYxOUCQbKCPUI6h/6JQV9GAEEDCE45l3pEhKiVoAAzADK7AByRgbMLWIlyfISjEdyY5zbgC+gJNc3YELEEXJKzCqlHfgJ6O5p5RuJRuuWlpF1awqDFX1WdKqknynH3EvokueeTy2iAUfFH6okosdO3DtEReKKao8SANm1YyA7bx0EVb/CpU/+tPA4JW88xPke06sYQ5Cvp9vix/OOT3nHzVcTn4z+v2mvj9bwNE624XJCh3sP7Blvs89r/M53zu/dIhntRq8ldx3n/IfcbQQ589/kuQrcFL3o1IWrKNaWlxeRiuK9E1+ttT2Ryt+AVgpvMAK6aihAAAAAElFTkSuQmCC\" alt=\"$\\sigma$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> supports two operations: <img src=\"data:image/png;base64,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\" alt=\"$\\sigma[x]$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\"> which denotes the\nvalue that <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABUAAAASCAYAAAC0EpUuAAABHklEQVQ4y6WUYY3DMAyFP08HoBh6DIohY7CjUAjlMAjDUAg5Cj0GHYVC8P04Z/KsrOl0liLV8fPTSxo/VJW4gARkYLO1AGPA9MAMrBFTI7wZILm9CVBgtnwwohT6FJhqhPmF+tWaRiMcKoQKLPHIa43Q6tk13kJtc7U5FtIO6eoa+1CbrD8DXdm87Kk0TCHcxakqJ/7iDnzxIkRkcOk3jfgAUNWfBi6579wiPXEszu8oFbuvfZBIAd1V9fPfSkUkvaPy6PHPR+5TRPrHD209DxvZ8py6Bu5SrrMzY9iAa8U0mu+zeIHLn+ZWA/jqZ/qIykLqZ3oOKjczEPVKAuHs+wrpGE2imEvxAneaa8AskfDhpzb7izPlXDGNGqZqQL/kpe0Z3roFswAAAABJRU5ErkJggg==\" alt=\"$x$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> maps to under <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAASCAYAAACw50UTAAABEUlEQVQ4y5WU623DMAyEPwUZwDOoG2QGdwN3BXcD76AR2hWSDYxOUCQbKCPUI6h/6JQV9GAEEDCE45l3pEhKiVoAAzADK7AByRgbMLWIlyfISjEdyY5zbgC+gJNc3YELEEXJKzCqlHfgJ6O5p5RuJRuuWlpF1awqDFX1WdKqknynH3EvokueeTy2iAUfFH6okosdO3DtEReKKao8SANm1YyA7bx0EVb/CpU/+tPA4JW88xPke06sYQ5Cvp9vix/OOT3nHzVcTn4z+v2mvj9bwNE624XJCh3sP7Blvs89r/M53zu/dIhntRq8ldx3n/IfcbQQ589/kuQrcFL3o1IWrKNaWlxeRiuK9E1+ttT2Ryt+AVgpvMAK6aihAAAAAElFTkSuQmCC\" alt=\"$\\sigma$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">; <img src=\"data:image/png;base64,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\" alt=\"$\\sigma[x \\mapsto V]$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\">, which \nproduces a new store in which <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABUAAAASCAYAAAC0EpUuAAABHklEQVQ4y6WUYY3DMAyFP08HoBh6DIohY7CjUAjlMAjDUAg5Cj0GHYVC8P04Z/KsrOl0liLV8fPTSxo/VJW4gARkYLO1AGPA9MAMrBFTI7wZILm9CVBgtnwwohT6FJhqhPmF+tWaRiMcKoQKLPHIa43Q6tk13kJtc7U5FtIO6eoa+1CbrD8DXdm87Kk0TCHcxakqJ/7iDnzxIkRkcOk3jfgAUNWfBi6579wiPXEszu8oFbuvfZBIAd1V9fPfSkUkvaPy6PHPR+5TRPrHD209DxvZ8py6Bu5SrrMzY9iAa8U0mu+zeIHLn+ZWA/jqZ/qIykLqZ3oOKjczEPVKAuHs+wrpGE2imEvxAneaa8AskfDhpzb7izPlXDGNGqZqQL/kpe0Z3roFswAAAABJRU5ErkJggg==\" alt=\"$x$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> maps to value <img src=\"data:image/png;base64,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\" alt=\"$V$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> and is everywhere \nelse the same as <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAASCAYAAACw50UTAAABEUlEQVQ4y5WU623DMAyEPwUZwDOoG2QGdwN3BXcD76AR2hWSDYxOUCQbKCPUI6h/6JQV9GAEEDCE45l3pEhKiVoAAzADK7AByRgbMLWIlyfISjEdyY5zbgC+gJNc3YELEEXJKzCqlHfgJ6O5p5RuJRuuWlpF1awqDFX1WdKqknynH3EvokueeTy2iAUfFH6okosdO3DtEReKKao8SANm1YyA7bx0EVb/CpU/+tPA4JW88xPke06sYQ5Cvp9vix/OOT3nHzVcTn4z+v2mvj9bwNE624XJCh3sP7Blvs89r/M53zu/dIhntRq8ldx3n/IfcbQQ589/kuQrcFL3o1IWrKNaWlxeRiuK9E1+ttT2Ryt+AVgpvMAK6aihAAAAAElFTkSuQmCC\" alt=\"$\\sigma$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">. \n</p>\n<p class=\"indent para-continue\">Now, we can redefine strongest postcondition for assignment to eliminate the\nuse of existential quantification and model the operation of an interpreter,\nby separating out the notion of the store:\n</p>\n<div class=\"mathpre para-block input-mathpre\"><img src=\"data:image/png;base64,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\" alt=\"\\[\\begin{mdMathprearray}%\n1.\\mdMathspace{1}&amp;\\mdMathspace{1}\\mathid{SP}(&lt;\\sigma,\\mdMathspace{1}\\mathid{P}_\\mathid{c}&gt;,\\mdMathspace{1}\\mathid{x}\\mdMathspace{1}:=\\mdMathspace{1}\\mathid{E})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{1}&lt;\\sigma[\\mathid{x}\\mdMathspace{1}\\mapsto \\mathid{eval}(\\sigma,\\mathid{E})],\\mdMathspace{1}\\mathid{P}_\\mathid{c}&gt;\\\\\n\\end{mdMathprearray}\\]\" class=\"math-display math\" style=\"vertical-align:-0.000em;height:1.509em\"></div>\n<p>where <img src=\"data:image/png;base64,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\" alt=\"$eval(\\sigma,E)$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\"> evaluates expression <img src=\"data:image/png;base64,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\" alt=\"$E$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> under the store <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAASCAYAAACw50UTAAABEUlEQVQ4y5WU623DMAyEPwUZwDOoG2QGdwN3BXcD76AR2hWSDYxOUCQbKCPUI6h/6JQV9GAEEDCE45l3pEhKiVoAAzADK7AByRgbMLWIlyfISjEdyY5zbgC+gJNc3YELEEXJKzCqlHfgJ6O5p5RuJRuuWlpF1awqDFX1WdKqknynH3EvokueeTy2iAUfFH6okosdO3DtEReKKao8SANm1YyA7bx0EVb/CpU/+tPA4JW88xPke06sYQ5Cvp9vix/OOT3nHzVcTn4z+v2mvj9bwNE624XJCh3sP7Blvs89r/M53zu/dIhntRq8ldx3n/IfcbQQ589/kuQrcFL3o1IWrKNaWlxeRiuK9E1+ttT2Ryt+AVgpvMAK6aihAAAAAElFTkSuQmCC\" alt=\"$\\sigma$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">\n(where every occurrence of a free variable\n<img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAASCAYAAAC5DOVpAAABG0lEQVQ4y42U4U3EMAyFv54YIGKEjlCxAR2hYoTeBl0B3QiFDegIMAJhgzICGcH8wD6ZqEkuUqQkfnp99nONiGAbGIEIJGAHFh/PN7ACAqwigg/MStRnwEuBaNK47cECg6oJDrwoKBXIhoxsskAExgy8GbCS5uLIRqtTOgAmBe2NukXF9SfgDDzjVtd1ExD0ulJfH/zJ/7ZChlKKZkhF2QzEf266YHBEsUbkXL+ICKcD2U/u3ErRav6O5loqqNygqvfmHSkbfGEb6wy8XW+VRlxuUJaAoaTswZ2/apK0fT5F5Iq7yzDBnX8aKb4Cj/4hV+bV3FdUrcCLV1VyM/mxUvgfD/uvNQ3m7H0DtqIhha8HVbDbGNL+m2vu/gKZZu4Djud7fwAAAABJRU5ErkJggg==\" alt=\"$v$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> in <img src=\"data:image/png;base64,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\" alt=\"$E$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> is replaced by the value <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEEAAAArCAYAAAAnmOV5AAACwUlEQVRo3u2a0ZGbMBCG//VQALkOQjog6eC4DkjSAXTgKyFjd4BTQWJ3cE4FOdwB10EOd7B5yGpmowEkbDD3oJ3RGAuEVt+upF3ZxMy4tRBRDGDfcWvPzLtb9xlhOck66l6W6HOFZWXLzKRKOVdHzHzWfQF4Z+4tDeFNSIAQIAQIAUKAECAECAFCgBAgXAGBiGIiKojoiYhaImLP0hJRPmEGmBFRLe9tiGjteL4SPSrfxKKzAFgD4CtKPvDuWJ7Z9D2jni0A1AAS+V4NtQWQW3qkDh2qqCfv/gUgVentAUAjDR+slLQE8Gqnp8x8msADUgAbAO+Z+SzVjXwWAB490vEEwMnbE2SQtRBq+6wpCrCvNS/1BNEls+r2pm9PL85cnmDfeFKNE4eCjYE1BwTxtrajvpW2jQfA3nFoCCvlemvl5g/M7DrlOchnLFNoaikBfLOmRy7Km7VhSI7i6c7TqkitAxvTmJmPHkr+Udd3AM4TQ/hhBqLka4cR+qRxrgUagsxxIxtPJT/MuXcz86FjwTbb7snDwh87IA7GCeZs7+zpBWbVha/LTSBf1LXP/p/JGueGQESJGtBxhFJZz5Y0l5QK+s6xtSYA7nwNutIWBfDbN4IbaZUpJB1hqBLAT98XRxYE3wDns7rezT16CZqM+Lh4AeB+TO7wMhBtoSeiNAvpVkVyc8onX0PJNvo8JmKNADxbi50LxHcVGj/eaCroOOTVQ7/7MS9fiSWP1rzro1zINnWWHOJWcrJikt7sEcBudN4iIWTiCoFVvtC4QuqpcoeecLkayBfqC3SoulLQWqefar+9KFmaEEKq8prCqt/j3y/auAqC8ohKrN1KqYVyPAWASyGodmudvIl+xRU6/H+eIJFfiTcqsn5tpYQzxgAhQAgQAoQAIUAIEGaXaOH+c+usYIk/cy4OIbEOdZb4Myf+AmvwBtGadeckAAAAAElFTkSuQmCC\" alt=\"$\\sigma[v]$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\">). \nThis is the standard substitution rule of a standard operational semantics. \n</p>\n<p class=\"indent para-continue\">We also redefine the rule for the <strong>if-then-else</strong> statement so\nthat it chooses which branch to take and appends the appropriate\nsymbolic expression (predicate) to the path-condition <img src=\"data:image/png;base64,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\" alt=\"$P_c$\" class=\"math-inline math\" style=\"vertical-align:-0.159em;height:0.885em\">:\n</p>\n<div class=\"mathpre para-block input-mathpre\"><img src=\"data:image/png;base64,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\" alt=\"\\[\\begin{mdMathprearray}%\n4.\\mdMathspace{1}&amp;\\mdMathspace{1}\\mathid{SP}(&lt;\\sigma,\\mdMathspace{1}\\mathid{P}_\\mathid{c}&gt;,\\mdMathspace{1}\\mathkw{if}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{S}_1\\mdMathspace{1}\\mathkw{else}\\mdMathspace{1}\\mathid{S}_2\\mdMathspace{1}\\mathkw{end})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{2}\\mdMathbr{}\n\\mdMathindent{3}&amp;\\mdMathspace{3}\\mathkw{let}\\mdMathspace{1}\\mathid{choice}\\mdMathspace{1}=\\mdMathspace{1}\\mathid{eval}(\\sigma,\\mathid{E})\\mdMathspace{1}\\mathkw{in}\\mdMathbr{}\n\\mdMathindent{3}&amp;\\mdMathspace{3}\\mathkw{if}\\mdMathspace{1}\\mathid{choice}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{SP}(&lt;\\sigma,\\mdMathspace{1}\\mathid{P}_\\mathid{c}\\mdMathspace{1}::\\mdMathspace{1}\\mathid{expr}(\\mathid{choice})\\mdMathspace{1}&gt;,\\mathid{S}_1)\\mdMathspace{1}\\mdMathbr{}\n\\mdMathindent{3}&amp;\\mdMathspace{3}\\mathkw{else}\\mdMathspace{1}\\mathid{SP}(&lt;\\sigma,\\mdMathspace{1}\\mathid{P}_\\mathid{c}\\mdMathspace{1}::\\mdMathspace{1}\\neg \\mathid{expr}(\\mathid{choice})\\mdMathspace{1}&gt;,\\mathid{S}_2)\n\\end{mdMathprearray}\\]\" class=\"math-display math\" style=\"vertical-align:-0.000em;height:5.386em\"></div>\n<p>The other strongest postcondition rules remain unchanged.\n</p><h3 id=\"sec-summing-it-up\"   style=\"bookmark:3.3.&#8194;Summing it up\"><span class=\"heading-before\"><span class=\"heading-label\">3.3</span>.&#8194;</span>Summing it up</h3>\n<p>We have shown how the symbolic predicate transformer <img src=\"data:image/png;base64,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\" alt=\"$SP$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> can \nbe refined into an symbolic interpreter operating over the\nsymbolic types defined in the previous section.\nIn the case when every input variable is symbolic and every\noperator is redefined, the path-condition is equivalent to the\n<em>strongest postcondition</em> of the execution path <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\">. \nThis guarantees that the path-condition for <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> is <em>sound</em>.\nIn the case where a subset of the input variables are symbolic\nand/or not all operators are redefined, the path-condition of <img src=\"data:image/png;base64,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\" alt=\"$p$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> \nis not guaranteed to be sound. We leave it as an exercise to the\nreader to establish sufficient conditions under which the \nuse of concrete values in place of symbolic expressions is \nguaranteed to result in sound path-conditions. \n</p>\n<p class=\"indent\">This section does not address the compilation of a symbolic\nexpression to the (logic) language of an underlying ATP, nor the\nlifting of a satsifying assignment to a formula back to the\nlevel of the source language. This is best done for a particular\nsource language and ATP, as detailed in the next section. \n</p><h2 id=\"sec-impl\"   style=\"bookmark:4.&#8194;Architecture of PyExZ3\"><span class=\"heading-before\"><span class=\"heading-label\">4</span>.&#8194;</span>Architecture of PyExZ3</h2>\n<p>In this section we present the high-level architecture\nof a simple DSE tool for the Python language, written in Python, called&nbsp;<a href=\"http://www.github.com/thomasjball/pyexz3\">PyExZ3</a>. \nFigure&nbsp;<a href=\"#fig-arch\" title=\"Classes in PyExZ3\" class=\"localref\" style=\"target-element:figure\"><span class=\"figure-label\">6</span></a>\nshows the class diagram (dashed edges are &#8220;has-a&#8221; relationships; solid edges\nare &#8220;is-a&#8221; relationships) of the tool. \n</p>\n<figure id=\"fig-arch\" class=\"figure align-center\"    style=\"page-align:here\">\n<p data-line=\"1\"><span data-line=\"1\"></span><img src=\"data:image/png;base64,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title=\"arch\" alt=\"arch\" style=\"width:100%\"><span data-line=\"1\"></span>\n</p>\n<hr  class=\"figureline madoko\" data-line=\"2\">\n\n<div data-line=\"3\"><span data-line=\"3\"></span><fig-caption class=\"figure-caption\"><span class=\"caption-before\"><strong>Figure&#160;<span class=\"figure-label\">6</span>.</strong> </span>Classes in PyExZ3</fig-caption><span data-line=\"3\"></span>\n</div></figure><h3 id=\"sec-loadingtesting-the-code-under-test\"   style=\"bookmark:4.1.&#8194;Loading/testing the code under test\"><span class=\"heading-before\"><span class=\"heading-label\">4.1</span>.&#8194;</span>Loading/testing the code under test</h3>\n<p>The <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Loader</span></code> class takes as input the name of a Python file (e.g., <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>foo</span><span class='token delimiter python'>.</span><span class='token identifier python'>py</span></code>) \nto import. The loader expects to find a function named\n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>foo</span></code> inside the file <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>foo</span><span class='token delimiter python'>.</span><span class='token identifier python'>py</span></code>, which will serve as the starting point\nfor symbolic execution. Thc <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>FunctionInvocation</span></code> class\nwraps this starting point. By default, each parameter to <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>foo</span></code> is\na <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicInteger</span></code> unless there is decorator <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token delimiter python'>@</span><span class='token identifier python'>symbolic</span></code> specifying\nthe type to use for a particular argument.\n</p>\n<p class=\"indent\">The loader provides the capability to reload the\nfile/module <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>foo</span><span class='token delimiter python'>.</span><span class='token identifier python'>py</span></code> so that the function <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>foo</span></code> can be \nreexecuted within the same process from the same initial\nstate with different inputs (see the class <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>ExplorationEngine</span></code>)\nvia the <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>FunctionInvocation</span></code> class. \n</p>\n<p class=\"indent\">Finally, the loader looks for specially named functions <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>expected_result</span></code>\n(<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>expected_result_set</span></code>) in file <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>foo</span><span class='token delimiter python'>.</span><span class='token identifier python'>py</span></code> to use as a test oracle after\nthe path exploration (by <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>ExplorationEngine</span></code>) has completed. These\nfunctions are expected to return a list of values to check\nagainst the list of return values collected from the executions of \nthe <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>foo</span></code> function.\nThe presence of the function <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>expected_result</span></code> (<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>expected_result_set</span></code>) \nyields a comparison of the two lists as bags (sets). We use such weaker\ntests, rather than list equality, because the order in which paths\nare explored by the <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>ExplorationEngine</span></code> can easily change due to small\ndifferences in the input programs. \n</p><h3 id=\"sec-symbolic-types\"   style=\"bookmark:4.2.&#8194;Symbolic types\"><span class=\"heading-before\"><span class=\"heading-label\">4.2</span>.&#8194;</span>Symbolic types</h3>\n<p>Python supports multiple inheritance and, more importantly,\nallows user-defined classes to inherit \nfrom its built-in types (such as <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>object</span></code> and <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>int</span></code>).\nWe use these two features two implement\nsymbolic versions of Python objects and integers, \nfollowing the instrumented type approach defined in Section&nbsp;<a href=\"#sec-semantics\" title=\"2.&#8194;Instrumented Types\" class=\"localref\" style=\"target-element:h1\"><span class=\"heading-label\">2</span></a>. \n</p>\n<p class=\"indent\">The abstract class <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicType</span></code> contains the \nsymbolic expression tree and provides basic functions for constructing \nand accessing the tree.  This class does double duty, as it is used\nto represent the (typed) symbolic constants associated with the parameters to the \nfunction, as well as the expression trees (per Section&nbsp;<a href=\"#sec-semantics\" title=\"2.&#8194;Instrumented Types\" class=\"localref\" style=\"target-element:h1\"><span class=\"heading-label\">2</span></a>). Recall\nthat the symbolic constants only appear as leaves of expression trees.\nThis means that the expression tree stored in a <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicType</span></code> will\nhave instances of a <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicType</span></code> as some of its leaves, namely\nthose leaves representing the symbolic constants.\n<!-- Need to say why this is important -->\nThe abstract class provides an <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>unwrap</span></code> method which returns\nthe pair of concrete value and expression tree associated with\nthe <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicType</span></code>, as well as a <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>wrap</span></code> method that takes a \npair of concrete value and expression tree and creates a <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicType</span></code>\nencapsulating them. \n</p>\n<p class=\"indent\">The class <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicObject</span></code> inherits from both <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>object</span></code> and <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicType</span></code> and\noverrides the basic comparison operations (<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__eq__</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__neq__</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__lt__</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__le__</span></code>,\n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__gt__</span></code>, and <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__ge__</span></code>).\nThe class <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicInteger</span></code> inherits from both <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>int</span></code> and <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicObject</span></code>\nand overrides a number of <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>int</span></code>&#39;s arithmetic methods\n(<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__add__</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__sub__</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__mul__</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__mod__</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__floordiv_</span></code>)\nand bitwise methods\n(<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__and__</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__or__</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__xor__</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__lshift__</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__rshift__</span></code>).\n</p><h3 id=\"sec-tracing-control-flow\"   style=\"bookmark:4.3.&#8194;Tracing control-flow\"><span class=\"heading-before\"><span class=\"heading-label\">4.3</span>.&#8194;</span>Tracing control-flow</h3>\n<p>As Python interprets a program, it will evaluate expressions, \nsubstituting the value of a variable in its place in an \nexpression, applying operators (methods) to \nparameter values and assigning the return values of methods\nto variables. Value of type <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicInteger</span></code> will simply flow\nthrough this interpretation, without necessitating any change \nto the program or the interpreter. This takes care\nof the case of the strongest-postcondition rule for assignment,\nas elaborated in Section&nbsp;<a href=\"#sp-refined\" title=\"3.2.&#8194;From  to DSE\" class=\"localref\" style=\"target-element:h2\"><span class=\"heading-label\">3.2</span></a>.\n</p>\n<p class=\"indent\">The strong-postcondition rule for a conditional test requires\na little more work. In Python, any object can be tested in\nan <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token keyword python'>if</span></code> or <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token keyword python'>while</span></code> condition or as the operand of a boolean operation\n(<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token keyword python'>and</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token keyword python'>or</span></code>, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token keyword python'>not</span></code>)\nThe Python base class <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>object</span></code> provides a method named <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__bool__</span></code> that\nthe Python runtime calls whenever it needs to perform such a conditional test.\nThis hook provides us what we need to trace the conditional\ncontrol-flow of a Python execution.  We override this method in the class\n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicObject</span></code> in order to inform the <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>PathToConstraint</span></code> object (defined\nlater) of the symbolic expression for the conditional (as captured by\nthe <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicInteger</span></code> subclass). \n</p>\n<p class=\"indent\">Note that the use of this hook in combination with the\ntainted types will only trace those conditionals\nin a Python execution whose values inherit from <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicObject</span></code>; \nby definition, &#8220;untainted&#8221; conditionals do not depend on symbolic \ninputs so there is no value in adding them to the path-condition.\n</p><h3 id=\"sec-recording-path-conditions\"   style=\"bookmark:4.4.&#8194;Recording path-conditions\"><span class=\"heading-before\"><span class=\"heading-label\">4.4</span>.&#8194;</span>Recording path-conditions</h3>\n<p>A <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Predicate</span></code> records a conditional (more precisly the symbolic expression\nfound in <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicInteger</span></code>) and\nwhich way it evaluated in an execution.  A <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Constraint</span></code>\nhas a <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Predicate</span></code>, a parent <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Constraint</span></code> and a set\nof <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Constraint</span></code> children. <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Constraints</span></code> form a tree, where\neach path starting from the root of the tree represents\na path-condition. The tree represents all path-conditions that have\nbeen explored so far.\n</p>\n<p class=\"indent\">The class <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>PathToConstraint</span></code> has a reference to the root of \nthe tree of <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Constraint</span></code>s\nand is responsible for installing a new <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Constraint</span></code> in the tree\nwhen notified by the overrided <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token predefined python'>__bool__</span></code> method of <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicObject</span></code>.\n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>PathToConstraint</span></code> also tracks whether or not the current execution\nis following an existing path in the tree and grows the\ntree as needed. In fact, it actually\ntracks whether or not the current execution follows a particular\n<em>expected path</em> in the tree.\n</p>\n<p class=\"indent\">The expected path is the result\nof the <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>ExplorationEngine</span></code> picking a constraint <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA30lEQVQ4y52UURHCMAyGv3AI6CFhEtAwLZMwD5UwDZUwDUgoEigOwksHIUdhJXd52d98f65NhqryLYERSEAGSs0LMANBVflVnAEFIjAYLQBThYUWYKnFxRY7g/I0+HBgNYDQMFGTyYvJiGMDEBxktOJkhHXHZcfNyNLLry6a0AqZDaD0ACwkG0jshRxEJAADr1jpjIMDAFz/gZzsB1Xthvh3z92XCuejqt5F5F5hXSEiCbhttGi6CTs7SNtQ2mHLe57YbPfyNidVHNzqB6dNVc9+oj85TXVWivsJxdY6PAB/ne4jt7G24QAAAABJRU5ErkJggg==\" alt=\"$c$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> in the tree,\nand asking the ATP if the path-condition consisting of the prefix\nof predicates up to but not including <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA30lEQVQ4y52UURHCMAyGv3AI6CFhEtAwLZMwD5UwDZUwDUgoEigOwksHIUdhJXd52d98f65NhqryLYERSEAGSs0LMANBVflVnAEFIjAYLQBThYUWYKnFxRY7g/I0+HBgNYDQMFGTyYvJiGMDEBxktOJkhHXHZcfNyNLLry6a0AqZDaD0ACwkG0jshRxEJAADr1jpjIMDAFz/gZzsB1Xthvh3z92XCuejqt5F5F5hXSEiCbhttGi6CTs7SNtQ2mHLe57YbPfyNidVHNzqB6dNVc9+oj85TXVWivsJxdY6PAB/ne4jt7G24QAAAABJRU5ErkJggg==\" alt=\"$c$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> in the tree, \nfollowed by the negation of <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA30lEQVQ4y52UURHCMAyGv3AI6CFhEtAwLZMwD5UwDZUwDUgoEigOwksHIUdhJXd52d98f65NhqryLYERSEAGSs0LMANBVflVnAEFIjAYLQBThYUWYKnFxRY7g/I0+HBgNYDQMFGTyYvJiGMDEBxktOJkhHXHZcfNyNLLry6a0AqZDaD0ACwkG0jshRxEJAADr1jpjIMDAFz/gZzsB1Xthvh3z92XCuejqt5F5F5hXSEiCbhttGi6CTs7SNtQ2mHLe57YbPfyNidVHNzqB6dNVc9+oj85TXVWivsJxdY6PAB/ne4jt7G24QAAAABJRU5ErkJggg==\" alt=\"$c$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">&#39;s predicate is satisfiable. If the \nATP returns &#8220;satisfiable&#8221; (with a new input <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAwAAAAcCAYAAABVo158AAABDUlEQVQ4y92U23GDQAxFj7aCrYES4hYowZMSSAduIeMSNiXEJVAD6cCkg5AOlB+tLQSG5Dea0cCuXlfSBVSVLQU6oACNqrLnXAA1vagqiW15du9fwG6Fs2XvgayqiBl+LYk/yr8LEJFWRAYRmUTkKiKnRUSgwHCjwH3L59luzPgETHU5dneygGktYADasOVL5VG4pwWuK7SopJvZEvBieH3znTuW2PfRY3cQa4W86CE4N865X9g3KK1xEI8CprVx+qZ9s0cg2/FtlRshe+/gNPFnMINkmavzEJxzXWB68MG/BiDdDeIeFdwgmlhhDM86iAK8q+oYK/iFZdMS+0luWiNwAD6AT6PHt6oefMUfyjONtCS7Ic0AAAAASUVORK5CYII=\" alt=\"$i$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">), then the assumption\nis that path-condition prefix is sound (that is, the execution of\nthe program on input <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAwAAAAcCAYAAABVo158AAABDUlEQVQ4y92U23GDQAxFj7aCrYES4hYowZMSSAduIeMSNiXEJVAD6cCkg5AOlB+tLQSG5Dea0cCuXlfSBVSVLQU6oACNqrLnXAA1vagqiW15du9fwG6Fs2XvgayqiBl+LYk/yr8LEJFWRAYRmUTkKiKnRUSgwHCjwH3L59luzPgETHU5dneygGktYADasOVL5VG4pwWuK7SopJvZEvBieH3znTuW2PfRY3cQa4W86CE4N865X9g3KK1xEI8CprVx+qZ9s0cg2/FtlRshe+/gNPFnMINkmavzEJxzXWB68MG/BiDdDeIeFdwgmlhhDM86iAK8q+oYK/iFZdMS+0luWiNwAD6AT6PHt6oefMUfyjONtCS7Ic0AAAAASUVORK5CYII=\" alt=\"$i$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> will follow the prefix). \n</p>\n<p class=\"indent\">However, it is possible \nfor the path-condition to be unsound and for \nthe executed path to diverge early \nfrom the expected path, due to the fact that not every operation\nhas a symbolic encoding.  The tool simply reports the divergence\nand continues to process the execution as usual (as a diverging\npath may lead to some other interesting part of the code). \n</p><h3 id=\"sec-from-symbolic-types-to-z3\"   style=\"bookmark:4.5.&#8194;From symbolic types to Z3\"><span class=\"heading-before\"><span class=\"heading-label\">4.5</span>.&#8194;</span>From symbolic types to Z3</h3>\n<p>As we have explained DSE, the symbolic expressions are \nrepresented at the level of the source language. As detailed later\nin Section&nbsp;<a href=\"#sec-int2z3\" title=\"5.&#8194;From Python Integers to Z3 Arithmetic\" class=\"localref\" style=\"target-element:h1\"><span class=\"heading-label\">5</span></a>, we must translate \nfrom the source language to the input language of an\nautomated theorem prover (ATP), in this case&nbsp;<a href=\"http://z3.codeplex.org/\">Z3</a>.  This\nseparation of languages is quite useful, as we may have\nthe need to translate a given symbolic expression\nto the ATP&#39;s language multiple times, to make use of different\nfeatures of the underlying ATP. \nFurthermore, this separation\nof concerns allows us to easily retarget the DSE tool to a\ndifferent ATP. \n</p>\n<p class=\"indent\">The base class <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Z3Expression</span></code> represents a Z3 formula. The two \nsubclasses <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Z3Integer</span></code> and <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Z3BitVector</span></code> represent different ways \nto model arithmetic reasoning about integers in Z3. We will describe\nthe details of these encodings in Section&nbsp;<a href=\"#sec-int2z3\" title=\"5.&#8194;From Python Integers to Z3 Arithmetic\" class=\"localref\" style=\"target-element:h1\"><span class=\"heading-label\">5</span></a>.\n</p>\n<p class=\"indent\">The class <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Z3Wrapper</span></code> is responsible for performing the\ntranslation from the source language (Python) to Z3&#39;s input language, \ninvoking Z3, and lifting a Z3 answer back to the level of Python. \nThe <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>findCounterexample</span></code> method does all the work, taking as\ninput a list of <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Predicate</span></code>s (called <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>assertions</span></code>) \nas well as a single <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Predicate</span></code> (called\nthe <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>query</span></code>). The <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>assertions</span></code> represent a path-condition\nprefix derived from the <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Constraint</span></code> tree that we wish the next\nexecution to follow, while <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>query</span></code> represents the predicate\nfollowing the prefix in the tree that we will negate.\n</p>\n<p class=\"indent para-continue\">The method constructs the formula\n</p>\n<div class=\"equation align-center para-block\" style=\"line-adjust:0\"><span class=\"equation-before\"><span class=\"equation-label\">(4)</span></span>\n\n<div class=\"math para-block input-math\"   style=\"line-adjust:0\"><img src=\"data:image/png;base64,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\" alt=\"\\[(\\bigwedge_{a \\in asserts} a) \\wedge \\neg query\\]\" class=\"math-display math\" style=\"vertical-align:-0.000em;height:2.342em\"></div></div>\n<p>and asks Z3 if it is satisfiable. The method performs\na standard syntactic &#8220;cone of influence&#8221; (CIF)\nreduction on the <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>asserts</span></code> with respect to the \n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>query</span></code> to shrink the size of the formula. For example,\nif <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>asserts</span></code> is the set of predicates <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAdQAAAArCAYAAADBlHkUAAANSElEQVR42u2d7ZXiuBKGXxEBzUZwIQN6bgQXMoDdCBYygLMR7KEzMBPBDJ0BbAQ77gzMRrBABro/XJpWa/wp25LlKZ3D6e4Z29jS4/pSqSSklKjahBArAEsAUwCfAByllHs0aEKIuZTyDdx63YQQcwBXKeUjtPtmvpgpZoqZctFGFW9yJ4S4AzgBmACIAPyviTIVQoyFEAkpZ279bxMAsRBiGsiLxXwxU8wUM2UzzrEQIhFCnIUQkRBi0ZpCFUKcABzoz2cp5VpK+drEQqOH/QfAq5TylRnof5NSXoiD3gtA5ouZYqaYKVumpJTPUsoZgD0p6LMQ4i6E2JRyUhTyFULsNGW6lVIeW4IzIbd8yQiE1YQQEYAFGVePnt4j8+Wmn1cAtuSxTQDcAFwARHUMbmaKW5+Zons60Z8vRZHZMg/1D/r5aFGZqrDxmvEJ0gLc0q+nnr6QzJcDb40UzGcAZxJaT0jzK1TILWKmuA2BKYpIKP23o3na3INzPwAkfeKi46p+AGzoeqs2rscfPx8AcxrHXc/ui/lyN/Z3AOOcY3Z0TJJ3DDPFn5CY0q4lyVvOPK4s5Kv+89JGqIMSm65Syme2yYK3KE8AVgCe+hKmY76chT6nANZFc4kaH69SyjUzxS1kpoQQY1L4hfpw5LDTDgDGSCd6uYXf1Dh+7slLyXy56eMpKZiyxJw/6eeqRpYkM8VM9ZKpqsrYiYeqafcrZU9xG8bLEFMo5Nnnujzmy1k/K3lw1Oaoyry7cR35wUwxU31lqoo+dOWhquSmiBEaVFMW49bzfTBf3Qu+lfZnXPG0b/RzQQqKmeI2aKZcKVS1fufIGGVbw1Q8YxHSfWshmo3nW2G+uufqN+33a8Vz9DDZgplyMuZTIcQhkGIZg2Nq5GCAF+SiB1dizJHAOyANLR0QZlr+Gz3LxlMfMl9uuNKF163iObqQXDJTTtoCaUZsIoQ49VyxDo4pFx7qWn8gbt+tyIgE3o76Zl1lDqGH7Zsxzq4b89UxVxRaG+d4CUXtX+33T8xU943qBTwDeEWa3apK6C16xuogmXKhUNVAnlngibkQ4ox0LdUGaQWQZyp1FWo5s3OGtenDyj0zV51xNTH+rupN6EJyzEw5U6pvtKxkhjRkvUBaPi825i19tkEyNer4RR/jvZD0twbXWZCVdadPbLruZJ2fqKhx5jEeBd6CMs1iGtBXADMp5dJVJiMJ3Ujrn4TCguZ41W0XfQw8WLmN+XLQR6Fz1cYzT34GptSYaPzsqnynVpC9Nb6klFeKTjwBeEGa6apkpG/ZOEymCipDjPFeGeJsWV1ipa7RoEJFpARGRtWMk1bF4m4cE8Fz5RXyFhKtHyNUrPLR8n2c1DgCmBp9dKaxvtMxc4vrq3M3jp+rMV+u+ihkrkhZS+0zrlllqPYYhcgUyaVE8aPJoEPJeQetn+YdP99O61s1NeBDJgXHVBV9OKqo/W0n5pUlcLW0FiOC85l2EVBGwAtdc0WW1l9It5O7aOcpC+w3D56D2u4uon7cI63UsXWZ5KDVx1whXee1lFJetX5Uc2uxZjFOLL5KXdN1NZlGfDnuo5C5svUmbg2+MyimaI7yDwq1q3MT+lnmDS70cG3H4eAXqpG7pfE5ALhTZrDL6EvITE1sQr66W/y35RfPbBUyAbooWLyrOicC8NUA8deM4zoPFRGUkiC9Id2h54kgdpotSC9HTON4KUhMORhj/a0B5K4zCq358tBHg+CqBSaHytQBaQKYft5/KyoPVWzdWRKUlPJIBSvW9L07UqynUPan9cDUtez8IoWqCxfbtVjTBlbFCcULcacGzHr7k16IC4DfHXiBWZmVs7Z26LFsf2kvclFmm64c3iwF9MOT8GvCl+s+GgpX1pZ8zX4Lhim1zEaPohme57XkXNUurgeI9rZ+RroE5YKPmcFzZuoHowkAxnlZ06O80BJ1LMgafjR8+EdNYbICcMsANDM0o4fo9LAGhe86EX6UwHLCx8zKZR8ydmn81MtwLOmDTy280LeyUEjHL9cjgD4KmSsfXnBITG1hVFQiGaaMtaJqS3oEzltWsZRSldObIU1uWyDdMq2rzODgmCJDVr2DmZ78KONlPpMmvtKL3MQaHlt6ENcij8GwnJxadUZm5Yo6+JmU9wWeG4VADoanj4ov9JeAXg5rvjz1Uchc3Vq4xmPATH3Bj1E8PXejSIbq86fe5QdlBq+RZgYf0V1mcJBMUd9siZPErEo1opc5pjkalX6/p9BS0wG28iBoHdVbFQhdWXVa8sqZIDsiXaKw9lnEO6Pp0D8qjOFK73fL7/zXEEZ991B99FHIXD1y+r1LARoMUxQ2fRgGm2Imd4qAjnM+f1rxmR7akps99UtEy4FWPzNT5GQ+kTOnqlJJIcRhRAc80wFLsowPZJE0jaHfOhrvpQcPdYH3MPMb0k1mr+hfqzz3bay5uyC8dhtAH4XA1S3Hiytrsx54nD5klp4UWRTudTaV0ESxkk74qo398mdmSivruUC6vncmpRRSyv3IsEgu5NKu6SVvK37etpX5fbLfVZajUdJrjvf5hd6U9DKEfxXvvS1P/xfPQnMcQB8FyxW9Yw8Lb0Ifl29DZarEYPtacNy6S7ZakCfmfH5rJVJDZYqmRHd0jZmUcq8bwKO8EIZmuTdJo340cOfzHshbVlxGSa850pJeSU9Kek2N+y3rn7Y8fdu58qbNhi9ffRQyV6bwmlr0dTxgpkyDTUX2LiUGf6/mT3UZS4pDzed3VSI1KKYokVGN2TorklS0bEYPVdjuWH/twEOtlBVHc1OdpH1nlPSakuFx91zSa5rR90VNhaYe+nwd9V0dD8k627Zhs+HLVx+FzBXwMXFrXJMLG2MkJKbyIhpvFSMlvVCmQogVJcad4aZEamhMbTVZkPndowpg1bEe8q7RZup7VQ/1hI7XsFGYfC+lFJrRoSbuDx7qvl6rKguKOoxzQidbzcquY/ldPT3vJIA+Cpkr4GPosuoc2veEG4t54ZCYyjPY/q4ox7yGe4UQG0qMO9GYvSCtwLXueD4/NKZK1yiPil7qFjosqauQqTKMssoPGQJOdei1IHtujrRkobN1exklvVTlkchV5RHDiiwbv22BJb2paf1N6oBKIaWxD7489lGwXGny4JihDHK9He1PmwhXMExVNN7M5iOx0pSzeaUs9y5yU/rOlNW4Ni0GXHL+HPWLH0fIKX6Mj0Wk44JrxABWnguYL+g+1P2eoBVd7/B747Ixw4+FqVfGmCWWRaNXFY5N6hzfNl+++ih0rrQNMyoVGCev6/sGFg0KkQfBVE7x9mnOMVNYFnhvaQx1OZrA8QYEITBlqw87VajGNRYVjz/rwsKA8K4Bey/YNeTkU5lmvKD6M52r9kUXAgFUWsx48eeGQXPoQgBB28mjaAy75MtHHw2BqxxDY16wo4kS1uMulVpfmMrZpWuT8/9JG7K15n1NDWfFu9PRV6b6rlCVNX2oaeFFRocnCnBkbIukWe69Uaa+gNbgOyt4tO+PtX9TAnmnbwtmaZEnFls2yaZKoC5fPvpoaIJSeeiqX4z+O5n92sDLC4qpDCV/VwqCFKnadk83hHZDNLxCZqrvCvVQ9xoEZExA3mHsUVlwzKKPytRHyIUEzVnrnzjHYj7QPWT2c40QfVTz+KQN4W/Dl+s+Gmooj4RUjI/7a55aUGjBMmUYOieNGzVXOdYURGf7n+ZMDcwDkI+9YKrPCnXaVihmSB96sczNflcBPkcC+43Jo6ZCn/kaHldDZ6rN8HTOvZubz0/5vbBnqq0NxtvK5LpSFtu4T9VffDdaGmFmcC5DegbKMp1SxrXNOrVJ04w75mtYXA2dKeN6XWzDt8CPm89ff+Z3ogWmKreRo2dSRSLWrEozX+AjbUawDezWt8b41m1ztLNZN/M1HK6CZIoKfcSqSHrBoWq5xwPpvs1djHlwm8/3nKnqyptc2TxI1H+qvfKaWAkJWQlPPNCDsfzu9BI/WZw7B/CZNmZo416YL2bKG1NUqk+vMT7LOGaF9+pAa9/7JjNTta9Tqg+reqhtVDpSVsKGh3gQkG6Qztf9bnmJP1q20JkvZsonU3ohiNeMZxsD+Ex/vrAyDYapVj3UO93Mo6l216y4TwD+w17EIKy+q403QJ7EKcuKZ76YqRCZouMmSLNIjxnKNCal+yKl3PNo958p4zoq2QsAXmkzi9oeqgKjrcl5tdP5gYc6aEgPtlYfCZcTupnvZL6YKV9M7Ulh3vT7oh1K/iFlu2RlGgZTBdEKAPiSe1SNRc4JWki/xvui5wUvcwgy/VxVHNk0WA+26vD+mC9mygtT+Lj2VC1ZOsNjaT9mqnnf42OVpcLCQYUhX03TRzSPoDLTXpukYpPlsEG6x96VbalgLL4xWdtf+5w5ynwxU8wUM9WUKZpG2OJ9Dr00XF9JoWo3+SvSNW1qW6tISvliebMRWXwzRiAYUGOk8xHrAO6V+WKmmClmyuZ8ld39QLqmOSLlXJqXUVmhdvTgOwC/8LxCEJAGN1bMF48P3zMzZePh2iY1/h91yI6Gqn8CUQAAAABJRU5ErkJggg==\" alt=\"$\\{ (x&lt;a), (a&lt;0), (y&gt;0) \\}$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\"> and\nthe query is <img src=\"data:image/png;base64,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\" alt=\"$(x=0)$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\">, then the CIF yields the\nset  <img src=\"data:image/png;base64,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\" alt=\"$\\{ (x&lt;a), (a&lt;0) \\}$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\">, which does not include the predicate <img src=\"data:image/png;base64,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\" alt=\"$(y&gt;0)$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\">,\nas the variable <img src=\"data:image/png;base64,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\" alt=\"$y$\" class=\"math-inline math\" style=\"vertical-align:-0.210em;height:0.676em\"> is not in the set of variables (transitively)\nrelated to variable <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABUAAAASCAYAAAC0EpUuAAABHklEQVQ4y6WUYY3DMAyFP08HoBh6DIohY7CjUAjlMAjDUAg5Cj0GHYVC8P04Z/KsrOl0liLV8fPTSxo/VJW4gARkYLO1AGPA9MAMrBFTI7wZILm9CVBgtnwwohT6FJhqhPmF+tWaRiMcKoQKLPHIa43Q6tk13kJtc7U5FtIO6eoa+1CbrD8DXdm87Kk0TCHcxakqJ/7iDnzxIkRkcOk3jfgAUNWfBi6579wiPXEszu8oFbuvfZBIAd1V9fPfSkUkvaPy6PHPR+5TRPrHD209DxvZ8py6Bu5SrrMzY9iAa8U0mu+zeIHLn+ZWA/jqZ/qIykLqZ3oOKjczEPVKAuHs+wrpGE2imEvxAneaa8AskfDhpzb7izPlXDGNGqZqQL/kpe0Z3roFswAAAABJRU5ErkJggg==\" alt=\"$x$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">.\n</p>\n<p class=\"indent\">If the formula is satisfiable a model is requested\nfrom Z3 and lifted back to Python&#39;s type universe.  Note that\nbecause of the CIF reduction, the model may not mention certain\ninput variables, in which case we simply keep their values from\nthe execution from which the <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>asserts</span></code> and <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token identifier python'>query</span></code> were derived. \n</p><h3 id=\"sec-putting-it-all-together\"   style=\"bookmark:4.6.&#8194;Putting it all together\"><span class=\"heading-before\"><span class=\"heading-label\">4.6</span>.&#8194;</span>Putting it all together</h3>\n<p>The class <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>ExplorationEngine</span></code> ties everything together. It kicks off\nan execution of the Python code under test using <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>FunctionInvocation</span></code>.\nAs the Python code executes, building symbolic expressions via <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>SymbolicType</span></code>\nand its subclasses, callbacks to <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>PathToConstraint</span></code> create\na path-condition, represented by <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Constraint</span></code> and <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Predicate</span></code>. \nNewly discovered <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Constraints</span></code> are added to the end of a deque maintained by\n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>ExplorationEngine</span></code>.\n</p>\n<p class=\"indent\">Given the first seed execution, <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>ExplorationEngine</span></code> starts the work of \nexploring paths in a breadth-first fashion. It removes a <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Constraint</span></code> <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA30lEQVQ4y52UURHCMAyGv3AI6CFhEtAwLZMwD5UwDZUwDUgoEigOwksHIUdhJXd52d98f65NhqryLYERSEAGSs0LMANBVflVnAEFIjAYLQBThYUWYKnFxRY7g/I0+HBgNYDQMFGTyYvJiGMDEBxktOJkhHXHZcfNyNLLry6a0AqZDaD0ACwkG0jshRxEJAADr1jpjIMDAFz/gZzsB1Xthvh3z92XCuejqt5F5F5hXSEiCbhttGi6CTs7SNtQ2mHLe57YbPfyNidVHNzqB6dNVc9+oj85TXVWivsJxdY6PAB/ne4jt7G24QAAAABJRU5ErkJggg==\" alt=\"$c$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">\nfrom the front of its deque and, if <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA30lEQVQ4y52UURHCMAyGv3AI6CFhEtAwLZMwD5UwDZUwDUgoEigOwksHIUdhJXd52d98f65NhqryLYERSEAGSs0LMANBVflVnAEFIjAYLQBThYUWYKnFxRY7g/I0+HBgNYDQMFGTyYvJiGMDEBxktOJkhHXHZcfNyNLLry6a0AqZDaD0ACwkG0jshRxEJAADr1jpjIMDAFz/gZzsB1Xthvh3z92XCuejqt5F5F5hXSEiCbhttGi6CTs7SNtQ2mHLe57YbPfyNidVHNzqB6dNVc9+oj85TXVWivsJxdY6PAB/ne4jt7G24QAAAABJRU5ErkJggg==\" alt=\"$c$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> has not been already &#8220;processed&#8221;, \nuses <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Z3Wrapper</span></code> to find a new input (as discussed in the previous section)\nwhere <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA30lEQVQ4y52UURHCMAyGv3AI6CFhEtAwLZMwD5UwDZUwDUgoEigOwksHIUdhJXd52d98f65NhqryLYERSEAGSs0LMANBVflVnAEFIjAYLQBThYUWYKnFxRY7g/I0+HBgNYDQMFGTyYvJiGMDEBxktOJkhHXHZcfNyNLLry6a0AqZDaD0ACwkG0jshRxEJAADr1jpjIMDAFz/gZzsB1Xthvh3z92XCuejqt5F5F5hXSEiCbhttGi6CTs7SNtQ2mHLe57YbPfyNidVHNzqB6dNVc9+oj85TXVWivsJxdY6PAB/ne4jt7G24QAAAABJRU5ErkJggg==\" alt=\"$c$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> is the query (to be negated) and the path to <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA30lEQVQ4y52UURHCMAyGv3AI6CFhEtAwLZMwD5UwDZUwDUgoEigOwksHIUdhJXd52d98f65NhqryLYERSEAGSs0LMANBVflVnAEFIjAYLQBThYUWYKnFxRY7g/I0+HBgNYDQMFGTyYvJiGMDEBxktOJkhHXHZcfNyNLLry6a0AqZDaD0ACwkG0jshRxEJAADr1jpjIMDAFz/gZzsB1Xthvh3z92XCuejqt5F5F5hXSEiCbhttGi6CTs7SNtQ2mHLe57YbPfyNidVHNzqB6dNVc9+oj85TXVWivsJxdY6PAB/ne4jt7G24QAAAABJRU5ErkJggg==\" alt=\"$c$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> in the \n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Constraint</span></code> tree forms the assertions. \n</p>\n<p class=\"indent\">A <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Constraint</span></code> <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA30lEQVQ4y52UURHCMAyGv3AI6CFhEtAwLZMwD5UwDZUwDUgoEigOwksHIUdhJXd52d98f65NhqryLYERSEAGSs0LMANBVflVnAEFIjAYLQBThYUWYKnFxRY7g/I0+HBgNYDQMFGTyYvJiGMDEBxktOJkhHXHZcfNyNLLry6a0AqZDaD0ACwkG0jshRxEJAADr1jpjIMDAFz/gZzsB1Xthvh3z92XCuejqt5F5F5hXSEiCbhttGi6CTs7SNtQ2mHLe57YbPfyNidVHNzqB6dNVc9+oj85TXVWivsJxdY6PAB/ne4jt7G24QAAAABJRU5ErkJggg==\" alt=\"$c$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> in the tree is considered &#8220;processed&#8221; if an execution \nhas covered <img src=\"data:image/png;base64,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\" alt=\"$c&#39;$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.833em\">, a sibling of <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA30lEQVQ4y52UURHCMAyGv3AI6CFhEtAwLZMwD5UwDZUwDUgoEigOwksHIUdhJXd52d98f65NhqryLYERSEAGSs0LMANBVflVnAEFIjAYLQBThYUWYKnFxRY7g/I0+HBgNYDQMFGTyYvJiGMDEBxktOJkhHXHZcfNyNLLry6a0AqZDaD0ACwkG0jshRxEJAADr1jpjIMDAFz/gZzsB1Xthvh3z92XCuejqt5F5F5hXSEiCbhttGi6CTs7SNtQ2mHLe57YbPfyNidVHNzqB6dNVc9+oj85TXVWivsJxdY6PAB/ne4jt7G24QAAAABJRU5ErkJggg==\" alt=\"$c$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\"> in the tree that represents\nthe negation of the predicate associated with <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA30lEQVQ4y52UURHCMAyGv3AI6CFhEtAwLZMwD5UwDZUwDUgoEigOwksHIUdhJXd52d98f65NhqryLYERSEAGSs0LMANBVflVnAEFIjAYLQBThYUWYKnFxRY7g/I0+HBgNYDQMFGTyYvJiGMDEBxktOJkhHXHZcfNyNLLry6a0AqZDaD0ACwkG0jshRxEJAADr1jpjIMDAFz/gZzsB1Xthvh3z92XCuejqt5F5F5hXSEiCbhttGi6CTs7SNtQ2mHLe57YbPfyNidVHNzqB6dNVc9+oj85TXVWivsJxdY6PAB/ne4jt7G24QAAAABJRU5ErkJggg==\" alt=\"$c$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">, or if constraint <img src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAA30lEQVQ4y52UURHCMAyGv3AI6CFhEtAwLZMwD5UwDZUwDUgoEigOwksHIUdhJXd52d98f65NhqryLYERSEAGSs0LMANBVflVnAEFIjAYLQBThYUWYKnFxRY7g/I0+HBgNYDQMFGTyYvJiGMDEBxktOJkhHXHZcfNyNLLry6a0AqZDaD0ACwkG0jshRxEJAADr1jpjIMDAFz/gZzsB1Xthvh3z92XCuejqt5F5F5hXSEiCbhttGi6CTs7SNtQ2mHLe57YbPfyNidVHNzqB6dNVc9+oj85TXVWivsJxdY6PAB/ne4jt7G24QAAAABJRU5ErkJggg==\" alt=\"$c$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.468em\">\nhas been removed from the deque. \n</p><h2 id=\"sec-int2z3\"   style=\"bookmark:5.&#8194;From Python Integers to Z3 Arithmetic\"><span class=\"heading-before\"><span class=\"heading-label\">5</span>.&#8194;</span>From Python Integers to Z3 Arithmetic</h2>\n<p>In languages such as C and Java, integers are finite-precision,\ngenerally limited to the size of a machine word (32 or 64 bits, for example).\nFor such languages, satisfiability of finite-precision integer arithmetic\nis decidable and can be reduced to Z3&#39;s theory of bit-vectors, where\neach arithmetic operation is encoded by a circuit. This translation permits \nreasoning about non-linear arithmetic problems, such as \n<img src=\"data:image/png;base64,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\" alt=\"$\\exists x,y,z : x*z + y \\leq (z/y)+5$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\">.\n</p>\n<p class=\"indent\">Python (3.0) integers, however, are not finite-precision. They are only\nlimited by the size of machine memory. This means, for example, that\nPython integers don&#39;t overflow or underflow. It also means that\nwe can&#39;t hope to decide algorithmically whether or not a given\nequation over integer variables has a solution in general. Hibert&#39;s famous\n10th problem and its solution by Matiyasevich tells us that it is\nundecidable whether or not a polynomial\nequation of the form <img src=\"data:image/png;base64,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\" alt=\"$p(x_1, \\ldots, x_n) = 0$\" class=\"math-inline math\" style=\"vertical-align:-0.280em;height:1.119em\"> with integer coefficients\nhas an solution in the integers.\n</p>\n<p class=\"indent\">This means that we will resort to heuristic approaches in our use\nof the&nbsp;<a href=\"http://z3.codeplex.org/\">Z3</a> ATP.  The special case of linear integer arithmetic (LIA)\nis decidable and supported by Z3. In order to deal with non-linear operations,\nwe use uninterpreted functions (UF). Thus, if Z3 returns\n&#8220;unsatisfiable&#8221; we know that there is no solution, but if the\nZ3 &#8220;satisfiable&#8221;, we must treat the answer as a &#8220;don&#39;t know&#8221;. \nThe class <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Z3Integer</span></code> is used to translate a symbolic expression\ninto the theory LIA+UF and check for unsatisfiability. We leave it as an\nimplementation exercise to check if a symbolic expression can\nbe converted to LIA (without the use of UF) in order to make\nuse of &#8220;satisfiable&#8221; answers from the LIA solver.\n</p>\n<p class=\"indent\">If the translation to <code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Z3Integer</span></code> does not return &#8220;unsatisfiable&#8221;, we\nuse Z3&#39;s bit vector decision procedure (via the class\n<code class=\"code code1 language-python lang-python python highlighted\" ><span class='token namespace identifier python'>Z3BitVector</span></code>) to heuristically\ntry to find satisfiable answers, even in the presence of non-linear \narithmetic. We start with bitvectors of size <img src=\"data:image/png;base64,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\" alt=\"$N=32$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.754em\"> and <em>bound</em> the values\nof the symbolic constants to fit within 8 bits in order to find \nsatisfiable solutions\nwith small values. Also, because Python integers do not overflow/underflow, \nthe bound helps us reserve space in the bitvector to allow the\nresults of operations to exceed the bound while not overflowing\nthe bitvector. As long as Z3 returns &#8220;unsatisfiable&#8221; we increase\nthe bound. If the bound reaches <img src=\"data:image/png;base64,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\" alt=\"$N$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">, we increase <img src=\"data:image/png;base64,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\" alt=\"$N$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> by 8 bits,\nleaving the bound where it is and continue. \n</p>\n<p class=\"indent\">If Z3 returns\n&#8220;satisfiable&#8221;, it may be the case that Z3 found a solution\nthat involved overflow in the bitvector world of arithemetic\n(modulo <img src=\"data:image/png;base64,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\" alt=\"$2^N-1$\" class=\"math-inline math\" style=\"vertical-align:-0.084em;height:0.937em\">). Therefore,\nthe solution is validated back in the\nPython world by evaluating the formula under that solution\nusing Python semantics. \nIf the formula does not evaluate to the same\nvalue in both worlds, then we increase <img src=\"data:image/png;base64,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\" alt=\"$N$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\"> by 8 bits (to \ncreate a gap between the bound and <img src=\"data:image/png;base64,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\" alt=\"$N$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.728em\">) and continue to search\nfor a solution.\n</p>\n<p class=\"indent\">The process terminates when we find a valid satisfying solution \nor <img src=\"data:image/png;base64,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\" alt=\"$N=64$\" class=\"math-inline math\" style=\"vertical-align:-0.000em;height:0.781em\"> and the bound reaches 64 (in which case, we return &#8220;don&#39;t know&#8221;).\n</p><h2 id=\"sec-extensions\"   style=\"bookmark:6.&#8194;Extensions\"><span class=\"heading-before\"><span class=\"heading-label\">6</span>.&#8194;</span>Extensions</h2>\n<p>We have presented the basics of dynamic symbolic execution \n(for Python).\nA more thorough treatment would deal with other data types besides\nintegers, such as Python dictionaries, strings and lists, each\nof which presents their own challenges for symbolic reasoning. \nThere are many other interesting challenges in DSE, such\nas dealing with user-defined classes (rather than built-in types\nas done here) and multi-threaded execution. \n</p><h2 id=\"sec-acknowledgements\"   style=\"bookmark:7.&#8194;Acknowledgements\"><span class=\"heading-before\"><span class=\"heading-label\">7</span>.&#8194;</span>Acknowledgements</h2>\n<p>Many thanks to the students of the 2014 Marktoberdorf Summer School\non Dependable Software Systems Engineering\nfor their questions and feedback about the first author&#39;s lectures on dynamic\nsymbolic execution. The following students of the summer school\nhelpfully provided tests for the&nbsp;<a href=\"http://www.github.com/thomasjball/pyexz3\">PyExZ3</a> tool: Daniel Darvas,\nDamien Rusinek, Christian Dehnert and Thomas Pani. Thanks also to Peter\nChapman for his contributions. \n</p><!-- \n# From Python Dictionaries to Z3 Maps { #sec-dict2z3 }\n\nPython dictionaries, which map keys to values,\nintroduce mutation into the picture. The methods are:\n\n- Length:     `__length__(self)` - will be maintained concretely, as cardinality constraints\nare beyond the scope of what we can handle\n- Get:     `__getitem__(self,key)` - Select\n- Set:     `__setitem__(self,key,value)` - Update\n- Lookup:     `__contains__(self,key)` - \n- Delete:     `__delitem__(self,key)` - Update \n\n\nA dictionary can be mutated via the `__setitem__` and '__delitem__` methods.\nThe keys stored  in Python dictionary must be hashable values. \nAll of Python's immutable\nbuilt-in objects (such as integers, tuples and strings) are hashable. \nUser-defined classes give rise to hashable mutable objects, where\nthe hash value is the object id (an immutable field of the object).\n\nIdeally, symbolic expression trees should contain only\nimmutable values, as they are a pure function of the initial\n(immutable) state of a program.  However, the reality is that\nthe values associated with keys in a dictionary can be mutable.\nFor example, a dictionary itself can be a value in another\ndictionary.    \n\nAs we will see, for the purposes of modelling a dictionary symbolically, \nhowever, the object id of the value in a (key,value) pair is all that\nneeds to be recorded. The object id is immutable, as mentioned before. \nThus, we can represent the state of the dictionary via an append-only log \nof updates where the parameters \nto each update are immutable values (set and delete are the update\noperations).  Each lookup/get/length operation hangs off of a particular\nprefix of the log. \n\nWe model the dictionary as initially\nempty, containing no mapping (rejects all lookups and get, length is zero).\n\n\n\nThe essential axioms\n\n* Initially empty\n* Select\n* Update\n\nInverting select means that we need disequality on values in\nthe dictionary (SymbolicObject)\n\nEquality of dictionaries (extensional arrays)\n\n-->\n\n<h2 id=\"sec-references\" >References</h2>\n<div class=\"tex input-tex\">\n<div class=\"bibliography bib-numeric\"   data-line=\"1\" style=\"bibstyle:plainnat\">\n\n<div id=\"cadare05\" class=\"bibitem\"   data-line=\"4\" style=\"searchterm:Cristian+Cadar+Dawson+Engler+Execution+generated+test+cases+make+systems+code+crash+itself+_Proceedings+International+SPIN+Workshop_+pages++\"><span data-line=\"5\"></span><span class=\"bibitem-before\">[<span class=\"bibitem-label\">1</span>]&#160;&#160;</span><span data-line=\"5\"></span>Cristian Cadar and Dawson<span data-line=\"5\"></span>&#160;<span data-line=\"5\"></span>R. Engler.\n<span data-line=\"6\"></span><span class=\"newblock\"></span><span data-line=\"6\"></span> Execution generated test cases: How to make systems code crash\n  itself.\n<span data-line=\"8\"></span><span class=\"newblock\"></span><span data-line=\"8\"></span> In <span data-line=\"8\"></span><em>Proceedings of 12th International SPIN Workshop</em><span data-line=\"8\"></span>, pages\n  2<span data-line=\"9\"></span>&#8211;<span data-line=\"9\"></span>23, 2005.<span data-line=\"9\"></span>&nbsp;<a href=\"http://www.bing.com/search?q=Cristian+Cadar+Dawson+Engler+Execution+generated+test+cases+make+systems+code+crash+itself+_Proceedings+International+SPIN+Workshop_+pages++\" class=\"bibsearch\">&#128270;</a></div>\n<div id=\"cadars13\" class=\"bibitem\"   data-line=\"12\" style=\"searchterm:Symbolic+execution+software+testing+three+decades+later++Cristian+Cadar+Koushik+\"><span data-line=\"13\"></span><span class=\"bibitem-before\">[<span class=\"bibitem-label\">2</span>]&#160;&#160;</span><span data-line=\"13\"></span>Cristian Cadar and Koushik Sen.\n<span data-line=\"14\"></span><span class=\"newblock\"></span><span data-line=\"14\"></span> Symbolic execution for software testing: three decades later.\n<span data-line=\"15\"></span><span class=\"newblock\"></span><span data-line=\"15\"></span> <span data-line=\"15\"></span><em>Communications of the ACM</em><span data-line=\"15\"></span>, 56<span data-line=\"15\"></span><span style=\"penalty:0\"></span><span data-line=\"15\"></span> (2):<span data-line=\"15\"></span><span style=\"penalty:0\"></span><span data-line=\"15\"></span> 82<span data-line=\"15\"></span>&#8211;<span data-line=\"15\"></span>90,\n  2013.<span data-line=\"16\"></span>&nbsp;<a href=\"http://www.bing.com/search?q=Symbolic+execution+software+testing+three+decades+later++Cristian+Cadar+Koushik+\" class=\"bibsearch\">&#128270;</a></div>\n<div id=\"cadargpde06\" class=\"bibitem\"   data-line=\"19\" style=\"searchterm:Cristian+Cadar+Vijay+Ganesh+Peter+Pawlowski+David+Dill+Dawson+Engler+automatically+generating+inputs+death+_Proceedings+Conference+Computer+Communications+Security_+pages++\"><span data-line=\"20\"></span><span class=\"bibitem-before\">[<span class=\"bibitem-label\">3</span>]&#160;&#160;</span><span data-line=\"20\"></span>Cristian Cadar, Vijay Ganesh, Peter<span data-line=\"20\"></span>&#160;<span data-line=\"20\"></span>M. Pawlowski, David<span data-line=\"20\"></span>&#160;<span data-line=\"20\"></span>L. Dill, and Dawson<span data-line=\"20\"></span>&#160;<span data-line=\"20\"></span>R.\n  Engler.\n<span data-line=\"22\"></span><span class=\"newblock\"></span><span data-line=\"22\"></span> EXE: automatically generating inputs of death.\n<span data-line=\"23\"></span><span class=\"newblock\"></span><span data-line=\"23\"></span> In <span data-line=\"23\"></span><em>Proceedings of the 13th ACM Conference on Computer and\n  Communications Security</em><span data-line=\"24\"></span>, pages 322<span data-line=\"24\"></span>&#8211;<span data-line=\"24\"></span>335, 2006.<span data-line=\"24\"></span>&nbsp;<a href=\"http://www.bing.com/search?q=Cristian+Cadar+Vijay+Ganesh+Peter+Pawlowski+David+Dill+Dawson+Engler+automatically+generating+inputs+death+_Proceedings+Conference+Computer+Communications+Security_+pages++\" class=\"bibsearch\">&#128270;</a></div>\n<div id=\"clarke76\" class=\"bibitem\"   data-line=\"27\" style=\"searchterm:+system+generate+test+data+symbolically+execute+programs++Lori+Clarke+\"><span data-line=\"28\"></span><span class=\"bibitem-before\">[<span class=\"bibitem-label\">4</span>]&#160;&#160;</span><span data-line=\"28\"></span>Lori<span data-line=\"28\"></span>&#160;<span data-line=\"28\"></span>A. Clarke.\n<span data-line=\"29\"></span><span class=\"newblock\"></span><span data-line=\"29\"></span> A system to generate test data and symbolically execute programs.\n<span data-line=\"30\"></span><span class=\"newblock\"></span><span data-line=\"30\"></span> <span data-line=\"30\"></span><em>IEEE Transactions on Software Engineering</em><span data-line=\"30\"></span>, 2<span data-line=\"30\"></span><span style=\"penalty:0\"></span><span data-line=\"30\"></span>\n  (3):<span data-line=\"31\"></span><span style=\"penalty:0\"></span><span data-line=\"31\"></span> 215<span data-line=\"31\"></span>&#8211;<span data-line=\"31\"></span>222, 1976.<span data-line=\"31\"></span>&nbsp;<a href=\"http://www.bing.com/search?q=+system+generate+test+data+symbolically+execute+programs++Lori+Clarke+\" class=\"bibsearch\">&#128270;</a></div>\n<div id=\"demourab08\" class=\"bibitem\"   data-line=\"34\" style=\"searchterm:+efficient+solver++Leonardo+Mendon+Moura+Nikolaj+rner+\"><span data-line=\"35\"></span><span class=\"bibitem-before\">[<span class=\"bibitem-label\">5</span>]&#160;&#160;</span><span data-line=\"35\"></span>Leonardo<span data-line=\"35\"></span>&#160;<span data-line=\"35\"></span>Mendon<span data-line=\"35\"></span>&#231;<span data-line=\"35\"></span>a de<span data-line=\"35\"></span>&#160;<span data-line=\"35\"></span>Moura and Nikolaj Bj<span data-line=\"35\"></span>&#248;<span data-line=\"35\"></span>rner.\n<span data-line=\"36\"></span><span class=\"newblock\"></span><span data-line=\"36\"></span> Z3: an efficient SMT solver.\n<span data-line=\"37\"></span><span class=\"newblock\"></span><span data-line=\"37\"></span> In <span data-line=\"37\"></span><em>Proceedings of the 14th International Conference of Tools\n  and Algorithms for the Construction and Analysis of Systems</em><span data-line=\"38\"></span>, pages 337<span data-line=\"38\"></span>&#8211;<span data-line=\"38\"></span>340,\n  2008.<span data-line=\"39\"></span>&nbsp;<a href=\"http://www.bing.com/search?q=+efficient+solver++Leonardo+Mendon+Moura+Nikolaj+rner+\" class=\"bibsearch\">&#128270;</a></div>\n<div id=\"dijkstra76\" class=\"bibitem\"   data-line=\"42\" style=\"searchterm:+Discipline+Programming_++Edsger+Dijkstra+\"><span data-line=\"43\"></span><span class=\"bibitem-before\">[<span class=\"bibitem-label\">6</span>]&#160;&#160;</span><span data-line=\"43\"></span>Edsger<span data-line=\"43\"></span>&#160;<span data-line=\"43\"></span>W. Dijkstra.\n<span data-line=\"44\"></span><span class=\"newblock\"></span><span data-line=\"44\"></span> <span data-line=\"44\"></span><em>A Discipline of Programming</em><span data-line=\"44\"></span>.\n<span data-line=\"45\"></span><span class=\"newblock\"></span><span data-line=\"45\"></span> Prentice-Hall, 1976.<span data-line=\"45\"></span>&nbsp;<a href=\"http://www.bing.com/search?q=+Discipline+Programming_++Edsger+Dijkstra+\" class=\"bibsearch\">&#128270;</a></div>\n<div id=\"godefroid11\" class=\"bibitem\"   data-line=\"48\" style=\"searchterm:Higher+order+test+generation++Patrice+Godefroid+\"><span data-line=\"49\"></span><span class=\"bibitem-before\">[<span class=\"bibitem-label\">7</span>]&#160;&#160;</span><span data-line=\"49\"></span>Patrice Godefroid.\n<span data-line=\"50\"></span><span class=\"newblock\"></span><span data-line=\"50\"></span> Higher-order test generation.\n<span data-line=\"51\"></span><span class=\"newblock\"></span><span data-line=\"51\"></span> In <span data-line=\"51\"></span><em>Proceedings of the ACM SIGPLAN Conference on Programming\n  Language Design and Implementation</em><span data-line=\"52\"></span>, pages 258<span data-line=\"52\"></span>&#8211;<span data-line=\"52\"></span>269, 2011.<span data-line=\"52\"></span>&nbsp;<a href=\"http://www.bing.com/search?q=Higher+order+test+generation++Patrice+Godefroid+\" class=\"bibsearch\">&#128270;</a></div>\n<div id=\"godefroidks05\" class=\"bibitem\"   data-line=\"55\" style=\"searchterm:DART+directed+automated+random+testing++Patrice+Godefroid+Nils+Klarlund+Koushik+\"><span data-line=\"56\"></span><span class=\"bibitem-before\">[<span class=\"bibitem-label\">8</span>]&#160;&#160;</span><span data-line=\"56\"></span>Patrice Godefroid, Nils Klarlund, and Koushik Sen.\n<span data-line=\"57\"></span><span class=\"newblock\"></span><span data-line=\"57\"></span> DART: directed automated random testing.\n<span data-line=\"58\"></span><span class=\"newblock\"></span><span data-line=\"58\"></span> In <span data-line=\"58\"></span><em>Proceedings of the ACM SIGPLAN Conference on Programming\n  Language Design and Implementation</em><span data-line=\"59\"></span>, pages 213<span data-line=\"59\"></span>&#8211;<span data-line=\"59\"></span>223, 2005.<span data-line=\"59\"></span>&nbsp;<a href=\"http://www.bing.com/search?q=DART+directed+automated+random+testing++Patrice+Godefroid+Nils+Klarlund+Koushik+\" class=\"bibsearch\">&#128270;</a></div>\n<div id=\"godefroidlm12\" class=\"bibitem\"   data-line=\"62\" style=\"searchterm:SAGE+whitebox+fuzzing+security+testing++Patrice+Godefroid+Michael+Levin+David+Molnar+\"><span data-line=\"63\"></span><span class=\"bibitem-before\">[<span class=\"bibitem-label\">9</span>]&#160;&#160;</span><span data-line=\"63\"></span>Patrice Godefroid, Michael<span data-line=\"63\"></span>&#160;<span data-line=\"63\"></span>Y. Levin, and David<span data-line=\"63\"></span>&#160;<span data-line=\"63\"></span>A. Molnar.\n<span data-line=\"64\"></span><span class=\"newblock\"></span><span data-line=\"64\"></span> SAGE: whitebox fuzzing for security testing.\n<span data-line=\"65\"></span><span class=\"newblock\"></span><span data-line=\"65\"></span> <span data-line=\"65\"></span><em>Communications of the ACM</em><span data-line=\"65\"></span>, 55<span data-line=\"65\"></span><span style=\"penalty:0\"></span><span data-line=\"65\"></span> (3):<span data-line=\"65\"></span><span style=\"penalty:0\"></span><span data-line=\"65\"></span> 40<span data-line=\"65\"></span>&#8211;<span data-line=\"65\"></span>44,\n  2012.<span data-line=\"66\"></span>&nbsp;<a href=\"http://www.bing.com/search?q=SAGE+whitebox+fuzzing+security+testing++Patrice+Godefroid+Michael+Levin+David+Molnar+\" class=\"bibsearch\">&#128270;</a></div>\n<div id=\"gupta00\" class=\"bibitem\"   data-line=\"69\" style=\"searchterm:Generating+test+data+branch+coverage++Neelam+Gupta+Aditya+Mathur+Mary+Soffa+\"><span data-line=\"70\"></span><span class=\"bibitem-before\">[<span class=\"bibitem-label\">10</span>]&#160;&#160;</span><span data-line=\"70\"></span>Neelam Gupta, Aditya<span data-line=\"70\"></span>&#160;<span data-line=\"70\"></span>P. Mathur, and Mary<span data-line=\"70\"></span>&#160;<span data-line=\"70\"></span>Lou Soffa.\n<span data-line=\"71\"></span><span class=\"newblock\"></span><span data-line=\"71\"></span> Generating test data for branch coverage.\n<span data-line=\"72\"></span><span class=\"newblock\"></span><span data-line=\"72\"></span> In <span data-line=\"72\"></span><em>Proceedings of the Automate Software Engineering\n  Conference</em><span data-line=\"73\"></span>, pages 219<span data-line=\"73\"></span>&#8211;<span data-line=\"73\"></span>228, 2000.<span data-line=\"73\"></span>&nbsp;<a href=\"http://www.bing.com/search?q=Generating+test+data+branch+coverage++Neelam+Gupta+Aditya+Mathur+Mary+Soffa+\" class=\"bibsearch\">&#128270;</a></div>\n<div id=\"king76\" class=\"bibitem\"   data-line=\"76\" style=\"searchterm:Symbolic+execution+program+testing++James+King+\"><span data-line=\"77\"></span><span class=\"bibitem-before\">[<span class=\"bibitem-label\">11</span>]&#160;&#160;</span><span data-line=\"77\"></span>James<span data-line=\"77\"></span>&#160;<span data-line=\"77\"></span>C. King.\n<span data-line=\"78\"></span><span class=\"newblock\"></span><span data-line=\"78\"></span> Symbolic execution and program testing.\n<span data-line=\"79\"></span><span class=\"newblock\"></span><span data-line=\"79\"></span> <span data-line=\"79\"></span><em>Communications of the ACM</em><span data-line=\"79\"></span>, 19<span data-line=\"79\"></span><span style=\"penalty:0\"></span><span data-line=\"79\"></span> (7):<span data-line=\"79\"></span><span style=\"penalty:0\"></span><span data-line=\"79\"></span>\n  385–394, 1976.<span data-line=\"80\"></span>&nbsp;<a href=\"http://www.bing.com/search?q=Symbolic+execution+program+testing++James+King+\" class=\"bibsearch\">&#128270;</a></div>\n<div id=\"korel90\" class=\"bibitem\"   data-line=\"83\" style=\"searchterm:Automated+software+test+data+generation++Bogdan+Korel+\"><span data-line=\"84\"></span><span class=\"bibitem-before\">[<span class=\"bibitem-label\">12</span>]&#160;&#160;</span><span data-line=\"84\"></span>Bogdan Korel.\n<span data-line=\"85\"></span><span class=\"newblock\"></span><span data-line=\"85\"></span> Automated software test data generation.\n<span data-line=\"86\"></span><span class=\"newblock\"></span><span data-line=\"86\"></span> <span data-line=\"86\"></span><em>IEEE Transactions on Software Engineering</em><span data-line=\"86\"></span>, 16<span data-line=\"86\"></span><span style=\"penalty:0\"></span><span data-line=\"86\"></span>\n  (8):<span data-line=\"87\"></span><span style=\"penalty:0\"></span><span data-line=\"87\"></span> 870<span data-line=\"87\"></span>&#8211;<span data-line=\"87\"></span>879, 1990.<span data-line=\"87\"></span>&nbsp;<a href=\"http://www.bing.com/search?q=Automated+software+test+data+generation++Bogdan+Korel+\" class=\"bibsearch\">&#128270;</a></div>\n<div id=\"korel92\" class=\"bibitem\"   data-line=\"90\" style=\"searchterm:Dynamic+method+software+test+data+generation++Bogdan+Korel+\"><span data-line=\"91\"></span><span class=\"bibitem-before\">[<span class=\"bibitem-label\">13</span>]&#160;&#160;</span><span data-line=\"91\"></span>Bogdan Korel.\n<span data-line=\"92\"></span><span class=\"newblock\"></span><span data-line=\"92\"></span> Dynamic method of software test data generation.\n<span data-line=\"93\"></span><span class=\"newblock\"></span><span data-line=\"93\"></span> <span data-line=\"93\"></span><em>Journal of Software Testing, Verification and Reliability</em><span data-line=\"93\"></span>,\n  2<span data-line=\"94\"></span><span style=\"penalty:0\"></span><span data-line=\"94\"></span> (4):<span data-line=\"94\"></span><span style=\"penalty:0\"></span><span data-line=\"94\"></span> 203<span data-line=\"94\"></span>&#8211;<span data-line=\"94\"></span>213, 1992.<span data-line=\"94\"></span>&nbsp;<a href=\"http://www.bing.com/search?q=Dynamic+method+software+test+data+generation++Bogdan+Korel+\" class=\"bibsearch\">&#128270;</a></div>\n<div id=\"senacav06\" class=\"bibitem\"   data-line=\"97\" style=\"searchterm:Koushik+Agha+CUTE+jcute+Concolic+unit+testing+explicit+path+model+checking+tools+_Proceedings+Computer+Aided+Verification+Conference_+pages++\"><span data-line=\"98\"></span><span class=\"bibitem-before\">[<span class=\"bibitem-label\">14</span>]&#160;&#160;</span><span data-line=\"98\"></span>Koushik Sen and Gul Agha.\n<span data-line=\"99\"></span><span class=\"newblock\"></span><span data-line=\"99\"></span> CUTE and jcute: Concolic unit testing and explicit path\n  model-checking tools.\n<span data-line=\"101\"></span><span class=\"newblock\"></span><span data-line=\"101\"></span> In <span data-line=\"101\"></span><em>Proceedings of 18th Computer Aided Verification Conference</em><span data-line=\"101\"></span>,\n  pages 419<span data-line=\"102\"></span>&#8211;<span data-line=\"102\"></span>423, 2006.<span data-line=\"102\"></span>&nbsp;<a href=\"http://www.bing.com/search?q=Koushik+Agha+CUTE+jcute+Concolic+unit+testing+explicit+path+model+checking+tools+_Proceedings+Computer+Aided+Verification+Conference_+pages++\" class=\"bibsearch\">&#128270;</a></div></div></div>\n<div class=\"madokologo block\" style=\"text-align:right;font-size:xx-small;margin-top:4em\"><span data-line=\"1\"></span>Created with&nbsp;<a href=\"https://www.madoko.net\">Madoko.net</a>.</div>\n</body>\n\n</html>\n"
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\\abovedisplayskip%\n}\n\\newcommand\\scriptsize{\\@setfontsize\\scriptsize\\@viiipt{9.5}}\n\\newcommand\\tiny{\\@setfontsize\\tiny\\@vipt\\@viipt}\n\\newcommand\\large{\\@setfontsize\\large\\@xiipt{14}}\n\\newcommand\\Large{\\@setfontsize\\Large\\@xivpt{18}}\n\\newcommand\\LARGE{\\@setfontsize\\LARGE\\@xviipt{22}}\n\\newcommand\\huge{\\@setfontsize\\huge\\@xxpt{25}}\n\\newcommand\\Huge{\\@setfontsize\\Huge\\@xxvpt{30}}\n\n\\normalsize\n\n% Customization of fonts\n\\renewcommand\\sldefault{it}\n\\renewcommand\\bfdefault{b}\n\\let\\slshape\\itshape\n%\n\n% ********************* DIMENSIONS:\n% TEXT DIMENSIONS\n\\setlength\\parindent{18\\p@}\n\\@settopoint\\parindent\n\\setlength\\textwidth{124mm}\n\\@settopoint\\textwidth\n\\setlength\\textheight{200mm}\n\\@settopoint\\textheight\n\\setlength\\columnsep{10mm}\n\\@settopoint\\columnsep\n\\setlength\\columnwidth{95mm}\n\\@settopoint\\columnwidth\n\\setlength\\columnseprule{0\\p@}\n\\hoffset -0.5cm\n\\voffset -1cm\n\n% HEADS:\n\\setlength\\headheight{12\\p@}\n\\setlength\\headsep   {15\\p@}\n\\setlength\\topskip   {10\\p@}\n\\setlength\\footskip  {25\\p@}\n\\setlength\\maxdepth  {.5\\topskip}\n% SIDE MARGINS\n\\setlength\\oddsidemargin   {0mm}\n\\setlength\\evensidemargin  {0mm}\n\\setlength\\topmargin       {10mm}\n\\@settopoint\\topmargin\n% TEXT PARAMETERS\n\\setlength\\lineskip{1\\p@}\n\\setlength\\normallineskip{1\\p@}\n\\renewcommand\\baselinestretch{}\n\\setlength\\parskip{0\\p@}\n\n% Center on A4:\n\n\\def\\paper@width {210mm}\n\\def\\paper@height{297mm}\n\n\\hoffset=-1in\n\\voffset=-1in\n\n\\@tempdima=\\paper@width\n\\advance\\@tempdima by-\\textwidth\n\\divide\\@tempdima by2\n\\setlength\\evensidemargin  {\\@tempdima}%\n\\setlength\\oddsidemargin   {\\@tempdima}%\n\n\\@tempdima=\\paper@height\n\\advance\\@tempdima by-\\textheight\n\\advance\\@tempdima by-\\headsep\n\\advance\\@tempdima by-\\headheight\n\\divide\\@tempdima by2\n\\setlength\\topmargin  {\\@tempdima}%\n\n\n\n\n% BREAKS\n\\setlength\\smallskipamount{6\\p@ \\@plus 1\\p@ \\@minus 1\\p@}\n\\setlength\\medskipamount{12\\p@ \\@plus 3\\p@ \\@minus 3\\p@} \n\\setlength\\bigskipamount{24pt \\@plus 3\\p@ \\@minus 3\\p@}  \n% PAGE-BREAKING PENALTIES\n\\clubpenalty=4000\n\\widowpenalty=4000\n\\displaywidowpenalty=50\n\\predisplaypenalty=0   % Breaking before a math display.\n% \\postdisplaypenalty  % Breaking after a math display.\n% \\interlinepenalty    % Breaking at a line within a paragraph.\n% \\brokenpenalty       % Breaking after a hyphenated line.\n\\pretolerance=100    % Badness tolerance for the first pass (before hyphenation)\n\\tolerance=800       % Badness tolerance after hyphenation\n\\hbadness=800        % Badness above which bad hboxes will be shown\n\\emergencystretch=3\\p@\n\\hfuzz=1\\p@           % do not be to critical about boxes\n\n%\n\\doublehyphendemerits=0\n\\adjdemerits=0\n\\brokenpenalty=0\n\\interlinepenalty=0\n%\n\\if@twocolumn\n \\setlength\\marginparsep {10\\p@}\n\\else\n  \\setlength\\marginparsep{7\\p@}\n\\fi\n\\setlength\\marginparpush{5\\p@}\n\n% FOOTNOTES\n\\setlength\\footnotesep{6.65\\p@}\n\\setlength{\\skip\\footins}{12\\p@ \\@plus 6\\p@}\n% FLOATS\n\\setlength\\floatsep    {15\\p@ \\@plus 10\\p@ \\@minus 4\\p@}\n\\setlength\\textfloatsep{12\\p@ \\@plus 6\\p@ \\@minus 4\\p@}\n\\setlength\\intextsep   {12\\p@ \\@plus 6\\p@ \\@minus 4\\p@}\n\\setlength\\dblfloatsep    {15\\p@ \\@plus 10\\p@ \\@minus 4\\p@}\n\\setlength\\dbltextfloatsep{12\\p@ \\@plus 12\\p@ \\@minus 4\\p@}\n%  For floats on a separate float page or column:\n\\setlength\\@fptop{0\\p@ \\@plus 1fil}\n\\setlength\\@fpsep{8\\p@ \\@plus 1000fil}\n\\setlength\\@fpbot{0\\p@ \\@plus 1fil}\n\\setlength\\@dblfptop{0\\p@ \\@plus 1fil}\n\\setlength\\@dblfpsep{8\\p@ \\@plus 1000fil}\n\\setlength\\@dblfpbot{0\\p@ \\@plus 1fil}\n%\n\\setcounter{topnumber}{5}\n\\renewcommand\\topfraction{.90}\n\\setcounter{bottomnumber}{5}\n\\renewcommand\\bottomfraction{.90}\n\\setcounter{totalnumber}{10}\n\\renewcommand\\textfraction{.10}\n\\renewcommand\\floatpagefraction{.9}\n\\setcounter{dbltopnumber}{5}\n\\renewcommand\\dbltopfraction{.99}\n\\renewcommand\\dblfloatpagefraction{.8}\n%\n% PENALTIES\n\\@lowpenalty   51\n\\@medpenalty  151\n\\@highpenalty 301\n\\@beginparpenalty -\\@lowpenalty\n\\@endparpenalty   -\\@lowpenalty\n\\@itempenalty     -\\@lowpenalty\n% LISTS\n\\setlength\\partopsep{0\\p@}\n\\def\\@listI{\\leftmargin\\leftmargini\n            \\parsep 0\\p@ \\@plus2\\p@ \\@minus\\p@\n            \\topsep 9\\p@ \\@plus2\\p@ \\@minus2\\p@\n            \\partopsep\\p@\n            \\itemsep 1\\p@ \\@plus.5\\p@ \\@minus1\\p@}\n\\let\\@listi\\@listI\n\\@listi\n\\def\\@listii {\\leftmargin\\leftmarginii\n              \\labelwidth\\leftmarginii\n              \\advance\\labelwidth-\\labelsep\n              \\topsep    4\\p@ \\@plus2\\p@ \\@minus\\p@\n              \\parsep    0\\p@ \\@plus1\\p@  \\@minus\\p@\n              \\itemsep   \\parsep}\n\\def\\@listiii{\\leftmargin\\leftmarginiii\n              \\labelwidth\\leftmarginiii\n              \\advance\\labelwidth-\\labelsep\n              \\topsep    2\\p@ \\@plus\\p@\\@minus\\p@\n              \\parsep    \\z@\n              \\partopsep \\p@ \\@plus\\z@ \\@minus\\p@\n              \\itemsep   \\topsep}\n\\def\\@listiv {\\leftmargin\\leftmarginiv\n              \\labelwidth\\leftmarginiv\n              \\advance\\labelwidth-\\labelsep}\n\\def\\@listv  {\\leftmargin\\leftmarginv\n              \\labelwidth\\leftmarginv\n              \\advance\\labelwidth-\\labelsep}\n\\def\\@listvi {\\leftmargin\\leftmarginvi\n              \\labelwidth\\leftmarginvi\n              \\advance\\labelwidth-\\labelsep}\n%\n\\DeclareMathSizes{\\@xivpt}{\\@xivpt}{\\@xpt}{\\@viiipt}\n\\DeclareMathSizes{12}{12}{\\@viiipt}{\\@viipt}\n%\n% ******************** HEADINGS\n%\n% normal heading\n\\def\\ps@headings{%\n      \\let\\@oddfoot\\@empty\\let\\@evenfoot\\@empty\n      \\def\\@evenhead{\\footnotesize\\rlap{\\thepage}\\hfill\\textit{\\leftmark}\\hfill}%\n      \\def\\@oddhead{\\footnotesize\\hfill\\textit{\\rightmark}\\hfill\\llap{\\thepage}}%\n}%\n% empty RH\n\\def\\ps@empty{\\let\\@mkboth\\@gobbletwo\n     \\def\\@oddhead{\\hfill}\\def\\@oddfoot{}\n\\let\\@evenhead\\@oddhead\\let\\@evenfoot\\@oddfoot}\n%\n% RH  with pagenumber at bottom\n\\def\\ps@plain{\\let\\@mkboth\\@gobbletwo\n     \\def\\@oddhead{\\hfill}\\def\\@oddfoot{}\n\\let\\@evenhead\\@oddhead\n  \\def\\@oddfoot{\\hfill\\footnotesize\\thepage\\hfill}\n  \\let\\@evenfoot\\@oddfoot\n}\n% First page RH\n\\def\\ps@copyright{\\let\\@mkboth\\@gobbletwo\n  \\def\\@evenhead{\\parbox[t]{.75\\textwidth}{\\footnotesize\\raggedright\\itshape\\titleheadline}\\hfill\\footnotesize\\thepage}%\n  \\def\\@oddhead {\\parbox[t]{.75\\textwidth}{\\footnotesize\\raggedright\\itshape\\titleheadline}\\hfill\\footnotesize\\thepage}%\n  \\let\\@oddfoot\\relax%\n  \\let\\@evenfoot\\@oddfoot%\n}\n%\n% HEADLINE: Book Title  \n%           Book Editors\n%           IOS Press, 0000\n%\n\\def\\titleheadline{%\n   \\book@title\\\\\n   \\book@editors\\\\\n   \\@publisher, \\the\\@pubyear}\n%\n\\def\\@copyright{\\@issn/\\the@copyear/\\$\\@price\\ \\copyright@sign\\\n\\the\\@pubyear\\@copyrightowner}%\n%\n\n% ************************ FOOTNOTE\n%\n\\newcommand\\@makefntext[1]{%\n    \\parindent1em\\@makefnmark #1}\n\\def\\@makefnmark{\\@textsuperscript{\\normalfont\\@thefnmark}}%\n%\n% ************************ Counters\n\\setcounter{secnumdepth}{3}\n\\newcounter {section}\n\\newcounter {subsection}[section]\n\\newcounter {subsubsection}[subsection]\n\\newcounter {paragraph}[subsubsection]\n\\newcounter {subparagraph}[paragraph]\n\\renewcommand \\thesection {\\@arabic\\c@section}\n\\renewcommand\\thesubsection   {\\thesection.\\@arabic\\c@subsection}\n\\renewcommand\\thesubsubsection{\\thesubsection .\\@arabic\\c@subsubsection}\n\\renewcommand\\theparagraph    {\\thesubsubsection.\\@arabic\\c@paragraph}\n\\renewcommand\\thesubparagraph {\\theparagraph.\\@arabic\\c@subparagraph}\n%\n% ******************** Sectioning commands\n\\def\\no@harm{\\let\\thanks=\\@gobble \\let\\\\=\\@empty}\n%**************** Section commands\n\\def\\nohyphen{\\pretolerance=10000 \\tolerance=10000\n\\hyphenpenalty=10000 \\exhyphenpenalty=10000}\n\\newcommand\\section{\\@startsection {section}{1}{\\z@}%\n                                   {-\\bigskipamount}%\n                                   {\\medskipamount}%\n                                   {\\normalsize\\bfseries\\nohyphen\\raggedright}}\n\\newcommand\\subsection{\\@startsection {subsection}{2}{\\z@}%\n                                   {-\\medskipamount}%\n                                   {\\medskipamount}%\n                                   {\\normalsize\\itshape\\nohyphen\\raggedright}}\n\\newcommand\\subsubsection{\\@startsection{subsubsection}{3}{\\z@}%\n                                     {-\\medskipamount}%\n                                     {\\smallskipamount}%\n                                     {\\normalsize\\itshape\\nohyphen\\raggedright}}\n\\newcommand\\paragraph{\\@startsection{paragraph}{4}{\\z@}%\n                                    {\\smallskipamount}%\n                                    {-1em}%\n                                    {\\normalsize\\itshape}}\n\\newcommand\\subparagraph{\\@startsection{subparagraph}{5}{\\z@}%\n                                       {0.1pt}%\n                                       {-1em}%\n                                       {\\normalsize\\itshape}}\n% Format for the counter:\n\\def\\@seccntformat#1{\\csname the#1\\endcsname.\\enspace}\n%\n\\def\\appendix{\\par\n   \\setcounter{section}{0}%\n   \\setcounter{subsection}{0}%\n   \\gdef\\thesection{\\Alph{section}}}\n%\n\\def\\acknowledgements{\\section*{\\acknowledgementsname}%\n  \\typeout{\\acknowledgementsname}}\n%\n\\def\\notes{\\section*{Notes}\\footnotesize}\n\\def\\endnotes{\\par \\vskip 6pt plus12pt minus2pt\\relax}\n%****************** LISTS\n\\if@twocolumn\n  \\setlength\\leftmargini  {2em}\n\\else\n  \\setlength\\leftmargini  {2.5em}\n\\fi\n\\leftmargin  \\leftmargini\n\\setlength\\leftmarginii  {2.2em}\n\\setlength\\leftmarginiii {1.87em}\n\\setlength\\leftmarginiv  {1.7em}\n\\if@twocolumn\n  \\setlength\\leftmarginv  {.5em}\n  \\setlength\\leftmarginvi {.5em}\n\\else\n  \\setlength\\leftmarginv  {1em}\n  \\setlength\\leftmarginvi {1em}\n\\fi\n\\setlength  \\labelsep  {.4em}\n\\setlength  \\labelwidth{\\leftmargini}\n\\addtolength\\labelwidth{-\\labelsep}\n%\n\\renewcommand\\theenumi{\\@arabic\\c@enumi}\n\\renewcommand\\theenumii{\\@alph\\c@enumii}\n\\renewcommand\\theenumiii{\\@roman\\c@enumiii}\n\\renewcommand\\theenumiv{\\@Alph\\c@enumiv}\n\\newcommand\\labelenumi{\\theenumi.}\n\\newcommand\\labelenumii{(\\theenumii)}\n\\newcommand\\labelenumiii{\\theenumiii.}\n\\newcommand\\labelenumiv{\\theenumiv.}\n\\renewcommand\\p@enumii{\\theenumi}\n\\renewcommand\\p@enumiii{\\theenumi(\\theenumii)}\n\\renewcommand\\p@enumiv{\\p@enumiii\\theenumiii}\n%\n\\def\\setenumlabel#1{\\gdef\\max@enumlabel{#1}}\n\\setenumlabel{1.}\n%\n\\def\\enumerate{\\@ifnextchar[{\\enumerate@}{\\enumerate@[\\max@enumlabel]}}\n%\n\\def\\enumerate@[#1]{\\ifnum \\@enumdepth >4 \\@toodeep\\else\n \\advance\\@enumdepth \\@ne\n \\edef\\@enumctr{enum\\romannumeral\\the\\@enumdepth}%\n \\list {\\csname label\\@enumctr\\endcsname}%\n {\\usecounter{\\@enumctr}\\def\\makelabel##1{{\\hfill\\rm ##1}}\n\\settowidth{\\labelwidth}{#1}\n\\advance\\labelwidth by\\parindent \\labelsep=0.5em\n \\leftmargin\\z@ \\rightmargin\\z@ \\itemindent=\\labelwidth\n        \\advance\\itemindent\\labelsep\n        \\leftmargin=\\the\\itemindent\\itemindent=\\z@\n        \\partopsep\\z@ \\topsep\\smallskipamount \\parsep\\z@ \\itemsep\\z@ %\\@rightskip\\z@ plus 1fil\n \\listparindent\\z@}\\fi\\setenumlabel{1.}}\n\n%%%%%%%%%%%%%%%%%%%%%% ITEMIZE\n\\newcommand\\labelitemi{\\normalfont\\bfseries \\textbullet}\n\\newcommand\\labelitemii{\\textasteriskcentered}\n\\newcommand\\labelitemiii{\\textasteriskcentered}\n\\newcommand\\labelitemiv{\\textperiodcentered}\n\n\\let\\@itemize@indent\\parindent\n%\n\\def\\itemize{\\@ifnextchar[{\\itemize@}{\\itemize@[]}}\n\\def\\itemize@[#1]{\\ifnum \\@itemdepth >4 \\@toodeep\\else\n  \\advance\\@itemdepth \\@ne\n  \\edef\\@itemitem{labelitem\\romannumeral\\the\\@itemdepth}%\n  \\if.#1. \\else\\def\\@@tempa{#1}\\edef\\@itemitem{@@tempa}\\fi\\list\n{\\csname\\@itemitem\\endcsname}{\\settowidth{\\labelwidth}\n                {\\csname\\@itemitem\\endcsname}\n                \\def\\makelabel##1{##1}\\labelsep=0.5em%ST\n\t\t\\itemindent=\\labelwidth \\advance\\itemindent\\labelsep\n                \\advance\\itemindent\\@itemize@indent\n\t\t\\leftmargin\\the\\itemindent \\itemindent=\\z@\n                \\partopsep\\z@ \\topsep\\smallskipamount \\parsep\\z@ %\\@rightskip\\z@ plus 1fil\n\t\t\\itemsep\\z@ \\listparindent\\z@} \\fi}\n%\n\\newenvironment{description}\n               {\\list{}{\\labelwidth\\z@ \\itemindent-\\leftmargin\n                        \\let\\makelabel\\descriptionlabel}}\n               {\\endlist}\n\\newcommand*\\descriptionlabel[1]{\\hspace\\labelsep\n                                \\normalfont\\bfseries #1}\n\\newenvironment{verse}\n               {\\let\\\\\\@centercr\n                \\list{}{\\itemsep      \\z@\n                        \\itemindent   -1.5em%\n                        \\listparindent\\itemindent\n                        \\rightmargin  \\leftmargin\n                        \\advance\\leftmargin 1.5em}%\n                \\item\\relax}\n               {\\endlist}\n\n\\newenvironment{quotation}\n               {\\list{}{\\small\\listparindent2mm%\n                        \\itemindent\\z@   %\n                        \\rightmargin\\z@   \\leftmargin\\parindent%\n                        \\partopsep\\z@ \\topsep\\smallskipamount \\parsep\\z@%\n                        }%\n                \\item[\\Q@strut]\\relax}\n               {\\endlist}\n\\def\\Q@strut{\\leavevmode\\hbox{\\vrule height9pt depth1pt width0pt}}\n\n\\newenvironment{quote}\n               {\\list{}{\\listparindent\\z@%\n                        \\itemindent    \\listparindent%\n                        \\rightmargin\\z@   \\leftmargin 1.5em%\n                        \\partopsep\\z@ \\topsep6pt \\parsep\\z@%\n                        }%\n                \\item[\\Q@strut]\\relax}\n               {\\endlist}\n%\n%************************** TABULAR\n\\let\\savehline\\hline\n   \\def\\thline{\\noalign{\\vskip3pt}\\savehline\\noalign{\\vskip3pt}}%\n   \\def\\fhline{\\noalign{\\vskip1pt}\\savehline\\noalign{\\vskip7pt}}%\n   \\def\\bhline{\\noalign{\\vskip3pt}\\noalign{\\global\\arrayrulewidth=1\\p@}\\savehline\\noalign{\\global\\arrayrulewidth=.5\\p@}\\noalign{\\vskip3pt}}%\n   \\def\\lhline{\\noalign{\\vskip3pt}\\noalign{\\global\\arrayrulewidth=.3\\p@}\\savehline\\noalign{\\global\\arrayrulewidth=.5\\p@}\\noalign{\\vskip3pt}}\n%\n%************************** MATH SETTINGS\n\\setlength\\mathindent{2em}\n\\setlength\\arraycolsep{1.2\\p@}\n\\setlength\\tabcolsep{6\\p@}\n\\setlength\\arrayrulewidth{.4\\p@}\n\\setlength\\doublerulesep{2\\p@}\n\\setlength\\tabbingsep{\\labelsep}\n\\setlength\\jot{6\\p@}\n\\skip\\@mpfootins = \\skip\\footins\n\\setlength\\fboxsep{3\\p@}\n\\setlength\\fboxrule{.4\\p@}\n\\if@seceqn\n\\@addtoreset {equation}{section}\n\\renewcommand\\theequation{\\thesection.\\@arabic\\c@equation}\n\\else\n\\renewcommand\\theequation{\\@arabic\\c@equation}\n\\fi\n%******* TABLES, FIGURES, ALGORITHM\n\\newcounter{figure}\n\\if@secfloat\n \\@addtoreset{figure}{section}\n \\renewcommand \\thefigure {\\thesection.\\@arabic\\c@figure}\n\\else\n \\renewcommand \\thefigure {\\@arabic\\c@figure}\n\\fi\n\\def\\fps@figure{tbp}\n\\def\\ftype@figure{1}\n\\def\\ext@figure{lof}\n\\def\\fnum@figure{\\figurename~\\thefigure.}\n\\newenvironment{figure}\n               {\\let\\@makecaption\\@makefigurecaption\\let\\@floatboxreset\\@figureboxreset\\@float{figure}}\n               {\\end@float}\n\\newenvironment{figure*}\n               {\\let\\@makecaption\\@makefigurecaption\\let\\@floatboxreset\\@figureboxreset\\@dblfloat{figure}}\n               {\\end@dblfloat}\n\n\\def\\@figureboxreset{%\n        \\reset@font%\n        \\centering%\n        \\@setnobreak%\n        \\@setminipage%\n}\n\n\n\\long\\def\\@makefigurecaption#1#2{\\footnotesize%\n \\vskip\\abovecaptionskip\n\\setbox\\@tempboxa\\hbox{\\textbf{#1}\\enspace #2}%\n  \\ifdim \\wd\\@tempboxa >\\hsize\n    \\unhbox\\@tempboxa\\par\n  \\else\n    \\hbox to\\hsize{\\hfil\\box\\@tempboxa\\hfil}%\n  \\fi}\n%\n% TABLE\n\\newcounter{table}\n\\if@secfloat\n  \\@addtoreset{table}{section}\n\\renewcommand \\thetable{\\thesection.\\@arabic\\c@table}\n\\else\n  \\renewcommand \\thetable{\\@arabic\\c@table}\n\\fi\n\\def\\fps@table{tbp}\n\\def\\ftype@table{2}\n\\def\\ext@table{lot}\n\\def\\fnum@table{\\tablename~\\thetable.}\n%\n\\newenvironment{table}\n               {\\let\\@makecaption\\@maketablecaption%\n               \\let\\@floatboxreset\\@tableboxreset\\@float{table}}\n               {\\end@float}\n\\newenvironment{table*}\n               {\\let\\@makecaption\\@maketablecaption%\n               \\let\\@floatboxreset\\@tableboxreset\\@dblfloat{table}}\n               {\\end@dblfloat}\n%\n\\def\\@tableboxreset{%\n        \\reset@font%\n        \\centering\\footnotesize%\n        \\def\\arraystretch{1.2}\n        \\@setnobreak%\n        \\@setminipage%\n}\n\n\\newlength\\abovecaptionskip\n\\newlength\\belowcaptionskip\n\\setlength\\abovecaptionskip{8\\p@}\n\\setlength\\belowcaptionskip{3\\p@}\n%\n\\newdimen\\tablewidth \\tablewidth\\textwidth\n\\newdimen\\saved@tablewidth \\saved@tablewidth\\textwidth\n%\n\\long\\def\\@maketablecaption#1#2{%\n \\begingroup%\n    \\footnotesize%\n    \\global\\setbox\\@tempboxa\\hbox{\\textbf{#1}\\enspace #2}%\n \\endgroup%\n \\centering%\n \\ifdim \\wd\\@tempboxa>\\tablewidth %\n    \\parbox[t]{\\tablewidth}{\\footnotesize\\textbf{#1}\\enspace #2\\vphantom{Ay}\\par}%\n \\else\n    \\hbox to\\hsize{\\hfill\\box\\@tempboxa\\vphantom{Ay}\\hfill}%\n \\fi%\n \\global\\saved@tablewidth\\tablewidth%\n \\global\\tablewidth\\hsize\\vskip\\belowcaptionskip}\n%\n%\n%%****************** Algorithm\n\\newcounter{algorithm}\n\\if@secfloat\n  \\@addtoreset{algorithm}{section}\n\\renewcommand \\thealgorithm{\\thesection.\\@arabic\\c@algorithm}\n\\else\n  \\renewcommand \\thealgorithm{\\@arabic\\c@algorithm}\n\\fi\n\\def\\fps@algorithm{tbp}\n\\def\\ftype@algorithm{4}\n\\def\\ext@algorithm{loa}\n\\def\\fnum@algorithm{\\algorithmname~\\thealgorithm.}\n%\n\\newenvironment{algorithm}\n               {\\let\\@makecaption\\@makealgorithmcaption%\n               \\let\\@floatboxreset\\@algorithmboxreset\\@float{algorithm}}\n               {\\end@float}\n\\newenvironment{algorithm*}\n               {\\let\\@makecaption\\@makealgorithmcaption%\n               \\let\\@floatboxreset\\@algorithmboxreset\\@dblfloat{algorithm}}\n               {\\end@dblfloat}\n\n\\def\\@algorithmboxreset{%\n        \\reset@font%\n        \\centering\n        \\@setnobreak%\n        \\@setminipage%\n}\n\\long\\def\\@makealgorithmcaption#1#2{\\vskip 2ex \\small\n  \\hbox to \\hsize{\\parbox[t]{\\hsize}{{\\bf #1} #2}}}\n%\n%%%% Program Code:\n\\def\\programcode{%\n\\let\\@makealgorithmcaption\\@makefigurecaption\n\\def\\algorithmname{Program Code}}\n\n\n%********************* COMPATIBILITY WITH OLD LATEX:\n\\DeclareOldFontCommand{\\rm}{\\normalfont\\rmfamily}{\\mathrm}\n\\DeclareOldFontCommand{\\sf}{\\normalfont\\sffamily}{\\mathsf}\n\\DeclareOldFontCommand{\\tt}{\\normalfont\\ttfamily}{\\mathtt}\n\\DeclareOldFontCommand{\\bf}{\\normalfont\\bfseries}{\\mathbf}\n\\DeclareOldFontCommand{\\it}{\\normalfont\\itshape}{\\mathit}\n\\let\\sl\\it\n\\DeclareOldFontCommand{\\sc}{\\normalfont\\scshape}{\\@nomath\\sc}\n\\DeclareRobustCommand*\\cal{\\@fontswitch\\relax\\mathcal}\n\\DeclareRobustCommand*\\mit{\\@fontswitch\\relax\\mathnormal}\n%\n% *********** MATH\n%\n\\if@secthm\n \\@addtoreset{thm}{section}\n \\def\\thethm{\\thesection.\\arabic{thm}}\n\\else\n \\def\\thethm{\\arabic{thm}}\n\\fi\n%\n%***************************** BIBLIOGRAPHY\n\n\\newenvironment{thebibliography}[1]\n     {\\section*{\\refname}\\footnotesize\\rmfamily\\upshape%\n      \\list{\\@biblabel{\\@arabic\\c@enumiv}}%\n           {\\settowidth\\labelwidth{\\@biblabel{#1}}%\n            \\leftmargin\\labelwidth\n            \\setlength\\labelsep{8\\p@}\n            \\advance\\leftmargin\\labelsep\n            \\usecounter{enumiv}%\n            \\let\\p@enumiv\\@empty\n            \\renewcommand\\theenumiv{\\@arabic\\c@enumiv}}%\n      \\sloppy\n      \\clubpenalty4000\n      \\@clubpenalty \\clubpenalty\n      \\widowpenalty4000%\n      \\sfcode`\\.\\@m}\n     {\\def\\@noitemerr\n       {\\@latex@warning{Empty `thebibliography' environment}}%\n      \\endlist}\n%\n\\newcommand\\newblock{\\hskip .11em\\@plus.33em\\@minus.07em}\n%\n\\def\\@citex[#1]#2{%\n  \\let\\@citea\\@empty\n  \\@cite{\\@for\\@citeb:=#2\\do\n    {\\@citea\\def\\@citea{,\\penalty\\@m\\hskip.1pt}%\n     \\edef\\@citeb{\\expandafter\\@firstofone\\@citeb}%\n     \\if@filesw\\immediate\\write\\@auxout{\\string\\citation{\\@citeb}}\\fi\n     \\@ifundefined{b@\\@citeb}{\\mbox{\\reset@font\\bfseries ?}%\n       \\G@refundefinedtrue\n       \\@latex@warning\n         {Citation `\\@citeb' on page \\thepage \\space undefined}}%\n       {\\hbox{\\csname b@\\@citeb\\endcsname}}}}{#1}}\n\n%****************************** FRONTMATTER *\n%\n\\newtoks\\t@glob@notes\n\\newtoks\\t@loc@notes\n\\newcount\\note@cnt\n\\newcounter{author}\n\\newcount\\n@author\n\\def\\n@author@{}\n\\newcounter{address}\n%\n\\newcount\\sv@hyphenpenalty\n%\n\\newcount\\prev@elem \\prev@elem=0\n\\newcount\\cur@elem  \\cur@elem=0\n\\chardef\\e@pretitle=1\n\\chardef\\e@title=1\n\\chardef\\e@subtitle=1\n\\chardef\\e@author=2\n\\chardef\\e@address=3\n%\n\\newif\\if@newelem\n\\newif\\if@firstauthor\n\\newif\\if@preface\n\\newif\\if@hasabstract\n\\newif\\if@haskeywords\n%\n\\newbox\\fm@box\n\\newdimen\\fm@size\n\\newbox\\t@abstract\n\\newbox\\t@keywords\n%\n\\def\\add@tok#1#2{\\global#1\\expandafter{\\the#1#2}}\n\\def\\add@xtok#1#2{\\begingroup\n  \\no@harm\n  \\xdef\\@act{\\global\\noexpand#1{\\the#1#2}}\\@act\n\\endgroup}\n%\n\\def\\tailthanksref[#1]#2{\\noexpand\\pthanksref{#1}}\n\\def\\pthanksref#1{\\global\\advance\\note@cnt\\@ne\\ifnum\\note@cnt>\\@ne\n\\global\\t@loc@notes\\expandafter{\\the\\t@loc@notes\\note@sep}\\fi\n\\global\\t@loc@notes\\expandafter{\\the\\t@loc@notes#1}}\n%\n\\def\\beg@elem{\\global\\t@loc@notes={}\\global\\note@cnt\\z@}\n\\def\\@xnamedef#1{\\expandafter\\xdef\\csname #1\\endcsname}\n\\def\\no@harm{%\n  \\let\\\\=\\relax  \\let\\rm\\relax\n  \\let\\ss=\\relax \\let\\ae=\\relax \\let\\oe=\\relax\n  \\let\\AE=\\relax \\let\\OE=\\relax\n  \\let\\o=\\relax  \\let\\O=\\relax\n  \\let\\i=\\relax  \\let\\j=\\relax\n  \\let\\aa=\\relax \\let\\AA=\\relax\n  \\let\\l=\\relax  \\let\\L=\\relax\n  \\let\\d=\\relax  \\let\\b=\\relax \\let\\c=\\relax\n  \\let\\bar=\\relax\n  \\def\\protect{\\noexpand\\protect\\noexpand}}\n%\n\\def\\proc@elem#1#2{\\begingroup\n    \\no@harm\n    \\def\\thanks##1##{\\@gobble}%\n    \\def\\thanksref##1##{\\@gobble}%\n    \\@xnamedef{@#1}{#2}%\n  \\endgroup\n  \\prev@elem=\\cur@elem\n  \\cur@elem=\\csname e@#1\\endcsname\n  \\expandafter\\elem@nothanksref#2\\thanksref\\relax%\n  \\expandafter\\elem@nothanks#2\\thanks\\relax}\n%\n\\def\\elem@nothanksref#1\\thanksref{\\futurelet\\@peektok\\elem@thanksref}\n\\def\\elem@thanksref{\\ifx\\@peektok\\relax\n  \\else \\expandafter\\elem@morethanksref \\fi}\n\\def\\elem@morethanksref[#1]#2{\\add@thanks{#1}\\elem@nothanksref}\n%\n\\def\\elem@nothanks#1\\thanks{\\futurelet\\@peektok\\elem@thanks}\n\\def\\elem@thanks{\\ifx\\@peektok\\relax\n  \\else \\ifx\\@peektok[ \\expandafter\\expandafter\\expandafter\\elem@morethankse\n  \\else \\expandafter\\expandafter\\expandafter\\elem@morethanks \\fi\\fi}\n%\n\\def\\elem@morethankse[#1]#2{\\thanks@optarg[#1]{#2}\\add@thanks{#1}\\elem@nothanks}\n\\def\\elem@morethanks#1{\\thanks@optarg[]{#1}\\add@thanks{}\\elem@nothanks}\n%\n\\def\\add@thanks#1{%\n  \\global\\advance\\note@cnt\\@ne\n    \\ifnum\\note@cnt>\\@ne \\add@xtok\\t@loc@notes{\\note@sep}\\fi\n    \\ifx.#1.\\add@xtok\\t@loc@notes{\\thefootnote}\\else\n        \\add@xtok\\t@loc@notes{#1}\\fi%\n}\n\\def\\add@addressref#1{%\n  \\global\\advance\\note@cnt\\@ne\n    \\ifnum\\note@cnt>\\@ne \\add@xtok\\t@loc@notes{\\note@sep}\\fi\n    \\add@tok\\t@loc@notes{\\ref{#1}}%\n}\n\\def\\note@sep{,}\n%\n\\def\\thanks@optarg[#1]#2{%\n    \\ifx.#1.\\add@tok\\t@glob@notes{\\footnotetext}%\n    \\else\\add@tok\\t@glob@notes{\\freefootnotetext}\\fi%\n    \\refstepcounter{footnote}%\n    \\ifx.#1.\\add@xtok\\t@glob@notes{[\\the\\c@footnote]}%\n    \\else\\add@xtok\\t@glob@notes{[#1]}\\fi%\n    \\add@tok\\t@glob@notes{{#2}}%\n        \\ignorespaces}%\n%\n% FRONTMATTER\n%\n\\def\\artty#1{}\n%\n\\newdimen\\a@title@skip   \\a@title@skip=12\\p@\n\\newskip\\b@section@skip  \\b@section@skip=12\\p@ plus6\\p@ minus6\\p@%\n\\newskip\\b@pretitle@skip \\b@pretitle@skip=6\\p@\n%\n\\def\\frontmatter{%\n  \\let\\@corresp@note\\relax\n  \\global\\t@glob@notes={}\\global\\c@author\\z@\n  \\global\\c@address\\z@\n  \\global\\n@author=0\\n@author@\\relax\n  \\global\\advance\\n@author\\m@ne\n  \\global\\@firstauthortrue\n  \\global\\@hasabstractfalse\n  \\global\\@prefacefalse\n  \\parindent\\z@\n  \\open@fm \\ignorespaces}\n%\n\\def\\preface{\\@prefacetrue}\n%\n% ENDFRONTMATTER\n%\n\\def\\endfrontmatter{%\n  \\global\\n@author=\\c@author \\@writecount\n  \\global\\@topnum\\z@\n  \\ifx\\@firstpage\\@lastpage\n    \\gdef\\@pagerange{\\@firstpage}\n  \\else\n    \\gdef\\@pagerange{\\@firstpage--\\@lastpage}\n  \\fi\n%  \\thispagestyle{copyright}%\n  \\if@twocolumn\\else\\output@glob@notes\\fi\n  \\if@preface\n    \\@hasabstractfalse\n  \\fi\n  \\if@hasabstract\n    \\normal@text\n    \\vskip 18\\p@\n    \\centering\n    \\leavevmode\\box\\t@abstract\\par\n  \\fi\n  \\if@haskeywords\n    \\normal@text\n    \\if@hasabstract \\vskip6pt\\else\\vskip18pt\\fi    \n    \\centering\n    \\leavevmode\\box\\t@keywords\\par\n  \\fi\n  \\close@fm\n  \\if@twocolumn\\output@glob@notes\\fi\n  \\markboth{\\@runauthor\\@runtitle}{\\@runauthor\\@runtitle}%\n  \\global\\@prefacefalse\n  \\global\\leftskip\\z@\n  \\global\\@rightskip\\z@\n  \\global\\rightskip\\@rightskip\n%  \\global\\c@footnote=0\n  \\let\\title\\relax       \\let\\author\\relax\n  \\let\\address\\relax\n  \\let\\frontmatter\\relax \\let\\endfrontmatter\\relax\n  \\let\\@maketitle\\relax  \\let\\@@maketitle\\relax\n  \\normal@text}\n%\n% Dvieju koloneliu zurnale per visa lapo ploti eina\n% tik pretitle, title ir subtitle. Tam ivedame komanda\n% \\maketitle, kuri uzdaro box'a\n\n  \\def\\two@c@maketitle{%\n    \\global\\let\\close@fm\\relax%\n    \\vskip\\b@section@skip%\n    \\par \\egroup\n    \\emergencystretch=1pc \\twocolumn[\\unvbox\\fm@box]}\n%\n\\if@restonecol\n  \\let\\maketitle\\relax\n\\else\n  \\let\\maketitle\\two@c@maketitle\n\\fi\n%\n\n%\n\\newdimen\\t@xtheight\n\\def\\init@settings{\n\\splittopskip=\\topskip \\splitmaxdepth=\\maxdepth\n\\t@xtheight\\textheight \\advance\\t@xtheight-\\splittopskip}\n%\n\\def\\open@fm{\n  \\global\\setbox\\fm@box=\\vbox\\bgroup\n  \\hsize=\\textwidth\n  \\centering\n  \\sv@hyphenpenalty\\hyphenpenalty\n  \\hyphenpenalty\\@M}\n%\n\n\\def\\close@fm{%\n  \\vskip\\b@section@skip%\n  \\par \\egroup\n  \\if@twocolumn\\else%\n    \\fm@size=\\dp\\fm@box \\advance\\fm@size by \\ht\\fm@box\n    \\@whiledim\\fm@size>\\t@xtheight \\do{%\n      \\global\\setbox\\@tempboxa=\\vsplit\\fm@box to \\t@xtheight\n      \\unvbox\\@tempboxa \\newpage\n      \\fm@size=\\dp\\fm@box \\advance\\fm@size by \\ht\\fm@box}\n  \\fi%\n  \\if@twocolumn\n    \\emergencystretch=1pc \\twocolumn[\\unvbox\\fm@box]\n  \\else\n    \\unvbox\\fm@box\n  \\fi}\n%\n\\def\\output@glob@notes{\\bgroup\n  \\the\\t@glob@notes\n  \\egroup}\n%\n\\def\\justify@off{\\let\\\\=\\@normalcr\n  \\leftskip\\z@ \\@rightskip\\@flushglue \\rightskip\\@rightskip}\n\\def\\justify@on{\\let\\\\=\\@normalcr\n  \\parfillskip\\@flushglue%\n  \\leftskip\\z@ \\@rightskip\\z@ \\rightskip\\@rightskip}\n%\n\\def\\normal@text{\\global\\let\\\\=\\@normalcr\n  \\global\\leftskip\\z@ \\global\\@rightskip\\z@ \\global\\rightskip\\@rightskip\n  \\global\\parfillskip\\@flushglue}\n%\n\\def\\@writecount{\\write\\@mainaux{\\string\\global\n  \\string\\@namedef{n@author@}{\\the\\n@author}}%\n}\n%\n% TITLE\n\\def\\pretitle#1{%\n\\vspace*{\\b@pretitle@skip}\\pretitle@size#1\\par\\vskip6\\p@\\hrule\n\\vskip12\\p@}\n%\n\\def\\title#1{%\n  \\beg@elem\n  \\title@note@fmt\n  \\add@tok\\t@glob@notes\n    {\\title@note@fmt}%\n  \\proc@elem{title}{#1}%\n  \\def\\title@notes{\\the\\t@loc@notes}%\n  \\title@fmt{\\@title}{\\title@notes}%\n  \\ignorespaces}\n%\n\\newdimen\\@@topskip \\@@topskip=24\\p@\n%\n\\def\\title@fmt#1#2{%\n  \\vspace*{\\@@topskip}\n  {\\title@size #1\\hbox{$^{#2}$}\\par}%\n  \\vskip\\a@title@skip%\n  }\n\n%\n\\def\\subtitle#1{%\n  \\beg@elem\n  \\proc@elem{subtitle}{#1}%\n  \\def\\title@notes{\\the\\t@loc@notes}%\n  \\subtitle@fmt{\\@subtitle}{\\title@notes}%\n  \\ignorespaces}\n%\n%\n\\def\\subtitle@fmt#1#2{%\n  {\\subtitle@size #1\\,\\hbox{$^{\\mathrm{#2}}$}\\par}%\n  \\vskip\\a@title@skip%\n  }\n%\n\\def\\title@note@fmt{\\def\\thefootnote{\\arabic{footnote}}}\n%\n% AUTHOR\n%\n\\newdimen\\b@author@skip\n\\b@author@skip 12\\p@\n%\n\\def\\author{\\@ifnextchar[{\\author@optarg}{\\author@optarg[]}}\n%\n\\def\\author@optarg[#1]#2{\\stepcounter{author}%\n  \\beg@elem\\def\\degs##1{##1}\\def\\fnms##1{##1}\\def\\inits##1{##1}%\n        \\def\\snm##1{\\MakeUppercase{##1}}\\def\\roles##1{##1}%\n  \\if@firstauthor%\n  \\first@author \\global\\@firstauthorfalse \\fi%\n  \\@for\\@tempa:=#1\\do{\\expandafter\\add@addressref\\expandafter{\\@tempa}}%\n  \\proc@elem{author}{#2}%\n  \\author@fmt{\\the\\c@author}{\\the\\t@loc@notes}{\\@author}}%\n%\n%\\newbox\\author@box\n\n%\n\\def\\author@fmt#1#2#3{\\@newelemtrue\n  \\ifnum\\prev@elem=\\e@author \\global\\@newelemfalse \\fi\n  \\if@newelem \\author@fmt@init \\fi\n  \\edef\\@tempb{#2}\\ifx\\@tempb\\@empty\n    \\hbox{#3}\\else\n    \\hbox{#3\\,$^{\\mathrm{#2}}$}%\n  \\fi}\n%\n\\def\\first@author{\\author@note@fmt%\n  \\add@tok\\t@glob@notes%\n    {\\author@note@fmt}}%\n%\n\\def\\author@fmt@init{%\n  \\par\n  \\vskip \\b@author@skip\n  \\authors@size\\centering\n  \\leavevmode}\n%\n\\def\\and{\\unskip~and~}\n%\n\\def\\author@note@fmt{%\n  \\def\\thefootnote{\\arabic{footnote}}}\n%\n\\def\\sxarabic#1{%\n        \\expandafter\\ifcase\\value{#1} \\or *\\or **\\or *** \\or **** \\or *****\\fi\n}\n%\n% ADDRESS\n%\n\\def\\email#1{{e-mail:\\ #1}}\n%\n\\def\\address{\\@ifstar{\\address@star}%\n  {\\@ifnextchar[{\\address@optarg}{\\address@noptarg}}}\n%\n\\def\\address@optarg[#1]#2{\\refstepcounter{address}%\n  \\beg@elem\n  \\proc@elem{address}{#2}%\n  \\address@fmt{\\the\\c@address}{\\the\\t@loc@notes}{\\@address}\\label{#1}%\n  \\ignorespaces}\n%\n\\def\\address@noptarg#1{\\refstepcounter{address}%\n  \\beg@elem\n  \\proc@elem{address}{#1}%\n  \\address@fmt{\\z@}{\\the\\t@loc@notes}{\\@address}%\n  \\ignorespaces}\n%\n\\def\\address@star#1{%\n  \\beg@elem\n  \\proc@elem{address}{#1}%\n  \\address@fmt{\\m@ne}{\\the\\t@loc@notes}{\\@address}%\n  \\ignorespaces}\n%\n\\def\\theaddress{\\alph{address}}\n%\n\\def\\address@fmt#1#2#3{\\@newelemtrue\n  \\ifnum\\prev@elem=\\e@address \\@newelemfalse \\fi\n  \\if@newelem \\address@fmt@init \\fi\n  \\bgroup\\parskip\\z@\\noindent\\centering \\address@size\n  \\ifnum#1=\\z@\n    #3\\,$^{\\mathrm{#2}}$\\space%\n  \\else\n    \\ifnum#1=\\m@ne\n      $^{\\phantom{\\mathrm{\\theaddress}}\\,}$#3\\,$^{\\mathrm{#2}}$%\n    \\else\n      $^{\\mathrm{\\theaddress}\\,}$#3\\,$^{\\mathrm{#2}}$%\n    \\fi\n  \\fi\n  \\par\\egroup}\n%\n\\def\\address@fmt@init{%\n        \\def\\@currentlabel{\\theaddress}\n  \\par\n  \\vskip 2\\p@ plus 1\\p@ minus 1\\p@}\n%\n% ABSTRACT\n%\n\\def\\abstract{\\@ifnextchar[{\\@abstract}{\\@abstract[]}}\n\\def\\@abstract[#1]{%\n  \\global\\@hasabstracttrue\n  \\hyphenpenalty\\sv@hyphenpenalty\n  \\global\\setbox\\t@abstract=\\vbox\\bgroup\n  \\linewidth\\abstract@width\n  \\hsize\\abstract@width\n  \\justify@on\\abstract@size\\parindent 1em\n  \\abstract@indent\\textbf{\\abstractname}\\ignorespaces}\n\\def\\endabstract{\\par\\egroup}\n%\n% KEYWORDS\n\\def\\sep{\\unskip, }\n\\global\\@haskeywordsfalse\n\\newdimen\\dp@t@keywords\n\\def\\keyword{\\global\\@haskeywordstrue%\n  \\global\\setbox\\t@keywords=\\vbox\\bgroup%\n  \\hsize\\abstract@width%\n  \\justify@on\\abstract@size\\parindent 0\\p@\n  \\textbf{\\keywordsname}\\ignorespaces\n  }\n\\def\\endkeyword{\\par\\egroup\\global\\dp@t@keywords=\\dp\\t@keywords}\n\\def\\keywords#1{\\begin{keyword}#1\\end{keyword}}\n%\n% \n%\n% Running title\n\\def\\runningtitle#1{\\gdef\\@runtitle{#1}}   \\def\\@runtitle{}\n\\def\\runningauthor#1{{\\def\\etal{et al.}\\gdef\\@runauthor{#1\\@runsep}}} \\def\\@runauthor{}\n\\def\\runningsep#1{\\gdef\\@runsep{#1}}\n\\def\\@runsep{\\ /\\ }\n%\n\\def\\journal#1{\\gdef\\@journal{#1}}     \\@ifundefined{@journal}{\\gdef\\@journal{Journal not defined}}{}\n\\def\\volume#1{\\gdef\\@volume{#1}}       \\def\\@volume{0}\n\\def\\issue#1{\\gdef\\@issue{#1}}         \\def\\@issue{0}\n%\n%\n\\newcount\\@pubyear\n\\newcount\\@copyear\n\\@pubyear=\\number\\year\n\\@copyear\\@pubyear \n\\advance\\@copyear-2000\n\n\\def\\pubyear#1{\\global\\@pubyear#1\n  \\global\\@copyear\\@pubyear \n  \\global\\advance\\@copyear-2000%\n  \\ignorespaces}\n%\n\\def\\the@copyear{\\ifnum\\@copyear<10 0\\fi\\the\\@copyear}\n\n%\n\\pubyear{2003}\n%\n\\def\\firstpage#1{\\def\\@tempa{#1}\\ifx\\@tempa\\@empty\\else\n  \\gdef\\@firstpage{#1}\\gdef\\@lastpage{#1}%\n  \\global\\c@page=#1 \\ignorespaces\\fi\n  }\n\\def\\@firstpage{1}\n\\def\\lastpage#1{\\def\\@tempa{#1}\\ifx\\@tempa\\@empty\\else\n  \\gdef\\@lastpage{#1}\\ignorespaces\\fi}\n\\def\\@lastpage{0}\n\\def\\@pagerange{1--0}\n\n% Write the last page:\n\\def\\write@last@page{%\n\\write\\@mainaux{\\string\\global\\string\\@namedef{@lastpage}{\\the\\c@page}}}\n\n\\AtEndDocument{\\write@last@page}\n% SGML\n\\long\\def\\convertas#1#2{#2}\n\\def\\sday#1{#1}\\def\\smonth#1{#1}\\def\\syear#1{#1}\n\\def\\aid#1{\\gdef\\@aid{#1}}\n%\n\\def\\SSDI#1{\\gdef\\@ssdi{#1}} \\def\\@ssdi{000000-00}\n\\def\\issn#1{\\gdef\\@issn{#1}}\n\\def\\price#1{\\gdef\\@price{#1}}\n%\n\\def\\date#1{\\gdef\\@date{#1}}    \\def\\@date{\\today}\n%\n\n\\def\\empty@data{\\@nil}\n%\n\n%***************** BACKMATTER\n\\newcommand\\backmatter{\\goodbreak}\n\n%**************** INICIALIZATION\n\\newcommand\\refname{References}\n\\newcommand\\figurename{Figure}\n\\newcommand\\tablename{Table}\n\\newcommand\\algorithmname{Algorithm}\n\\newcommand\\appendixname{Appendix}\n\\newcommand\\abstractname{Abstract. }\n\\newcommand\\keywordsname{Keywords. }\n\\def\\acknowledgementsname{Acknowledgements}\n%\n\\def\\copyright@sign{\\copyright}\n%\n% DIMENSIONS\n\\def\\@articletypesize{\\large}\n\\def\\pretitle@size{\\LARGE}\n\\def\\title@size{\\huge}\n\\def\\subtitle@size{\\large\\itshape}\n\\def\\authors@size{\\normalsize}\n\\def\\abstract@size{\\footnotesize}\n\\def\\abstract@width{22pc}\n\\def\\abstract@indent{\\noindent}\n\\def\\address@size{\\normalsize\\itshape}\n% Block preparation of contents:\n\\def\\addcontentsline#1#2#3{}\n\\long\\def\\addtocontents#1#2{}\n%\n\\newcommand\\today{}\n\\edef\\today{\\ifcase\\month\\or\n  January\\or February\\or March\\or April\\or May\\or June\\or\n  July\\or August\\or September\\or October\\or November\\or December\\fi\n  \\space\\number\\day, \\number\\year}\n%\n\\@twosidetrue\n\\pagenumbering{arabic}\n\\frenchspacing\n\\init@settings\n\n\\if@twocolumn\\setlength\\tablewidth{\\columnwidth}\n\\else\\setlength\\tablewidth{\\textwidth}\\fi\n%\\pagestyle{headings}\n\\pagestyle{empty}\n\n\\endinput\n%%\n%% End of file `IOS-Book-Article.cls'.\n"
  },
  {
    "path": "marktoberdorf_paper/forPublisher/css.sty",
    "content": "%---------------------------------------------------------------------------\n%  Copyright 2013 Microsoft Corporation.\n% \n%  This is free software; you can redistribute it and/or modify it under the\n%  terms of the Apache License, Version 2.0. A copy of the License can be\n%  found in the file \"license.txt\" at the root of this distribution.\n%---------------------------------------------------------------------------\n\\NeedsTeXFormat{LaTeX2e}[1995/12/01]\n\n\\RequirePackage{iftex}\n\\RequirePackage{etoolbox}\n\\RequirePackage{xkeyval}\n\\RequirePackage[table]{xcolor}\n\\RequirePackage{mdframed}\n\\RequirePackage{graphicx}\n\\RequirePackage{tablefootnote}\n\n% font selection\n\\ifXeTeX\\RequirePackage{fontspec}\\else\n\\ifLuaTeX\\RequirePackage{fontspec}\\else\n\\providecommand{\\fontspec}[2][]{}\n\\fi\\fi\n\n\n% Define CSS 17 standard colors\n\\definecolor{Red}{HTML}{FF0000}\n\\definecolor{Lime}{HTML}{00FF00}\n\\definecolor{Blue}{HTML}{0000FF}\n\n\\definecolor{Yellow}{HTML}{FFFF00}\n\\definecolor{Cyan}{HTML}{00FFFF}\n\\definecolor{Magenta}{HTML}{FF00FF}\n\n\\definecolor{Navy}{HTML}{000080}\n\\definecolor{Maroon}{HTML}{800000}\n\\definecolor{Green}{HTML}{008000}\n\n\\definecolor{Teal}{HTML}{008080}\n\\definecolor{Purple}{HTML}{800080}\n\\definecolor{Olive}{HTML}{808000}\n\n\\definecolor{Black}{HTML}{000000}\n\\definecolor{Dimgray}{HTML}{696969}\n\\definecolor{Gray}{HTML}{808080}\n\\definecolor{Darkgray}{HTML}{A9A9A9}\n\\definecolor{Silver}{HTML}{C0C0C0}\n\\definecolor{Lightgray}{HTML}{D3D3D3}\n\\definecolor{Gainsboro}{HTML}{DCDCDC}\n\\definecolor{Floralwhite}{HTML}{FFFAF0}\n\\definecolor{Ivory}{HTML}{FFFFF0}\n\\definecolor{White}{HTML}{FFFFFF}\n\n\\definecolor{Orange}{HTML}{FFA500}\n\\definecolor{Aqua}{HTML}{00FFFF}\n\\definecolor{Fuchsia}{HTML}{FF00FF}\n\n% ---------------------------------------------\n% Basic LaTeX helpers\n% ---------------------------------------------\n\n% use '\\defcommand' to define a command no matter if it is predefined or not\n\\def\\defcommand{\\@ifstar\\defcommand@S\\defcommand@N}\n\\def\\defcommand@S#1{\\let#1\\outer\\renewcommand*#1}\n\\def\\defcommand@N#1{\\let#1\\outer\\renewcommand#1} \n\n\\def\\providecsgdef#1{\\ifcsdef{#1}{\\providecommand\\@foo}{\\csgdef{#1}}}\n\n\\providecommand\\@swap[2]{#2#1}\n\\providecommand\\@swaparg[2]{#2{#1}}\n\\providecommand\\expandnext[2]{\\expandafter\\@swaparg\\expandafter{#2}{#1}}\n\\newcommand\\@expandafter\\expandnext %legacy\n\\newcommand\\expandnextii[3]{\\expandnext{\\expandnext{#1}{#2}}{#3}}\n\\newcommand\\expandnextiii[4]{\\expandnext{\\expandnextii{#1}{#2}{#3}}{#4}}\n\\newcommand\\expandnextiv[5]{\\expandnext{\\expandnextiii{#1}{#2}{#3}{#4}}{#5}}   \n\n\\newcommand{\\eifstrequal}{\\expandafter\\ifstrequal\\expandafter}\n\\newcommand{\\eeifstrequal}[2]{\\expandnext{\\eifstrequal{#1}}{#2}}\n\n\\providecommand\\providelength[1]{%\n  \\begingroup\n    \\escapechar\\m@ne\n    \\xdef\\@gtempa{\\string#1}%\n  \\endgroup\n  \\@ifundefined{\\@gtempa}%\n    {\\newskip#1}%\n    {}%\n}\n\n% is a string an element of a list of (comma separated) strings\n\\newcommand{\\eifstrelement}[4]{%\n  \\def\\@found{}%\n  \\@for\\@ii:=#2\\do{%\n    \\eeifstrequal{\\@ii}{#1}{\\def\\@found{true}}{}%\n  }%\n  \\ifdefvoid{\\@found}{#4}{#3}%\n}\n\n% do two lists of strings intersect?\n\\newcommand{\\ifintersect}[4]{%\n  \\def\\@intersect{}%\n  \\@for\\@sname:=#1\\do{%\n  \t\\ifdefvoid{\\@intersect}{%\n    \t\\eifstrelement{\\@sname}{#2}{\\def\\@intersect{true}}{}%\n    }{}%\n  }%\n  \\ifdefvoid{\\@intersect}{#4}{#3}%\n}\n\n% get string head and tail\n\\def\\strsplit#1{\\expandafter\\strsplitx#1\\empty\\empty\\empty}\n\\def\\strsplitx#1#2\\empty{%\n\t\\edef\\strhead{#1}%\n\t\\edef\\strtail{#2}%\n}\n\n% normalize colors: to lowercase and then capitalize\n\\newcommand{\\cssDefNormalizeColor}[2]{%\n\t\\expandafter\\@cssNormColor#2\\empty{#1}\\empty%\n}\n\\def\\@cssNormColor#1#2\\empty#3\\empty{%\n\t\\uppercase{\\def\\@hd{#1}}\\lowercase{\\def\\@tl{#2}}%\n\t\\expandafter\\global\\expandafter\\edef\\csname #3\\endcsname{\\@hd\\@tl}%\n}\n\n\n% ---------------------------------------------------\n% Some TeX stuff to compose functions\n% ---------------------------------------------------\n\\newcommand{\\apptox}[2]{%  apptox{\\cmd1}{\\cmd2} == newcommand{\\cmd1'}[1]{\\cmd1{\\cmd2{#1}}}\n  \\providecommand{#1}[1]{##1}% define it if necessary (as identity)\n  \\protected@edef#1##1{#1{\\protect #2{##1}}}%\n}\n\n\\newcommand{\\pretox}[2]{%  pretox{\\cmd1}{\\cmd2} == newcommand{\\cmd1'}[1]{\\cmd2{\\cmd1{#1}}}\n  \\providecommand{#1}[1]{##1}%\n  \\protected@edef#1##1{\\protect #2{#1{##1}}}%\n}\n\n%-------------------------------------------------------------\n% Save footnotes inside mdframed and minipage environments\n%-------------------------------------------------------------\n\\newif\\if@saveFootnotes\n\\newcommand{\\cssSaveFootnotes}%\n\t{\\if@saveFootnotes\\else%\n\t\t\\let\\footnote\\tablefootnote%\n\t \\fi%\n\t \\@saveFootnotestrue}%\n\\newcommand{\\cssRestoreFootnotes}%\t \n\t{\\if@saveFootnotes\\else%\n \t\t\\tfn@tablefootnoteprintout% \n \t\t\\gdef\\tfn@fnt{0}%\n \t \\fi}% \n\n%-------------------------------------------------------------\n% Setup mdframed with default values\n%-------------------------------------------------------------\n\\newlength{\\cssPixel}\\setlength{\\cssPixel}{0.4pt}% assume 180 dpi\n\\mdfsetup{%\n\tleftmargin=0pt,%\n\trightmargin=0pt,%\n\tskipabove=0pt,%\n\tskipbelow=0pt,%\n\tinnertopmargin=0pt,%\n\tinnerbottommargin=0pt,%\n\tinnerleftmargin=0pt,%\n\tinnerrightmargin=0pt,%\n\tmiddlelinewidth=0pt,%\n\tlinewidth=0pt,%\n\touterlinewidth=0pt,innerlinewidth=0pt%\n}\n\n\n% ---------------------------------------------------\n% Basic command to process attributes passed to TeX\n% ---------------------------------------------------\n\\newif\\if@usewrap\n\\newcommand{\\@doBefore}{}\n\\newcommand{\\@doAfter}{}\n\\newcommand{\\@wrapCmd}[1]{#1}\n\n\\newcommand{\\@cssUseCmd}{\\renewcommand{\\@wrapCmd}[1]{##1}\\renewcommand{\\@doAfter}{}\\@usewraptrue}\n\\newcommand{\\@cssUseEnv}{\\renewcommand{\\@doBefore}{}\\renewcommand{\\@doAfter}{}\\@usewrapfalse}\n\n\\newcommand{\\@cssApplyCmd}[1]{{\\@wrapCmd{#1}}}\n\\newcommand{\\@cssApplyBefore}{\\@doBefore{}}\n\\newcommand{\\@cssApplyAfter}{\\@doAfter{}}\n\n\\newcommand{\\@cssProcessAttrs}[2]{%\n  \\setkeys*{cssx}{#1}\\setrmkeys*{csspre}\\setrmkeys*{css}\\setrmkeys*{csspost}% defaults\n  \\@cssApplyRulesFor{parentclass}{css}{\\cssParentClass}%\n  \\setkeys*{cssx}{#2}\\setrmkeys*{csspre}\\setrmkeys*{css}\\setrmkeys*{csspost}% regular\n  \\protected@edef\\cssParentClass{\\cssClass}%\n}\n\n\n\\newcommand{\\@cmdBefore}[2]{#1#2}\n\\newcommand{\\@cmdAfter}[2]{#2#1}\n\n\\newcommand{\\cssWrapCmd}[1]{\\apptox{\\@wrapCmd}{#1}}\n\\newcommand{\\cssDoBeforeX}[1]{#1}\n\\newcommand{\\cssDoAfterX}[1]{\\preto\\@doAfter{#1}}\n\\newcommand{\\cssDoBefore}[1]{\\if@usewrap\\cssWrapCmd{\\@cmdBefore{#1}}\\else #1\\fi}\n\\newcommand{\\cssDoAfter}[1]{\\if@usewrap\\cssWrapCmd{\\@cmdAfter{#1}}\\else\\preto\\@doAfter{#1}\\fi}\n%\\newcommand{\\cssDoBeforeX}[1]{\\if@usewrap\\cssWrapCmd{\\@cmdBefore{#1}}\\else #1\\fi}\n%\\newcommand{\\cssDoAfterX}[1]{\\if@usewrap\\cssWrapCmd{\\@cmdAfter{#1}}\\else\\preto\\@doAfter{#1}\\fi}\n\n\\newcommand{\\cssDoEnv}[1]{\\cssDoBefore{\\protect\\begin{#1}}\\cssDoAfter{\\protect\\end{#1}}}\n\\newcommand{\\cssDoEnvOpt}[2]{\\cssDoBefore{\\begin{#1}[#2]}\\cssDoAfter{\\end{#1}}}\n\\newcommand{\\cssDoEnvArg}[2]{\\cssDoBefore{\\begin{#1}{#2}}\\cssDoAfter{\\end{#1}}}\n\\newcommand{\\cssDoEnvArgII}[3]{\\cssDoBefore{\\begin{#1}{#2}{#3}}\\cssDoAfter{\\end{#1}}}\n\n\\newcommand{\\newKey}[4][]{\\define@key{#2}{#3}[#1]{#4}}\n\\newcommand{\\newLength}[2]{\\providelength{#1}\\setlength{#1}{#2}}\n\n\n\\newcommand{\\@cssReset}{}\n\\newcommand{\\cssAddReset}[1]{\\appto{\\@cssReset}{#1}}\n\\newcommand{\\cssNewResetCommand}[2]{\\newcommand{#1}{#2}\\cssAddReset{\\renewcommand{#1}{#2}}}\n\n\\newlength{\\cssFill}\n\\setlength{\\cssFill}{2sp plus 1fill minus 2sp} % make \\fill unequal to 0pt, and detectable as 2sp\n\n\\newcommand{\\cssNewLengthKey}[4][0pt]{%\n\t\\newLength{#4}{#1}%\n\t\\newKey{#2}{#3}{%\n\t\t\\ifstrequal{##1}{auto}{\\setlength{#4}{\\cssFill}}{\\setlength{#4}{##1}}%\n\t}%\n\t\\cssAddReset{\\setlength{#4}{#1}}%\n}\n\n\n\\newcommand{\\cssNewKeyNoReset}[4]{%\n\t\\newcommand{#3}{#4}%\n\t\\newKey{#1}{#2}{\\renewcommand{#3}{##1}}%\n}\n\n\\newcommand{\\cssNewKey}[4]{%\n\t\\cssNewResetCommand{#3}{#4}%\n\t\\newKey{#1}{#2}{%(#2=##1)%debug key setting\n\t\t\\renewcommand{#3}{##1}}%\n}\n\\newcommand{\\cssNewKeyX}[5]{%\n\t\\cssNewResetCommand{#3}{#4}%\n\t\\newKey{#1}{#2}{\\renewcommand{#3}{##1}#5{##1}}%\n}\n\\newcommand{\\cssNewListKey}[4]{%\n\t\\cssNewResetCommand{#3}{#4}%\n\t\\newKey{#1}{#2}{\\appto{#3}{,##1}}%\n}\n\\newcommand{\\cssNewListKeyX}[5]{%\n\t\\cssNewResetCommand{#3}{#4}%\n\t\\newKey{#1}{#2}{\\appto{#3}{,##1}#5{##1}}%\n}\n\\newcommand{\\cssNewPseudoKey}[3]{%\n\t\\newKey{#1}{#2}{\\setkeys{#1}{#3}}%\n}\n\n%-------------------------------------------------------------\n% css: display\n%-------------------------------------------------------------\n\\cssNewKey{css}{display}{\\cssDisplay}{block}\n\n%-------------------------------------------------------------\n% css: width, height, and margins\n%-------------------------------------------------------------\n\n\\cssNewLengthKey{css}{margin-left}{\\cssMarginLeft}\n\\cssNewLengthKey{css}{margin-right}{\\cssMarginRight}\n\\cssNewLengthKey{css}{margin-top}{\\cssMarginTop}\n\\cssNewLengthKey{css}{margin-bottom}{\\cssMarginBottom}\n\\cssNewLengthKey[1sp]{css}{width}{\\cssWidth}\n\\cssNewLengthKey[1sp]{css}{height}{\\cssHeight}\n\\cssNewKey{css}{vertical-align}{\\cssVerticalAlign}{}\n\n\\cssNewPseudoKey{css}{margin}{margin-top=#1,margin-bottom=#1,margin-left=#1,margin-right=#1}\n\n\n\\newcommand{\\@cssProcessMargins}{%\n\t\\eifstrequal{\\cssDisplay}{block}%\n\t\t{\\@cssBlockEndPar\\@cssBlockMargins}%\n\t{\\eifstrequal{\\cssDisplay}{inline-block}%\n\t\t{\\@cssBlockMargins}%\n\t\t{\\@cssInlineMargins}%\n\t}%\n}\n\n\\newcommand{\\@cssBlockEndPar}{%\n\t\\cssIfHasClass{para-continue,para-block}{}{\\cssDoAfterX{\\leavevmode\\par\\global\\hangindent=0pt\\relax}}%\n}\n\n\\newif\\if@hasdim\n\n\\newlength{\\cssHeightFull} % height including padding and border\n\\newlength{\\cssWidthFull}  % width including padding and border\n\\newLength{\\@cssMarginAfter}{0pt}\n\\newLength{\\@cssParSkip}{\\parskip}\n\\newLength{\\@cssParIndent}{\\parindent}\n\\newcommand{\\@cssFixMathSpacing}{\\strut\\vspace{-\\baselineskip}} % fixes weird abovedisplay skip spacing\n\\newcommand{\\@cssBlockMargins}{%\n\t\\@hasdimfalse\n\t\\ifdim\\cssWidth=1sp\\setlength{\\cssWidthFull}{1sp}\\else\\@hasdimtrue\\fi\n\t\\ifdim\\cssHeight=1sp\\setlength{\\cssHeightFull}{1sp}\\else\\@hasdimtrue\\fi\n\t\\if@hasdim%\n\t\t% set full height and width\n\t\t\\setlength{\\cssWidthFull}{\\dimexpr\\cssWidth+\\cssPaddingLeft+\\cssPaddingRight\\relax}%\n\t\t\\eifstrequal{\\cssBorderLeftStyle}{none}{}{\\addtolength{\\cssWidthFull}{\\cssBorderWidth}}%\n\t\t\\eifstrequal{\\cssBorderRightStyle}{none}{}{\\addtolength{\\cssWidthFull}{\\cssBorderWidth}}%\n\t\t\\setlength{\\cssHeightFull}{\\dimexpr\\cssHeight+\\cssPaddingTop+\\cssPaddingBottom\\relax}%\n\t\t\\eifstrequal{\\cssBorderTopStyle}{none}{}{\\addtolength{\\cssHeightFull}{\\cssBorderWidth}}%\n\t\t\\eifstrequal{\\cssBorderBottomStyle}{none}{}{\\addtolength{\\cssHeightFull}{\\cssBorderWidth}}%\t\t\n\t\t% set default width?\n\t\t\\ifdim\\cssWidth=1sp% in this case, cssWidthFull is just padding and borders\n\t\t\t\\setlength{\\cssWidth}{\\dimexpr\\linewidth-\\cssWidthFull-\\cssMarginLeft-\\cssMarginRight}%\n\t\t\t\\addtolength{\\cssWidthFull}{\\cssWidth}%\n\t\t\\fi%\n\t\t%minipage\t\t\t\t\n\t\t\\ifdim\\cssMarginTop=0pt\\else\\cssDoBeforeX{\\vspace{\\cssMarginTop}}\\fi\n\t\t\\ifdim\\cssMarginLeft=0pt\\else\\cssDoBeforeX{\\hspace*{\\cssMarginLeft}}\\fi\n\t\t\\setlength{\\@cssParIndent}{\\parindent}% save parskip and parindent since minipage resets it\n\t\t\\setlength{\\@cssParSkip}{\\parskip}%\n\t\t\\eifstrequal{\\cssVerticalAlign}{bottom}{\\def\\@cssValign{b}}%\n\t\t{\\eifstrequal{\\cssVerticalAlign}{center}{\\def\\@cssValign{c}}%\n\t\t{\\eifstrequal{\\cssVerticalAlign}{top}{\\def\\@cssValign{t}}%\n\t\t{\\def\\@cssValign{c}}}}% including `middle`\n\t\t\\ifdim\\cssHeight=1sp%\n\t   \t\t\\cssDoBeforeX{\\begin{minipage}[\\@cssValign]{\\cssWidthFull}}%\n\t\t\\else\n\t\t\t\\cssDoBeforeX{\\begin{minipage}[\\@cssValign][\\cssHeightFull]{\\cssWidthFull}}%\n\t\t\\fi\n\t\t\\cssDoBeforeX{\\cssSaveFootnotes\\setlength{\\parskip}{\\@cssParSkip}\\setlength{\\parindent}{\\@cssParIndent}}%\n\t\t%note: DoAfter prepends, so in opposite order\n\t\t\\ifdim\\cssMarginBottom=0pt\\else\\cssDoAfterX{\\vspace{\\cssMarginBottom}}\\fi\n\t\t\\ifdim\\cssMarginRight=0pt\\else\\cssDoAfterX{\\hspace*{\\cssMarginRight}}\\fi\n\t\t\\cssDoAfterX{\\end{minipage}\\cssRestoreFootnotes}%\n\t\\else\n\t\t% no height/width: trivlist\n\t\t\\@hasdimfalse\n\t\t\\ifdim\\cssMarginLeft=0pt\\else\\@hasdimtrue\\fi\n\t\t\\ifdim\\cssMarginRight=0pt\\else\\@hasdimtrue\\fi\n\t\t\\ifdim\\cssMarginTop=0pt\\else\\@hasdimtrue\\fi\n\t\t\\ifdim\\cssMarginBottom=0pt\\else\\@hasdimtrue\\fi\n\t  \\if@hasdim\n\t\t\t\\setlength{\\@cssMarginAfter}{\\dimexpr\\cssMarginBottom-\\cssMarginTop\\relax}%\n\t\t\t\\list{}{%\n\t\t\t\t\\leftmargin=\\cssMarginLeft%\n\t\t\t\t\\rightmargin=\\cssMarginRight%\n\t\t\t\t\\topsep=\\cssMarginTop%\n\t\t\t\t\\itemsep=0pt%\n\t\t\t\t\\parsep=0pt%\n\t\t\t\t\\parskip=0pt%\n\t\t\t\t\\partopsep=0pt%\t\n\t\t\t\t\\listparindent=\\parindent%\t\n\t\t\t}%\n\t\t\t\\cssDoAfterX{\\endlist}%\n\t\t\t\\ifdim\\@cssMarginAfter=0pt\\else\\cssDoAfter{\\vspace{\\@cssMarginAfter}}\\fi%\n\t\t\t\\eifstrequal{\\cssTextAlign}{left}% we need to do alignment here for inline cmds' with a display=block\n\t\t\t\t{\\raggedright}%\n\t\t\t{\\eifstrequal{\\cssTextAlign}{right}%\n\t\t\t\t{\\raggedleft}%\n\t\t\t{\\eifstrequal{\\cssTextAlign}{center}%\n\t\t\t\t{\\centering}%\n\t\t\t{}}}%\n\t\t\t\\cssIfHasClass{math-display}%\n\t\t\t\t{\\item\\@cssFixMathSpacing}%\n\t\t\t\t{\\item\\relax}%\t\n\t\t\\fi\n\t\\fi\n}\n\n\\newcommand{\\@cssHide}[1]{}\n\\newcommand{\\@cssInlineMargins}{%\n\t\\ifdim\\cssMarginLeft=0pt\\else\\cssDoBefore{\\hspace*{\\cssMarginLeft}}\\fi\n\t\\ifdim\\cssMarginRight=0pt\\else\\cssDoAfter{\\hspace*{\\cssMarginRight}}\\fi\n\t\\ifdim\\cssMarginBottom=0pt\\else\\cssDoBefore{\\rule[-\\cssMarginBottom]{0pt}{\\cssMarginBottom}}\\fi\n\t\\ifdim\\cssMarginTop=0pt\\else\\cssDoBefore{\\rule{0pt}{\\dimexpr\\baselineskip*0.7+\\cssMarginTop\\relax}}\\fi\n\t\\eifstrequal{\\cssDisplay}{hidden}{%\n\t\t\\cssWrapCmd{\\@cssHide}%\n\t}{}%\n}\n\n%-------------------------------------------------------------\n% css: Borders and padding \n%-------------------------------------------------------------\n\n\\cssNewLengthKey{css}{padding-left}{\\cssPaddingLeft}\n\\cssNewLengthKey{css}{padding-right}{\\cssPaddingRight}\n\\cssNewLengthKey{css}{padding-top}{\\cssPaddingTop}\n\\cssNewLengthKey{css}{padding-bottom}{\\cssPaddingBottom}\n\n\\newlength{\\cssBorderWidthTotal}\n\\cssNewLengthKey[\\cssPixel]{css}{border-width}{\\cssBorderWidth}\n\\cssNewKey{css}{border-color}{\\cssBorderColor}{black}\n\\cssNewKey{css}{border-top-style}{\\cssBorderTopStyle}{none}\n\\cssNewKey{css}{border-bottom-style}{\\cssBorderBottomStyle}{none}\n\\cssNewKey{css}{border-left-style}{\\cssBorderLeftStyle}{none}\n\\cssNewKey{css}{border-right-style}{\\cssBorderRightStyle}{none}\n\\cssNewKey{css}{background-color}{\\cssBackgroundColor}{white}\n\n\\cssNewPseudoKey{css}{padding}{padding-top=#1,padding-bottom=#1,padding-right=#1,padding-left=#1}\n\n\\cssNewPseudoKey{css}{border-style}%\n\t{border-top-style=#1,border-bottom-style=#1,border-left-style=#1,border-right-style=#1}\n\n\\newcommand{\\@cssProcessPadding}{%\n\t\\eifstrequal{\\cssDisplay}{block}%\n\t\t{\\@cssBlockPadding}%\n\t{\\eifstrequal{\\cssDisplay}{block-inline}%\n\t\t{\\@cssBlockPadding}%\n\t\t{\\@cssInlinePadding}%\n\t}}\n\n% Special math-framed environment that fixes vertical spacing around math display\n\\newenvironment{mdmathframed}[1][]%\n\t{\\begin{mdframed}[#1]\\@cssFixMathSpacing}%\n\t{\\@cssFixMathSpacing\\end{mdframed}}\n\n\n\\newif\\if@needframe\n\\newLength{\\@cssPaddingLength}{0pt}\n\\newcommand{\\@cssFramedArgs}{}\n\\newcommand{\\@cssBorderStyleAll}{}\n\\newcommand{\\@cssBlockPadding}{%\n\t\\@needframefalse%\n\t\\eifstrequal{\\cssBorderTopStyle}{none}{}{\\@needframetrue}%\n\t\\eifstrequal{\\cssBorderBottomStyle}{none}{}{\\@needframetrue}%\n\t\\eifstrequal{\\cssBorderLeftStyle}{none}{}{\\@needframetrue}%\n\t\\eifstrequal{\\cssBorderRightStyle}{none}{}{\\@needframetrue}%\n\t\\eifstrequal{\\cssBackgroundColor}{white}{}{\\@needframetrue}%\n\t\\ifdim\\cssPaddingTop=0pt\\else\\@needframetrue\\fi\n\t\\ifdim\\cssPaddingBottom=0pt\\else\\@needframetrue\\fi\n\t\\ifdim\\cssPaddingLeft=0pt\\else\\@needframetrue\\fi\n\t\\ifdim\\cssPaddingRight=0pt\\else\\@needframetrue\\fi\n\t\\strsplit{\\cssBackgroundColor}%\t\t\t\n\t  \\eifstrequal{\\strhead}{\\#}%\n\t  {\\definecolor{Temp}{HTML}{\\strtail}\\edef\\@@bcolor{Temp}}%\n\t  {\\cssDefNormalizeColor{@bcolor}{\\cssBackgroundColor}\\edef\\@@bcolor{\\@bcolor}}%\t\t\n\t%\\expandafter\\lowercase\\expandafter{\\expandafter\\def\\expandafter\\bcolor\\expandafter{\\cssBackgroundColor}}%\t\n\t\\if@needframe%\t\t\n\t\t\\cssDoAfter{\\cssRestoreFootnotes}% first, because post commands are pre-pended\n\t\t\\renewcommand{\\@cssFramedArgs}{%\n\t\t\tinnertopmargin=\\the\\cssPaddingTop,%\n\t\t\tinnerbottommargin=\\the\\cssPaddingBottom,%\n\t\t\tinnerleftmargin=\\the\\cssPaddingLeft,%\n\t\t\tinnerrightmargin=\\the\\cssPaddingRight,%\n\t\t\tlinewidth=\\the\\cssBorderWidth,%\n\t\t\tlinecolor=\\cssBorderColor,%\n\t\t\tbackgroundcolor=\\@@bcolor%\n\t\t}%\n\t\t\\setlength{\\cssBorderWidthTotal}{0pt}%\n\t\t\\eifstrequal{\\cssBorderTopStyle}{none}{\\appto{\\@cssFramedArgs}{,topline=false}}{}%\t\n\t\t\\eifstrequal{\\cssBorderBottomStyle}{none}{\\appto{\\@cssFramedArgs}{,bottomline=false}}{}%\t\n\t\t\\eifstrequal{\\cssBorderLeftStyle}{none}{\\appto{\\@cssFramedArgs}{,leftline=false}}%\n\t\t\t{\\addtolength{\\cssBorderWidthTotal}{\\cssBorderWidth}}%\t\n\t\t\\eifstrequal{\\cssBorderRightStyle}{none}{\\appto{\\@cssFramedArgs}{,rightline=false}}%\t\t\n\t\t\t{\\addtolength{\\cssBorderWidthTotal}{\\cssBorderWidth}}%\t\n\t\t\\cssIfHasClass{math-display}%\n\t\t\t{\\expandnext{\\cssDoEnvOpt{mdmathframed}}{\\@cssFramedArgs}}%\n\t\t\t{\\expandnext{\\cssDoEnvOpt{mdframed}}{\\@cssFramedArgs}}%\n\t\t% insert a minipage if height or width was set so the frame is as large\n\t\t\\@hasdimfalse\n\t\t\\ifdim\\cssWidth=1sp\\else\\@hasdimtrue\\fi\n\t\t\\ifdim\\cssHeight=1sp\\else\\@hasdimtrue\\fi\n\t\t\\if@hasdim%\n\t\t\t\\ifdim\\cssHeight=1sp%\t\t\t\t\t\n\t\t   \t\t\\cssDoBefore{\\begin{minipage}{\\cssWidth}}%\n\t\t\t\\else\n\t\t\t\t\\cssDoBefore{\\begin{minipage}[t][\\cssHeight]{\\cssWidth}}%\n\t\t    \\fi\n\t\t    \\cssDoBefore{\\setlength{\\parskip}{\\@cssParSkip}\\setlength{\\parindent}{\\@cssParIndent}}%\n\t\t\t%note: DoAfter prepends, so in opposite order\n\t\t\t\\cssDoAfter{\\end{minipage}}%\n\t\t\\fi    \n\t\t\\cssDoBefore{\\cssSaveFootnotes}%\n\t\\fi\n}\n\n\\newcommand{\\@robustFramebox}[2]{%\n\t\\eifstrequal{\\cssTextAlign}{center}{\\framebox[#1][c]{#2}}%\n\t{\\eifstrequal{\\cssTextAlign}{right}{\\framebox[#1][r]{#2}}%\n\t{\\framebox[#1][l]{#2}}}%\n}\n\n\\newcommand{\\@robustMakebox}[2]{%\n\t\\eifstrequal{\\cssDisplay}{table-cell}%\n\t\t{\\@robustTableParbox{#1}{#2}}%\n\t\t{\\eifstrequal{\\cssTextAlign}{center}{\\makebox[#1][c]{#2}}%\n\t\t {\\eifstrequal{\\cssTextAlign}{right}{\\makebox[#1][r]{#2}}%\n\t\t {\\makebox[#1][l]{#2}}}}%\n}\n\n\\newcommand{\\@robustRaisebox}[2]{%\n\t\\raisebox{#1}{#2}%\n}\n\n\\newcommand{\\@robustHeight}[1]{%\n\t\\eifstrequal{\\cssVerticalAlign}{top}%\n\t\t{\\raisebox{0pt}[0pt][\\cssHeight]{#1}}%\n\t{\\eifstrequal{\\cssVerticalAlign}{middle}%\n\t\t{\\raisebox{0pt}[0.5\\cssHeight][0.5\\cssHeight]{#1}}%\n\t{\\eifstrequal{\\cssVerticalAlign}{baseline}%\n\t\t{\\raisebox{0pt}[\\dimexpr\\cssHeight-\\depth\\relax][\\depth]{#1}}%\n\t\t{\\raisebox{0pt}[\\cssHeight][0pt]{#1}}% bottom\n\t}}%\n}\n\n\\newcommand{\\@robustTableParbox}[2]{%\n\t\\eifstrequal{\\cssVerticalAlign}{top}{\\def\\@cssValign{t}}%\n\t{\\eifstrequal{\\cssVerticalAlign}{center}{\\def\\@cssValign{c}}%\n\t{\\eifstrequal{\\cssVerticalAlign}{middle}{\\def\\@cssValign{c}}}%\n\t{\\def\\@cssValign{b}}}%\n\t\\ifdim\\cssHeight=1sp%\n\t\t\\parbox[\\@cssValign]{#1}{#2}%\n\t\\else%\n\t\t\\parbox[\\@cssValign][\\cssHeight]{#1}{#2}%\n\t\\fi%\n}\n\n\\newcommand{\\@cssInlinePadding}{%\n\t\\eifstrequal{\\cssBackgroundColor}{}{}%\n\t  {\\eifstrequal{\\cssBackgroundColor}{white}{}%\n\t\t{\\strsplit{\\cssBackgroundColor}%\t\t\t\n\t\t \\eifstrequal{\\strhead}{\\#}%\n\t\t \t{\\cssWrapCmd{\\protect\\colorbox[HTML]{\\strtail}}}%\n\t\t \t{\\cssWrapCmd{\\@robustColorbox{\\cssBackgroundColor}}}%\n\t\t}%\n\t  }%\n\t\\@needframefalse%\n\t\\eifstrequal{\\cssBorderTopStyle}{none}{}{\\@needframetrue}%\n\t\\eifstrequal{\\cssBorderBottomStyle}{none}{}{\\@needframetrue}%\n\t\\eifstrequal{\\cssBorderLeftStyle}{none}{}{\\@needframetrue}%\n\t\\eifstrequal{\\cssBorderRightStyle}{none}{}{\\@needframetrue}%\t\t\n\t\\if@needframe%\n\t\t\\setlength{\\fboxrule}{\\cssBorderWidth}%\n\t\t\\ifdim\\cssWidth=1sp%\n\t\t\t\\cssWrapCmd{\\fbox}%\n\t\t\\else\n\t\t    \\cssWrapCmd{\\@robustFramebox{\\cssWidth}}%\n\t\t\\fi\n\t\\else\n\t\t\\ifdim\\cssWidth=1sp\\else\n\t\t\t\\cssWrapCmd{\\@robustMakebox{\\cssWidth}}%\n\t\t\\fi\n\t\\fi\n\t% height?\n\t\\ifdim\\cssHeight=1sp\\else\\cssWrapCmd{\\@robustHeight}\\fi\n\t% raisebox?\n\t\\eifstrequal{\\cssDisplay}{inline}{%\n\t\t\\eifstrequal{\\cssVerticalAlign}{}{}%\n\t\t{\\eifstrequal{\\cssVerticalAlign}{top}{}%\n\t\t{\\eifstrequal{\\cssVerticalAlign}{bottom}{}%\n\t\t{\\eifstrequal{\\cssVerticalAlign}{middle}{}%\n\t\t{\\eifstrequal{\\cssVerticalAlign}{baseline}{}%\n\t\t{\\cssWrapCmd{\\@robustRaisebox{\\cssVerticalAlign}}%\n\t\t}}}}}%\n\t}{}%\n\t% padding\n\t\\if@needframe\n\t\t\\setlength{\\fboxsep}{\\cssPaddingTop}%  todo: define our own box so we can set paddingtop/bot separately\n\t\t\\ifdim\\cssPaddingBottom>\\fboxsep\\setlength{\\fboxsep}{\\cssPaddingBottom}\\fi\t\n\t\t\\ifdim\\cssPaddingLeft=\\fboxsep\\else\\hspace*{\\dimexpr\\cssPaddingLeft-\\fboxsep\\relax}\\fi\n\t\t\\ifdim\\cssPaddingRight=\\fboxsep\\else\\hspace*{\\dimexpr\\cssPaddingRight-\\fboxsep\\relax}\\fi\n\t\\else\n\t\t\\ifdim\\cssPaddingLeft=0pt\\else\\cssDoBefore{\\hspace*{\\cssPaddingLeft}}\\fi\n\t\t\\ifdim\\cssPaddingRight=0pt\\else\\cssDoAfter{\\hspace*{\\cssPaddingRight}}\\fi\n\t\t\\ifdim\\cssPaddingBottom=0pt\\else\\cssDoBefore{\\protect\\rule[-\\cssPaddingBottom]{0pt}{\\cssPaddingBottom}}\\fi\n\t\t\\ifdim\\cssPaddingTop=0pt\\else\\cssDoBefore{\\protect\\rule{0pt}{\\dimexpr\\cssPaddingTop+0.8em\\relax}}\\fi\n\t\\fi\n}\t\n\n%-------------------------------------------------------------\n% css: Textalign, textindent etc \n%-------------------------------------------------------------\n\n\\cssNewLengthKey[1sp]{css}{text-indent}{\\cssTextIndent}\n\\cssNewKey{css}{text-align}{\\cssTextAlign}{justify}\n\\cssNewLengthKey{css}{line-height}{\\cssLineHeight}\n\\cssNewKey{css}{float}{\\cssFloat}{}\n\n\\DeclareRobustCommand{\\@robustColor}[1]{%\n\t\\cssDefNormalizeColor{@fcolor}{#1}\\color{\\@fcolor}%\n}\n\\DeclareRobustCommand{\\@robustColorbox}[2]{%\n\t\\cssDefNormalizeColor{@bcolor}{#1}\\colorbox{\\@bcolor}{#2}%\n}\n\n\n\\newcommand{\\@cssProcessText}{%\n\t\\eifstrequal{\\cssDisplay}{block}%\n\t\t{\\@cssBlockText}%\n\t{\\eifstrequal{\\cssDisplay}{block-inline}%\n\t\t{\\@cssBlockText}%\n\t{\\eifstrequal{\\cssDisplay}{table-cell}%\n\t\t{\\@cssBlockText}%\n\t\t{\\@cssInlineText}%\n\t}}}\n\n\\newcommand{\\@cssBlockText}{%\n\t\\eifstrequal{\\cssId}{}{}{\\label{\\cssId}}% set 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    "path": "marktoberdorf_paper/forPublisher/dse.tex",
    "content": "\\documentclass{IOS-Book-Article}\n% generated by Madoko, version 0.9.3-beta\n%mdk-data-line={1}\n\n\n\\usepackage[heading-base=2]{madoko}\n\n\n\\begin{document}\n\n%mdk-begin-texraw\n%mdk-data-line={25}\n\\begin{frontmatter}              % The preamble begins here.\n%\\pretitle{Pretitle}\n\\title{Deconstructing Dynamic Symbolic Execution}\n%\\runningtitle{IOS Press Style Sample}\n%\\subtitle{Subtitle}\n\\author[A]{\\fnms{Thomas} \\snm{Ball}}\nand \n\\author[B]{\\fnms{Jakub} \\snm{Daniel}}\n\n\\runningauthor{Thomas Ball et al.}\n\\address[A]{Microsoft Research}\n\\address[B]{Charles University}\n\\begin{mdDiv}[class={abstract},elem={abstract},data-line={39}]%\n\\begin{mdP}[data-line={40}]%\n%mdk-data-line={40}\n{}Dynamic symbolic execution (DSE) is a well-known technique\nfor automatically generating tests to achieve higher levels\nof coverage in a program. Two keys ideas of DSE are\nto: (1) seed symbolic execution by executing a program on an\ninitial input; (2) use concrete values from the program\nexecution in place of symbolic expressions whenever symbolic\nreasoning is hard or not desired. We describe\nDSE for a simple core language and then present\na minimalist implementation of DSE for Python (in Python) \nthat follows this basic recipe. The code is available \nat https://www.github.com/thomasjball/PyExZ3/ (tagged %mdk-data-line={50}\n{}{\\textquotedblleft}v1.0{\\textquotedblright}%mdk-data-line={50}\n{}) \nand has been designed to make it easy to experiment with and\nextend.%\n\\end{mdP}%%\n\\end{mdDiv}%\n%mdk-begin-texraw\n%mdk-data-line={56}\n\\begin{keyword}\nSymbolic Execution, Automatic Test Generation, White-box Testing, Automated \nTheorem Provers\n\\end{keyword}\n\\end{frontmatter}\n\\thispagestyle{empty}\n\\pagestyle{empty}\n\\mdHxx[id=sec-intro,label={[1]\\{.heading-label\\}},toc={},data-line={70},caption={[[1]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Introduction},bookmark={1.{\\hspace{0.5em}}Introduction}]{%mdk-data-line={70}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{1}.{\\hspace{0.5em}}}%mdk-data-line={70}\n{}Introduction}%mdk-data-line={73}\n\\defcommand{\\mathkw}[1]{\\textbf{#1}}\n\\begin{mdP}[data-line={78}]%\n%mdk-data-line={78}\n{}Static, path-based symbolic execution explores one control-flow path\nat a time through a (sequential) program %mdk-data-line={79}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={79}\n{}, using an automated theorem\nprover (ATP) to determine if the current path %mdk-data-line={80}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={80}\n{} is feasible%mdk-data-line={80}\n{}{\\mdNbsp}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{clarke76}{}{\\mdSpan[class={bibitem-label}]{4}}, \\mdA[class={bibref,localref},target-element={bibitem}]{king76}{}{\\mdSpan[class={bibitem-label}]{11}}]}%mdk-data-line={80}\n{}. \nIdeally, symbolic execution of a path %mdk-data-line={81}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={81}\n{} through program\n%mdk-data-line={82}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={82}\n{} yields a logic formula %mdk-data-line={82}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\phi_p$}%mdk-data-line={82}\n{} that describes the set of inputs %mdk-data-line={82}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$I$}%mdk-data-line={82}\n{} (possibly empty)\nto program %mdk-data-line={83}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={83}\n{} such that for any %mdk-data-line={83}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$i \\in I$}%mdk-data-line={83}\n{}, the execution %mdk-data-line={83}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P(i)$}%mdk-data-line={83}\n{} follows path %mdk-data-line={83}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={83}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={85}]%\n%mdk-data-line={85}\n{}If the formula %mdk-data-line={85}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\phi_p$}%mdk-data-line={85}\n{} is unsatisfiable then %mdk-data-line={85}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$I$}%mdk-data-line={85}\n{} is empty and so path %mdk-data-line={85}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={85}\n{} is not feasible; \nif the formula is satisfiable then %mdk-data-line={86}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$I$}%mdk-data-line={86}\n{} is not empty and so path %mdk-data-line={86}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={86}\n{} is feasible.\nIn this case, a model of %mdk-data-line={87}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\phi_p$}%mdk-data-line={87}\n{} provides a witness %mdk-data-line={87}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$i \\in I$}%mdk-data-line={87}\n{}.  Thus, a model-generating ATP\ncan be used in conjunction with\nsymbolic execution to automatically generate tests to cover paths\nin a program. Combined with a search strategy, one gets, in the limit,\nan exhaustive white-box testing procedure, for which there are many\napplications%mdk-data-line={92}\n{}{\\mdNbsp}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{cadars13}{}{\\mdSpan[class={bibitem-label}]{2}}, \\mdA[class={bibref,localref},target-element={bibitem}]{cadargpde06}{}{\\mdSpan[class={bibitem-label}]{3}}, \\mdA[class={bibref,localref},target-element={bibitem}]{godefroidlm12}{}{\\mdSpan[class={bibitem-label}]{9}}]}%mdk-data-line={92}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={94}]%\n%mdk-data-line={94}\n{}The formula %mdk-data-line={94}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\phi_p$}%mdk-data-line={94}\n{} is called a %mdk-data-line={94}\n{}\\mdEm{path-condition}%mdk-data-line={94}\n{} of the path %mdk-data-line={94}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={94}\n{}. \nWe will see that a given path %mdk-data-line={95}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={95}\n{} can induce many different path-conditions.\nA path-condition %mdk-data-line={96}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\psi_p$}%mdk-data-line={96}\n{} for path %mdk-data-line={96}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={96}\n{} is %mdk-data-line={96}\n{}\\mdEm{sound}%mdk-data-line={96}\n{} if \nevery input assignment satisfying %mdk-data-line={97}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\psi_p$}%mdk-data-line={97}\n{} defines an\nexecution of program %mdk-data-line={98}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={98}\n{} that follows path %mdk-data-line={98}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={98}\n{}{\\mdNbsp}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{godefroid11}{}{\\mdSpan[class={bibitem-label}]{7}}]}%mdk-data-line={98}\n{}. \nBy its definition, the formula  %mdk-data-line={99}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\phi_p$}%mdk-data-line={99}\n{} is sound and the best representation of %mdk-data-line={99}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={99}\n{}\n(as for all sound path-conditions %mdk-data-line={100}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\psi_p$}%mdk-data-line={100}\n{}, we have that %mdk-data-line={100}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\psi_p \\implies \\phi_p$}%mdk-data-line={100}\n{}).\nIn practice, we attempt to compute sound under-approximations of %mdk-data-line={101}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\phi_p$}%mdk-data-line={101}\n{} \nsuch as %mdk-data-line={102}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\psi_p$}%mdk-data-line={102}\n{}. However, we also find it necessary (and useful) to \ncompute unsound path-conditions.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={105}]%\n%mdk-data-line={105}\n{}A path-condition can be translated into the input\nlanguage of an ATP, such as%mdk-data-line={106}\n{}{\\mdNbsp}\\mdA[data-linkid={z3}]{http://z3.codeplex.org/}{}{Z3}%mdk-data-line={106}\n{}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{demourab08}{}{\\mdSpan[class={bibitem-label}]{5}}]}%mdk-data-line={106}\n{}, which provides an answer\nof %mdk-data-line={107}\n{}{\\textquotedblleft}unsatisfiable{\\textquotedblright}%mdk-data-line={107}\n{}, %mdk-data-line={107}\n{}{\\textquotedblleft}satisfiable{\\textquotedblright}%mdk-data-line={107}\n{} or %mdk-data-line={107}\n{}{\\textquotedblleft}unknown{\\textquotedblright}%mdk-data-line={107}\n{}, due to theoretical or practical\nlimitations in automatically deciding satisfiability of various logics.\nIn the case that the ATP is able to prove %mdk-data-line={109}\n{}{\\textquotedblleft}satisfiable{\\textquotedblright}%mdk-data-line={109}\n{} we can query it for \nsatisfying model in order to generate test inputs. A path-condition \nfor %mdk-data-line={111}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={111}\n{} can be thought of as function from a\nprogram%mdk-data-line={112}\n{}{'}%mdk-data-line={112}\n{}s primary inputs to a Boolean output representing whether\nor not %mdk-data-line={113}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={113}\n{} is executed under a given input. Thus, we are asking\nthe ATP to invert a function when we ask it to decide\nthe satisfiability/unsatisfiability of a path-condition.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={117}]%\n%mdk-data-line={117}\n{}The static translation of a path %mdk-data-line={117}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={117}\n{} through a program %mdk-data-line={117}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={117}\n{} into \nthe most precise path-condition %mdk-data-line={118}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\phi_p$}%mdk-data-line={118}\n{} is not a simple task, as \nprogramming languages and their semantics are very complex.\nCompletely characterizing the set of inputs %mdk-data-line={120}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$I$}%mdk-data-line={120}\n{} that follow\npath %mdk-data-line={121}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={121}\n{} means providing a symbolic interpretation of \nevery operation in the language so that the\nATP can reason about it. For example, consider a method call in Python. \nPython%mdk-data-line={124}\n{}{'}%mdk-data-line={124}\n{}s algorithm for method resolution order (see%mdk-data-line={124}\n{}{\\mdNbsp}\\mdA[data-linkid={mro}]{https://www.python.org/download/releases/2.3/mro/}{}{MRO}%mdk-data-line={124}\n{})\ndepends on the inheritance hierarchy of the program, a directed, \nacyclic graph that can evolve during program execution. Symbolically\nencoding Python%mdk-data-line={127}\n{}{'}%mdk-data-line={127}\n{}s method resolution order is possible but non-trivial.\nThere are other reasons it is hard or undesirable to symbolically \nexecute various operations, as will be explained in detail later.%\n\\end{mdP}%\n\\mdHxxx[id=sec-dynamic-symbolic-execution,label={[1.1]\\{.heading-label\\}},toc={},data-line={131},caption={[[1.1]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Dynamic symbolic execution},bookmark={1.1.{\\hspace{0.5em}}Dynamic symbolic execution}]{%mdk-data-line={131}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{1.1}.{\\hspace{0.5em}}}%mdk-data-line={131}\n{}Dynamic symbolic execution}\\begin{mdP}[data-line={133}]%\n%mdk-data-line={133}\n{}\\mdEm{Dynamic}%mdk-data-line={133}\n{} symbolic execution (DSE) is a form of path-based\nsymbolic execution based on two insights. First, the approach \nstarts by executing program %mdk-data-line={135}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={135}\n{} on\nsome input %mdk-data-line={136}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$i$}%mdk-data-line={136}\n{}, seeding the symbolic execution process\nwith a feasible path%mdk-data-line={137}\n{}{\\mdNbsp}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{gupta00}{}{\\mdSpan[class={bibitem-label}]{10}}, \\mdA[class={bibref,localref},target-element={bibitem}]{korel90}{}{\\mdSpan[class={bibitem-label}]{12}}, \\mdA[class={bibref,localref},target-element={bibitem}]{korel92}{}{\\mdSpan[class={bibitem-label}]{13}}]}%mdk-data-line={137}\n{}. \nSecond,  DSE\nuses concrete values from the execution %mdk-data-line={139}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P(i)$}%mdk-data-line={139}\n{} in place of symbolic expressions \nwhenever symbolic reasoning is not possible or desired%mdk-data-line={140}\n{}{\\mdNbsp}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{cadare05}{}{\\mdSpan[class={bibitem-label}]{1}}, \\mdA[class={bibref,localref},target-element={bibitem}]{godefroidks05}{}{\\mdSpan[class={bibitem-label}]{8}}]}%mdk-data-line={140}\n{}.\nThe major benefit of DSE is to\nsimplify the construction of a symbolic execution tool by\nleveraging concrete execution behavior (given by\nactually running the program).\nAs DSE combines both \nconcrete and symbolic reasoning, it also has been called %mdk-data-line={146}\n{}{\\textquotedblleft}concolic{\\textquotedblright}%mdk-data-line={146}\n{} \nexecution%mdk-data-line={147}\n{}{\\mdNbsp}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{senacav06}{}{\\mdSpan[class={bibitem-label}]{14}}]}%mdk-data-line={147}\n{}.%\n\\end{mdP}%\n\\begin{mdDiv}[class={figure,floating,align-center},id=fig-dse,label={[1]\\{.figure-label\\}},elem={figure},toc-line={[1]\\{.figure-label\\}. Pseudo-code for dynamic symbolic execution},toc={tof},float-env={figure},float-name={Figure},caption={Pseudo-code for dynamic symbolic execution},data-line={149}]%\n\\begin{mdPre}[class={para-block,pre-fenced,pre-fenced3,language-python,lang-python,python,highlighted},language={python},data-line={150},data-line-code={151}]%\n\\mdPrecode[data-line={151}]{{\\preindent{2}}\\mdToken{Identifier,Python}{i}{\\prespace{1}}\\mdToken{Keyword,Python}{=}{\\prespace{1}}\\mdToken{Identifier,Python}{an}{\\prespace{1}}\\mdToken{Identifier,Python}{input}{\\prespace{1}}\\mdToken{Identifier,Python}{to}{\\prespace{1}}\\mdToken{Identifier,Python}{program}{\\prespace{1}}\\mdToken{Constructor,Identifier,Python}{P}\\prebr{}\n{\\preindent{2}}\\mdToken{Keyword,Python}{while}{\\prespace{1}}\\mdToken{Identifier,Python}{defined}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{i}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{5}}\\mdToken{Identifier,Python}{p}{\\prespace{1}}\\mdToken{Keyword,Python}{=}{\\prespace{1}}\\mdToken{Identifier,Python}{path}{\\prespace{1}}\\mdToken{Identifier,Python}{covered}{\\prespace{1}}\\mdToken{Identifier,Python}{by}{\\prespace{1}}\\mdToken{Identifier,Python}{execution}{\\prespace{1}}\\mdToken{Constructor,Identifier,Python}{P}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{i}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\prebr{}\n{\\preindent{5}}\\mdToken{Identifier,Python}{cond}{\\prespace{1}}\\mdToken{Keyword,Python}{=}{\\prespace{1}}\\mdToken{Identifier,Python}{pathCondition}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{p}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\prebr{}\n{\\preindent{5}}\\mdToken{Identifier,Python}{s}{\\prespace{1}}\\mdToken{Keyword,Python}{=}{\\prespace{1}}\\mdToken{Constructor,Identifier,Python}{ATP}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Namespace,Identifier,Python}{Not}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{cond}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\prebr{}\n{\\preindent{5}}\\mdToken{Identifier,Python}{i}{\\prespace{1}}\\mdToken{Keyword,Python}{=}{\\prespace{1}}\\mdToken{Identifier,Python}{s}\\mdToken{Delimiter,Python}{.}\\mdToken{Identifier,Python}{model}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%\n\\end{mdPre}%\n\\mdHr[class={figureline,madoko},data-line={158}]{}\\begin{mdDiv}[data-line={159}]%\n%mdk-data-line={159}\n{}\\mdSpan[class={figure-caption}]{\\mdSpan[class={caption-before}]{\\mdStrong{Figure{\\mdNbsp}\\mdSpan[class={figure-label}]{1}.} }Pseudo-code for dynamic symbolic execution}%mdk-data-line={159}\n{}%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[class={indent},data-line={160}]%\n%mdk-data-line={160}\n{}The pseudo-code of Figure%mdk-data-line={160}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-dse}{}{\\mdSpan[class={figure-label}]{1}}%mdk-data-line={160}\n{} shows the high level process\nof DSE. The variable %mdk-data-line={161}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{i}}%mdk-data-line={161}\n{} represents an input\nto program %mdk-data-line={162}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Constructor,Identifier,Python}{P}}%mdk-data-line={162}\n{}. Execution of program %mdk-data-line={162}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Constructor,Identifier,Python}{P}}%mdk-data-line={162}\n{} on the input %mdk-data-line={162}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{i}}%mdk-data-line={162}\n{}\ntraces  a path %mdk-data-line={163}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{p}}%mdk-data-line={163}\n{}, from which\na logical formula %mdk-data-line={164}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{pathCondition}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{p}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={164}\n{} is constructed.\nFinally, the ATP is called with the negation of the path-condition\nto find a new input (that hopefully will cover a new path).  This\npseudo-code elides a number of details that we will deal with later.%\n\\end{mdP}%\n\\begin{mdDiv}[class={figure,floating,align-center},id=fig-easy-dse,label={[2]\\{.figure-label\\}},elem={figure},toc-line={[2]\\{.figure-label\\}. Easy example: computing the maximum of four numbers in Python.},toc={tof},float-env={figure},float-name={Figure},caption={Easy example: computing the maximum of four numbers in Python.},data-line={169}]%\n\\begin{mdPre}[class={para-block,pre-fenced,pre-fenced3,language-python,lang-python,python,highlighted},language={python},data-line={170},data-line-code={171}]%\n\\mdPrecode[data-line={171}]{\\mdToken{Keyword,Python}{def}{\\prespace{1}}\\mdToken{Identifier,Python}{max2}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{s}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{t}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{4}}\\mdToken{Keyword,Python}{if}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{s}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textless}}{\\prespace{1}}\\mdToken{Identifier,Python}{t}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{8}}\\mdToken{Keyword,Python}{return}{\\prespace{1}}\\mdToken{Identifier,Python}{t}\\prebr{}\n{\\preindent{4}}\\mdToken{Keyword,Python}{else}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{8}}\\mdToken{Keyword,Python}{return}{\\prespace{1}}\\mdToken{Identifier,Python}{s}\\prebr{}\n\\prebr{}\n\\mdToken{Keyword,Python}{def}{\\prespace{1}}\\mdToken{Identifier,Python}{max4}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{4}}\\mdToken{Keyword,Python}{return}{\\prespace{1}}\\mdToken{Identifier,Python}{max2}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{max2}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{max2}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%\n\\end{mdPre}%\n\\mdHr[class={figureline,madoko},data-line={180}]{}\\begin{mdDiv}[data-line={181}]%\n%mdk-data-line={181}\n{}\\mdSpan[class={figure-caption}]{\\mdSpan[class={caption-before}]{\\mdStrong{Figure{\\mdNbsp}\\mdSpan[class={figure-label}]{2}.} }Easy example: computing the maximum of four numbers in Python.}%mdk-data-line={181}\n{}%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[class={indent,para-continue},data-line={182}]%\n%mdk-data-line={182}\n{}Consider the  Python function %mdk-data-line={182}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{max4}}%mdk-data-line={182}\n{} in Figure%mdk-data-line={182}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-easy-dse}{}{\\mdSpan[class={figure-label}]{2}}%mdk-data-line={182}\n{},\nwhich computes the maximum of four numbers via three calls to\nthe function %mdk-data-line={184}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{max2}}%mdk-data-line={184}\n{}. Suppose we execute %mdk-data-line={184}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{max4}}%mdk-data-line={184}\n{} with values\nof zero for all four arguments.  In this case, the \nexecution path %mdk-data-line={186}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={186}\n{} contains three comparisons (in the order %mdk-data-line={186}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textless}}{\\prespace{1}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={186}\n{}, \n%mdk-data-line={187}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textless}}{\\prespace{1}}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={187}\n{}, %mdk-data-line={187}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textless}}{\\prespace{1}}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={187}\n{}), all of which evaluate false.\nThus, the path-condition for path %mdk-data-line={188}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={188}\n{} is %mdk-data-line={188}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Keyword,Python}{not}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Keyword,Python}{not}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Keyword,Python}{not}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textless}}{\\prespace{1}}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={188}\n{}.\nNegating this condition yields %mdk-data-line={189}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{or}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{or}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={189}\n{}.\nTaking the execution ordering of the three comparisons into account, we\nderive three expressions from the negated path-condition to generate\nnew inputs that will explore execution prefixes of path %mdk-data-line={192}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={192}\n{} of increasing length:%\n\\end{mdP}%\n\\begin{mdUl}[class={ul,list-star,compact},elem={ul},data-line={194}]%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(1)]\\{.ul-li-label\\}},elem={li},data-line={194}]%\n%mdk-data-line={194}\n{}\\mdEm{length 0}%mdk-data-line={194}\n{}: %mdk-data-line={194}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={194}\n{}%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(2)]\\{.ul-li-label\\}},elem={li},data-line={195}]%\n%mdk-data-line={195}\n{}\\mdEm{length 1}%mdk-data-line={195}\n{}: %mdk-data-line={195}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Keyword,Python}{not}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={195}\n{}%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(3)]\\{.ul-li-label\\}},elem={li},data-line={196}]%\n%mdk-data-line={196}\n{}\\mdEm{length 2}%mdk-data-line={196}\n{}: %mdk-data-line={196}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Keyword,Python}{not}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Keyword,Python}{not}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={196}\n{}%\n\\end{mdLi}%%\n\\end{mdUl}%\n\\begin{mdP}[class={para-continue},data-line={198}]%\n%mdk-data-line={198}\n{}The purpose of taking execution order into account should be clear, as the\ncomparison %mdk-data-line={199}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={199}\n{} only executes in the case where %mdk-data-line={199}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Keyword,Python}{not}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Keyword,Python}{not}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={199}\n{}\nholds. Integer solutions to the above three systems of constraints are:%\n\\end{mdP}%\n\\begin{mdUl}[class={ul,list-star,compact},elem={ul},data-line={202}]%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(1)]\\{.ul-li-label\\}},elem={li},data-line={202}]%\n%mdk-data-line={202}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{a}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{b}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{2}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{c}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{d}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}}%mdk-data-line={202}\n{}%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(2)]\\{.ul-li-label\\}},elem={li},data-line={203}]%\n%mdk-data-line={203}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{a}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{b}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{c}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{d}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{3}}%mdk-data-line={203}\n{}%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(3)]\\{.ul-li-label\\}},elem={li},data-line={204}]%\n%mdk-data-line={204}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{a}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{b}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{c}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{2}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{d}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{0}}%mdk-data-line={204}\n{}%\n\\end{mdLi}%%\n\\end{mdUl}%\n\\begin{mdP}[data-line={206}]%\n%mdk-data-line={206}\n{}In the three cases above, we sought solutions that kept as many of\nthe variables as possible equal to the original input (in which \nall variables are equal to 0). Execution of the %mdk-data-line={208}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{max4}}%mdk-data-line={208}\n{} function \non the input corresponding to the first solution produces the path-condition\n%mdk-data-line={210}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{a}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{b}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Keyword,Python}{not}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{c}\\mdToken{Operator,Python}{{\\textless}}\\mdToken{Identifier,Python}{d}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Keyword,Python}{not}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{b}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textless}}{\\prespace{1}}\\mdToken{Identifier,Python}{c}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={210}\n{}, from which we can produce more\ninputs.   For this (loop-free function), there are a finite number\nof path-conditions. We leave it as an exercise to the reader to \nenumerate them all.%\n\\end{mdP}%\n\\mdHxxx[id=sec-leveraging-concrete-values-in-dse,label={[1.2]\\{.heading-label\\}},toc={},data-line={215},caption={[[1.2]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Leveraging concrete values in DSE},bookmark={1.2.{\\hspace{0.5em}}Leveraging concrete values in DSE}]{%mdk-data-line={215}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{1.2}.{\\hspace{0.5em}}}%mdk-data-line={215}\n{}Leveraging concrete values in DSE}\\begin{mdP}[data-line={217}]%\n%mdk-data-line={217}\n{}We now consider several situations where we can make use of concrete\nvalues in DSE. In the realm of (unbounded-precision) integer arithmetic \n(e.g., bignum integer arithmetic, as in Python 3.0 onwards),\nit is easy  to come up with  tiny programs that will be %mdk-data-line={220}\n{}\\mdEm{very difficult}%mdk-data-line={220}\n{},\nif not %mdk-data-line={221}\n{}\\mdEm{impossible}%mdk-data-line={221}\n{}, \nfor any symbolic execution tool to deal with, such as the function %mdk-data-line={222}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{fermat3}}%mdk-data-line={222}\n{}\nin Figure%mdk-data-line={223}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-fermat3}{}{\\mdSpan[class={figure-label}]{3}}%mdk-data-line={223}\n{}.%mdk-data-line={223}\n{} %mdk-data-line={223}\n{}%\n\\end{mdP}%\n\\begin{mdDiv}[class={figure,floating,align-center},id=fig-fermat3,label={[3]\\{.figure-label\\}},elem={figure},toc-line={[3]\\{.figure-label\\}. Hard example for symbolic execution},toc={tof},float-env={figure},float-name={Figure},caption={Hard example for symbolic execution},data-line={226}]%\n\\begin{mdPre}[class={para-block,pre-fenced,pre-fenced3,language-python,lang-python,python,highlighted},language={python},data-line={227},data-line-code={228}]%\n\\mdPrecode[data-line={228}]{\\mdToken{Keyword,Python}{def}{\\prespace{1}}\\mdToken{Identifier,Python}{fermat3}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{y}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{z}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{3}}\\mdToken{Keyword,Python}{if}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textgreater}}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{y}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textgreater}}{\\prespace{1}}\\mdToken{Number,Python}{0}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Identifier,Python}{z}{\\prespace{1}}\\mdToken{Operator,Python}{{\\textgreater}}{\\prespace{1}}\\mdToken{Number,Python}{0}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{6}}\\mdToken{Keyword,Python}{if}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{x}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{x}{\\prespace{1}}\\mdToken{Operator,Python}{+}{\\prespace{1}}\\mdToken{Identifier,Python}{y}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{y}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{y}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{z}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{z}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{z}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{10}}\\mdToken{Keyword,Python}{return}{\\prespace{1}}\\mdToken{String,Delim,Python,BracketOpen}{{\"}}\\mdToken{String,Python}{Fermat\\prespace{1}and\\prespace{1}Wiles\\prespace{1}were\\prespace{1}wrong!?!}\\mdToken{String,Delim,Python,BracketClose}{{\"}}\\prebr{}\n{\\preindent{3}}\\mdToken{Keyword,Python}{return}{\\prespace{1}}\\mdToken{Number,Python}{0}}%\n\\end{mdPre}%\n\\mdHr[class={figureline,madoko},data-line={234}]{}\\begin{mdDiv}[data-line={235}]%\n%mdk-data-line={235}\n{}\\mdSpan[class={figure-caption}]{\\mdSpan[class={caption-before}]{\\mdStrong{Figure{\\mdNbsp}\\mdSpan[class={figure-label}]{3}.} }Hard example for symbolic execution}%mdk-data-line={235}\n{}%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[class={indent},data-line={237}]%\n%mdk-data-line={237}\n{}Fermat%mdk-data-line={237}\n{}{'}%mdk-data-line={237}\n{}s Last Theorem, proved\nby Andrew Wiles in the late 20th century, states that no \nthree positive integers %mdk-data-line={239}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x$}%mdk-data-line={239}\n{}, %mdk-data-line={239}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$y$}%mdk-data-line={239}\n{}, and %mdk-data-line={239}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$z$}%mdk-data-line={239}\n{} can satisfy the equation\n%mdk-data-line={240}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x^n + y^n = z^n$}%mdk-data-line={240}\n{} for any integer value of %mdk-data-line={240}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$n$}%mdk-data-line={240}\n{} greater than two.\nThe function %mdk-data-line={241}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{fermat3}}%mdk-data-line={241}\n{} encodes this statement for %mdk-data-line={241}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$n=3$}%mdk-data-line={241}\n{}.   It\nis not reasonable to have a computer waste time trying to find\na solution that would cause %mdk-data-line={243}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{fermat3}}%mdk-data-line={243}\n{} to print the string \n%mdk-data-line={244}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{String,Delim,Python,BracketOpen}{{\"}}\\mdToken{String,Python}{Fermat\\prespace{1}and\\prespace{1}Wiles\\prespace{1}were\\prespace{1}wrong!?!}\\mdToken{String,Delim,Python,BracketClose}{{\"}}}%mdk-data-line={244}\n{}.  In cases of complex (non-linear)\narithmetic operations,\nsuch as %mdk-data-line={246}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{x}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{x}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{x}}%mdk-data-line={246}\n{}, we might choose to handle the operation concretely.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={248}]%\n%mdk-data-line={248}\n{}There are a number of ways to deal with the above issue: one is \nto recognize all non-linear terms in a symbolic expression and replace\nthem with their concrete counterparts during execution. For the %mdk-data-line={250}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{fermat3}}%mdk-data-line={250}\n{} \nexample, this would mean that during DSE the symbolic expression \n%mdk-data-line={252}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{x}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{x}{\\prespace{1}}\\mdToken{Operator,Python}{+}{\\prespace{1}}\\mdToken{Identifier,Python}{y}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{y}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{y}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{z}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{z}\\mdToken{Operator,Python}{*}\\mdToken{Identifier,Python}{z}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={252}\n{} would be reduced to the constant %mdk-data-line={252}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{False}}%mdk-data-line={252}\n{}\nby evaluation on the concrete values of variables %mdk-data-line={253}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{x}}%mdk-data-line={253}\n{}, %mdk-data-line={253}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{y}}%mdk-data-line={253}\n{} and %mdk-data-line={253}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{z}}%mdk-data-line={253}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={255}]%\n%mdk-data-line={255}\n{}Besides difficult operations (such as non-linear arithmetic),\nother examples of code that we might treat\nconcretely instead of symbolically include\nfunctions that are hard to invert, such as cryptographic hash functions,\nor low-level functions that we do not wish to test (such as \noperating system functions).  Consider\nthe code in Figure%mdk-data-line={261}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-hash}{}{\\mdSpan[class={figure-label}]{4}}%mdk-data-line={261}\n{}, which applies the function\n%mdk-data-line={262}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{unknown}}%mdk-data-line={262}\n{} to argument %mdk-data-line={262}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{x}}%mdk-data-line={262}\n{} and compares it to argument %mdk-data-line={262}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{y}}%mdk-data-line={262}\n{}.\nBy using the name %mdk-data-line={263}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{unknown}}%mdk-data-line={263}\n{} we simply mean to say that we \nwish to model this function as a black box, with no knowledge\nof how it operates internally.%\n\\end{mdP}%\n\\begin{mdDiv}[class={figure,floating,align-center},id=fig-hash,label={[4]\\{.figure-label\\}},elem={figure},toc-line={[4]\\{.figure-label\\}. Another hard example for symbolic execution},toc={tof},float-env={figure},float-name={Figure},caption={Another hard example for symbolic execution},data-line={267}]%\n\\begin{mdPre}[class={para-block,pre-fenced,pre-fenced3,language-python,lang-python,python,highlighted},language={python},data-line={268},data-line-code={269}]%\n\\mdPrecode[data-line={269}]{\\mdToken{Keyword,Python}{def}{\\prespace{1}}\\mdToken{Identifier,Python}{dart}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}\\mdToken{Delimiter,Python}{,}\\mdToken{Identifier,Python}{y}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{2}}\\mdToken{Keyword,Python}{if}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{unknown}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{y}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}\\mdToken{Keyword,Python,BracketOpen}{:}\\prebr{}\n{\\preindent{5}}\\mdToken{Keyword,Python}{return}{\\prespace{1}}\\mdToken{Number,Python}{1}\\prebr{}\n{\\preindent{2}}\\mdToken{Keyword,Python}{return}{\\prespace{1}}\\mdToken{Number,Python}{0}}%\n\\end{mdPre}%\n\\mdHr[class={figureline,madoko},data-line={274}]{}\\begin{mdDiv}[data-line={275}]%\n%mdk-data-line={275}\n{}\\mdSpan[class={figure-caption}]{\\mdSpan[class={caption-before}]{\\mdStrong{Figure{\\mdNbsp}\\mdSpan[class={figure-label}]{4}.} }Another hard example for symbolic execution}%mdk-data-line={275}\n{}%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[class={indent},data-line={276}]%\n%mdk-data-line={276}\n{}In such a case, we can use DSE to execute the function %mdk-data-line={276}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{unknown}}%mdk-data-line={276}\n{}\non a specific input (say %mdk-data-line={277}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Number,Python}{5013}}%mdk-data-line={277}\n{}) and observe its output\n(say %mdk-data-line={278}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Number,Python}{42}}%mdk-data-line={278}\n{}). That is, rather than execute %mdk-data-line={278}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{unknown}}%mdk-data-line={278}\n{} symbolically\nand invoke an ATP to invert the function%mdk-data-line={279}\n{}{'}%mdk-data-line={279}\n{}s path-condition, we\nsimply treat the call to %mdk-data-line={280}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{unknown}}%mdk-data-line={280}\n{} concretely, substituting\nits return value (in this case %mdk-data-line={281}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Number,Python}{42}}%mdk-data-line={281}\n{}) for the specialized expression \n%mdk-data-line={282}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{unknown}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Number,Python}{5013}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{y}}%mdk-data-line={282}\n{} to get the predicate %mdk-data-line={282}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Number,Python}{42}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{y}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={282}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={284}]%\n%mdk-data-line={284}\n{}Adding the constraint %mdk-data-line={284}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{5013}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={284}\n{} yields the sound but\nrather specific path-condition \n%mdk-data-line={286}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Number,Python}{5013}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Keyword,Python}{and}{\\prespace{1}}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Number,Python}{42}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{y}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={286}\n{}.\nNote that the path-condition %mdk-data-line={287}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Number,Python}{42}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{y}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={287}\n{} is not sound, as it admits\nany value for the variable %mdk-data-line={288}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{x}}%mdk-data-line={288}\n{}, which likely includes many values\nfor which %mdk-data-line={289}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{unknown}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{x}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}{\\prespace{1}}\\mdToken{Operator,Python}{==}{\\prespace{1}}\\mdToken{Identifier,Python}{y}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={289}\n{} is false.%\n\\end{mdP}%\n\\mdHxxx[id=sec-overview,label={[1.3]\\{.heading-label\\}},toc={},data-line={291},caption={[[1.3]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Overview},bookmark={1.3.{\\hspace{0.5em}}Overview}]{%mdk-data-line={291}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{1.3}.{\\hspace{0.5em}}}%mdk-data-line={291}\n{}Overview}\\begin{mdP}[class={para-continue},data-line={293}]%\n%mdk-data-line={293}\n{}This introduction elides many important \nissues that arise in implementing DSE for a real language, which we will \nfocus on in the remainder of the paper. These include how to:%\n\\end{mdP}%\n\\begin{mdUl}[class={ul,list-star,compact},elem={ul},data-line={297}]%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(1)]\\{.ul-li-label\\}},elem={li},data-line={297}]%\n%mdk-data-line={297}\n{}Identify the code under test %mdk-data-line={297}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={297}\n{} and the symbolic inputs to %mdk-data-line={297}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={297}\n{};%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(2)]\\{.ul-li-label\\}},elem={li},data-line={298}]%\n%mdk-data-line={298}\n{}Trace the control flow path %mdk-data-line={298}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={298}\n{} taken by execution %mdk-data-line={298}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P(i)$}%mdk-data-line={298}\n{};%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(3)]\\{.ul-li-label\\}},elem={li},data-line={299}]%\n%mdk-data-line={299}\n{}Reinterpret program operations to compute symbolic expressions;%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(4)]\\{.ul-li-label\\}},elem={li},data-line={300}]%\n%mdk-data-line={300}\n{}Generate a path-condition from %mdk-data-line={300}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={300}\n{} and the symbolic expressions;%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(5)]\\{.ul-li-label\\}},elem={li},data-line={301}]%\n%mdk-data-line={301}\n{}Generate a new input %mdk-data-line={301}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$i'$}%mdk-data-line={301}\n{} by negating (part of) the path-condition, translating\nthe path-condition to the input language of an ATP, invoking the ATP, and\nlifting a satisfying model (if any) back up to the source level;%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,compact-li},label={[(6)]\\{.ul-li-label\\}},elem={li},data-line={304}]%\n%mdk-data-line={304}\n{}Guide the search to expose new paths.%\n\\end{mdLi}%%\n\\end{mdUl}%\n\\begin{mdP}[data-line={306}]%\n%mdk-data-line={306}\n{}The rest of this paper is organized as follows. Section%mdk-data-line={306}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-semantics}{}{\\mdSpan[class={heading-label}]{2}}%mdk-data-line={306}\n{} \ndescribes an instrumented typing discipline where we lift each type (representing \na set of concrete values) to a symbolic type (representing\na set of pairs of concrete and symbolic values).\nSection%mdk-data-line={310}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-sp2dse}{}{\\mdSpan[class={heading-label}]{3}}%mdk-data-line={310}\n{} shows how strongest postconditions defines a symbolic\nsemantics for a small programming language and how strongest postconditions\ncan be refined to model DSE.\nSection%mdk-data-line={313}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-impl}{}{\\mdSpan[class={heading-label}]{4}}%mdk-data-line={313}\n{} describes an implementation of DSE for the Python language\nin the Python language that follows the instrumented semantics pattern closely\n(full implementation and tests available at%mdk-data-line={315}\n{}{\\mdNbsp}\\mdA[data-linkid={pyexz3}]{https://github.com/thomasjball/PyExZ3/}{}{PyExZ3}%mdk-data-line={315}\n{}, tagged %mdk-data-line={315}\n{}{\\textquotedblleft}v1.0{\\textquotedblright}%mdk-data-line={315}\n{}).\nSection%mdk-data-line={316}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-int2z3}{}{\\mdSpan[class={heading-label}]{5}}%mdk-data-line={316}\n{} describes the symbolic encoding of Python integer \noperations using two decision procedures of Z3: linear arithmetic with\nuninterpreted functions in place of non-linear operations;\nfixed-width bit-vectors with precise encodings of most operations.\nSection%mdk-data-line={320}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-extensions}{}{\\mdSpan[class={heading-label}]{6}}%mdk-data-line={320}\n{} offers a number of ideas for projects\nto extend the capabilities of%mdk-data-line={321}\n{}{\\mdNbsp}\\mdA[data-linkid={pyexz3}]{https://github.com/thomasjball/PyExZ3/}{}{PyExZ3}%mdk-data-line={321}\n{}.%\n\\end{mdP}%\n\\mdHxx[id=sec-semantics,label={[2]\\{.heading-label\\}},toc={},data-line={323},caption={[[2]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Instrumented Types},bookmark={2.{\\hspace{0.5em}}Instrumented Types}]{%mdk-data-line={323}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{2}.{\\hspace{0.5em}}}%mdk-data-line={323}\n{}Instrumented Types}\\begin{mdP}[data-line={325}]%\n%mdk-data-line={325}\n{}We are given a universe of classes/types %mdk-data-line={325}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U$}%mdk-data-line={325}\n{}; a type %mdk-data-line={325}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T \\in U$}%mdk-data-line={325}\n{} carries\nalong a  set of operations that apply to values of type %mdk-data-line={326}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={326}\n{},\nwhere an operation %mdk-data-line={327}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T$}%mdk-data-line={327}\n{}  takes an argument list of typed \nvalues as input (the first being of type %mdk-data-line={328}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={328}\n{}) and produces a single \ntyped value as output. Nullary (static) operations of type %mdk-data-line={329}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={329}\n{} can be \nused to create values of type %mdk-data-line={330}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={330}\n{} (such as constants, objects, etc.)%\n\\end{mdP}%\n\\begin{mdP}[class={indent,para-continue},data-line={332}]%\n%mdk-data-line={332}\n{}A program %mdk-data-line={332}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={332}\n{} has typed input variables\n%mdk-data-line={333}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$v_1 : T_1 \\ldots v_k : T_k$}%mdk-data-line={333}\n{} and a body from the language of statements %mdk-data-line={333}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$S$}%mdk-data-line={333}\n{}:%\n\\end{mdP}%\n\\begin{mdDiv}[class={mathpre,para-block,input-mathpre},elem={mathpre},data-line={335}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={336}\n\\begin{mdMathprearray}%mdk\n\\mathid{S}\\mdMathspace{1}\\rightarrow   &\\mdMathspace{1}\\mathid{v}\\mdMathspace{1}:=\\mdMathspace{1}\\mathid{E}\\mdMathbr{}\n\\mdMathindent{3}|\\mdMathspace{1}&\\mdMathspace{1}\\mathkw{skip}\\mdMathspace{1}\\mdMathbr{}\n\\mdMathindent{3}|\\mdMathspace{1}&\\mdMathspace{1}\\mathid{S}_1\\mdMathspace{1};\\mdMathspace{1}\\mathid{S}_2\\mdMathspace{1}\\mdMathbr{}\n\\mdMathindent{3}|\\mdMathspace{1}&\\mdMathspace{1}\\mathkw{if}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{S}_1\\mdMathspace{1}\\mathkw{else}\\mdMathspace{1}\\mathid{S}_2\\mdMathspace{1}\\mathkw{end}\\mdMathbr{}\n\\mdMathindent{3}|\\mdMathspace{1}&\\mdMathspace{1}\\mathkw{while}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{do}\\mdMathspace{1}\\mathid{S}\\mdMathspace{1}\\mathkw{end}\n\\end{mdMathprearray}%mdk\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[data-line={343}]%\n%mdk-data-line={343}\n{}The language of expressions (%mdk-data-line={343}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={343}\n{}) is defined by the application of operations\nto values, where constants (nullary operations) and \nprogram variables form the leaves \nof the expression tree and non-nullary operators \nform the interior nodes of the tree.\nFor now, we will consider all values to be immutable.\nThat is, the only source of mutation in the language is the \nassignment statement.%\n\\end{mdP}%\n\\begin{mdP}[class={indent,para-continue},data-line={352}]%\n%mdk-data-line={352}\n{}To introduce symbolic execution into the picture,\nwe can imagine that a type %mdk-data-line={353}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T \\in U$}%mdk-data-line={353}\n{} has\n(one or more) counterparts in a symbolic universe %mdk-data-line={354}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={354}\n{}. A type %mdk-data-line={354}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T' \\in U'$}%mdk-data-line={354}\n{}\nis a subtype of %mdk-data-line={355}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T \\in U$}%mdk-data-line={355}\n{} with two purposes:%\n\\end{mdP}%\n\\begin{mdUl}[class={ul,list-star,loose},elem={ul},data-line={357}]%\n\\begin{mdLi}[class={li,ul-li,list-star-li,loose-li},label={[(1)]\\{.ul-li-label\\}},elem={li},data-line={357}]%\n\\begin{mdP}[data-line={357}]%\n%mdk-data-line={357}\n{}First, a value of type %mdk-data-line={357}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}%mdk-data-line={357}\n{} represents a pair of values: \na concrete value %mdk-data-line={358}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={358}\n{} of (super)type %mdk-data-line={358}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={358}\n{} and a symbolic expression %mdk-data-line={358}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$e$}%mdk-data-line={358}\n{}. \nA symbolic expression is a tree\nwhose leaves are either nullary operators (i.e., constants) of a type in %mdk-data-line={360}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U$}%mdk-data-line={360}\n{}\nor are Skolem constants representing the (symbolic) inputs (%mdk-data-line={361}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$v_1 \\ldots v_k$}%mdk-data-line={361}\n{})\nto the program %mdk-data-line={362}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={362}\n{}, and whose interior nodes represent operations \nfrom types in %mdk-data-line={363}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U$}%mdk-data-line={363}\n{}. We refer to Skolem constants as %mdk-data-line={363}\n{}{\\textquotedblleft}symbolic constants{\\textquotedblright}%mdk-data-line={363}\n{}\nfrom this point on. Note that symbolic expressions do not contain\nreferences to program variables.%\n\\end{mdP}%%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,loose-li},label={[(2)]\\{.ul-li-label\\}},elem={li},data-line={367}]%\n\\begin{mdP}[data-line={367}]%\n%mdk-data-line={367}\n{}Second, the type %mdk-data-line={367}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}%mdk-data-line={367}\n{} redefines some of the operations %mdk-data-line={367}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T$}%mdk-data-line={367}\n{},\nnamely those for which we wish to compute symbolic expressions.\nAn operation %mdk-data-line={369}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T'$}%mdk-data-line={369}\n{} has the same parameter list as %mdk-data-line={369}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T$}%mdk-data-line={369}\n{}, allowing it\nto take inputs with types from both %mdk-data-line={370}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U$}%mdk-data-line={370}\n{} and %mdk-data-line={370}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={370}\n{}. The return type\nof %mdk-data-line={371}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T'$}%mdk-data-line={371}\n{} generally is from %mdk-data-line={371}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={371}\n{} (though it can be from %mdk-data-line={371}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U$}%mdk-data-line={371}\n{}). \nThus, %mdk-data-line={372}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T'$}%mdk-data-line={372}\n{} is a proper function subtype of %mdk-data-line={372}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T$}%mdk-data-line={372}\n{}. \nThe purpose of %mdk-data-line={373}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T'$}%mdk-data-line={373}\n{} is to:\n(1) perform operation %mdk-data-line={374}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o \\in T$}%mdk-data-line={374}\n{} on the concrete \nvalues associated with its inputs; \n(2) build a symbolic expression tree rooted at operation %mdk-data-line={376}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o$}%mdk-data-line={376}\n{} \nwhose children are the trees associated with the inputs to %mdk-data-line={377}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o$}%mdk-data-line={377}\n{}.%\n\\end{mdP}%%\n\\end{mdLi}%%\n\\end{mdUl}%\n\\begin{mdP}[data-line={379}]%\n%mdk-data-line={379}\n{}Figure%mdk-data-line={379}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-subtype}{}{\\mdSpan[class={figure-label}]{5}}%mdk-data-line={379}\n{} presents pseudo code for the instrumentation\nof a type %mdk-data-line={380}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={380}\n{} via a type %mdk-data-line={380}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}%mdk-data-line={380}\n{}.\nThe class %mdk-data-line={381}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdToken{Type,Identifier,Cpp}{Symbolic}}%mdk-data-line={381}\n{} is used to hold an expression tree (%mdk-data-line={381}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdToken{Type,Identifier,Cpp}{Expr}}%mdk-data-line={381}\n{}).\nGiven a class %mdk-data-line={382}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T \\in U$}%mdk-data-line={382}\n{}, a symbolic type %mdk-data-line={382}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T' \\in U'$}%mdk-data-line={382}\n{} is defined by inheriting \nfrom both %mdk-data-line={383}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={383}\n{} and %mdk-data-line={383}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdToken{Type,Identifier,Cpp}{Symbolic}}%mdk-data-line={383}\n{}. This ensures that a %mdk-data-line={383}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}%mdk-data-line={383}\n{} can be used\nwherever a %mdk-data-line={384}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={384}\n{} is expected.%\n\\end{mdP}%\n\\begin{mdDiv}[class={figure,floating,align-center},id=fig-subtype,label={[5]\\{.figure-label\\}},elem={figure},toc-line={[5]\\{.figure-label\\}. Type instrumentation to carry both concrete values and symbolic expressions.},toc={tof},float-env={figure},float-name={Figure},caption={Type instrumentation to carry both concrete values and symbolic expressions.},data-line={386}]%\n\\begin{mdPre}[class={para-block,pre-fenced,pre-fenced3,language-cpp2,lang-cpp2,cpp2,highlighted},data-line={387},data-line-code={388},language={cpp2}]%\n\\mdPrecode[data-line={388}]{{\\preindent{2}}\\mdToken{Keyword,Cpp}{class}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}}}{\\prespace{1}}\\mdToken{Operator,Cpp}{:}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T$}}}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{Symbolic}{\\prespace{1}}\\mdToken{Delimiter,Curly,Cpp,BracketOpen}{\\{}\\prebr{}\n{\\preindent{4}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}}}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{c}\\mdToken{Operator,Cpp}{:}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T$}}}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Identifier,Cpp}{e}\\mdToken{Operator,Cpp}{:}\\mdToken{Type,Identifier,Cpp}{Expr}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}{\\prespace{1}}\\mdToken{Operator,Cpp}{:}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T$}}}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{c}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Source,Cpp}{S}\\mdToken{Identifier,Cpp}{ymbolic}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{e}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}{\\prespace{1}}\\mdToken{Delimiter,Curly,Cpp,BracketOpen}{\\{}\\mdToken{Delimiter,Curly,Cpp,BracketClose}{\\}}\\prebr{}\n\\prebr{}\n{\\preindent{4}}\\mdToken{Keyword,Extra,Cpp}{override}{\\prespace{1}}\\mdToken{Identifier,Cpp}{o}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Keyword,Cpp}{this}\\mdToken{Operator,Cpp}{:}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T$}}}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Identifier,Cpp}{f1}\\mdToken{Operator,Cpp}{:}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T_1$}}}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}{\\prespace{1}}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Identifier,Cpp}{fk}\\mdToken{Operator,Cpp}{:}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T_k$}}}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}{\\prespace{1}}\\mdToken{Operator,Cpp}{:}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$R'$}}}{\\prespace{1}}\\mdToken{Delimiter,Curly,Cpp,BracketOpen}{\\{}\\prebr{}\n{\\preindent{6}}\\mdToken{Keyword,Extra,Cpp}{var}{\\prespace{1}}\\mdToken{Identifier,Cpp}{c}{\\prespace{1}}:={\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T$}}}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Identifier,Cpp}{o}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Keyword,Cpp}{this}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Identifier,Cpp}{f1}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}{\\prespace{1}}\\mdToken{Delimiter,Cpp}{,}\\mdToken{Identifier,Cpp}{fk}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}\\prebr{}\n{\\preindent{6}}\\mdToken{Keyword,Extra,Cpp}{var}{\\prespace{1}}\\mdToken{Identifier,Cpp}{e}{\\prespace{1}}:={\\prespace{1}}\\mdToken{Keyword,Cpp}{new}{\\prespace{1}}\\mdToken{Source,Cpp}{E}\\mdToken{Identifier,Cpp}{xpr}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$T$}}}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Identifier,Cpp}{o}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Identifier,Cpp}{expr}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{self}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Identifier,Cpp}{expr}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{f1}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Identifier,Cpp}{expr}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{fk}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}\\prebr{}\n{\\preindent{6}}\\mdToken{Keyword,Cpp}{return}{\\prespace{1}}\\mdToken{Keyword,Cpp}{new}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$R'$}}}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{c}\\mdToken{Delimiter,Cpp}{,}\\mdToken{Identifier,Cpp}{e}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}{\\prespace{1}}\\prebr{}\n{\\preindent{4}}\\mdToken{Delimiter,Curly,Cpp,BracketClose}{\\}}\\prebr{}\n{\\preindent{4}}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}\\prebr{}\n{\\preindent{2}}\\mdToken{Delimiter,Curly,Cpp,BracketClose}{\\}}\\prebr{}\n{\\preindent{2}}\\prebr{}\n{\\preindent{2}}\\mdToken{Keyword,Cpp}{class}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$R'$}}}{\\prespace{1}}\\mdToken{Operator,Cpp}{:}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$R$}}}\\mdToken{Delimiter,Cpp}{,}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{Symbolic}{\\prespace{1}}\\mdToken{Delimiter,Curly,Cpp,BracketOpen}{\\{}{\\prespace{1}}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Delimiter,Cpp}{.}{\\prespace{1}}\\mdToken{Delimiter,Curly,Cpp,BracketClose}{\\}}\\prebr{}\n{\\preindent{2}}\\prebr{}\n{\\preindent{2}}\\mdToken{Keyword,Extra,Cpp}{function}{\\prespace{1}}\\mdToken{Identifier,Cpp}{expr}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{v}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}{\\prespace{1}}\\mdToken{Operator,Cpp}{=}{\\prespace{1}}\\mdToken{Identifier,Cpp}{v}{\\prespace{1}}\\mdToken{Keyword,Extra,Cpp}{instanceof}{\\prespace{1}}\\mdToken{Type,Identifier,Cpp}{Symbolic}{\\prespace{1}}\\mdToken{Operator,Cpp}{?}{\\prespace{1}}\\mdToken{Identifier,Cpp}{v}\\mdToken{Delimiter,Cpp}{.}\\mdToken{Identifier,Cpp}{getExpr}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}{\\prespace{1}}\\mdToken{Operator,Cpp}{:}{\\prespace{1}}\\mdToken{Identifier,Cpp}{v}}%\n\\end{mdPre}%\n\\mdHr[class={figureline,madoko},data-line={403}]{}\\begin{mdDiv}[data-line={404}]%\n%mdk-data-line={404}\n{}\\mdSpan[class={figure-caption}]{\\mdSpan[class={caption-before}]{\\mdStrong{Figure{\\mdNbsp}\\mdSpan[class={figure-label}]{5}.} }Type instrumentation to carry both concrete values and symbolic expressions.}%mdk-data-line={404}\n{}%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[class={indent,para-continue},data-line={405}]%\n%mdk-data-line={405}\n{}A type such as %mdk-data-line={405}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}%mdk-data-line={405}\n{}  only can be constructed by providing a concrete value %mdk-data-line={405}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={405}\n{} of \ntype %mdk-data-line={406}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={406}\n{} and a symbolic expression %mdk-data-line={406}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$e$}%mdk-data-line={406}\n{} to the constructor for %mdk-data-line={406}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}%mdk-data-line={406}\n{}.\nThis will be done in exactly two places:%\n\\end{mdP}%\n\\begin{mdUl}[class={ul,list-star,loose},elem={ul},data-line={409}]%\n\\begin{mdLi}[class={li,ul-li,list-star-li,loose-li},label={[(1)]\\{.ul-li-label\\}},elem={li},data-line={409}]%\n\\begin{mdP}[data-line={409}]%\n%mdk-data-line={409}\n{}by the creation of symbolic constants associated with the primary\ninputs (%mdk-data-line={410}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$v_1 \\ldots v_k$}%mdk-data-line={410}\n{}) to the program;%\n\\end{mdP}%%\n\\end{mdLi}%\n\\begin{mdLi}[class={li,ul-li,list-star-li,loose-li},label={[(2)]\\{.ul-li-label\\}},elem={li},data-line={412}]%\n\\begin{mdP}[data-line={412}]%\n%mdk-data-line={412}\n{}by the instrumented operations as shown  in Figure%mdk-data-line={412}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-subtype}{}{\\mdSpan[class={figure-label}]{5}}%mdk-data-line={412}\n{}.%\n\\end{mdP}%%\n\\end{mdLi}%%\n\\end{mdUl}%\n\\begin{mdP}[data-line={414}]%\n%mdk-data-line={414}\n{}An instrumented operation %mdk-data-line={414}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$o$}%mdk-data-line={414}\n{} on arguments (%mdk-data-line={414}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdToken{Keyword,Cpp}{this}}%mdk-data-line={414}\n{}, %mdk-data-line={414}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{f1}}%mdk-data-line={414}\n{}, %mdk-data-line={414}\n{}{\\dots}%mdk-data-line={414}\n{}, %mdk-data-line={414}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{fk}}%mdk-data-line={414}\n{})\nfirst invokes its corresponding underlying\noperator %mdk-data-line={416}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T.o$}%mdk-data-line={416}\n{} on arguments (%mdk-data-line={416}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdToken{Keyword,Cpp}{this}}%mdk-data-line={416}\n{}, %mdk-data-line={416}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{f1}}%mdk-data-line={416}\n{}, %mdk-data-line={416}\n{}{\\dots}%mdk-data-line={416}\n{}, %mdk-data-line={416}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{fk}}%mdk-data-line={416}\n{}) to get concrete value %mdk-data-line={416}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{c}}%mdk-data-line={416}\n{}.\nIt then constructs a new expression tree %mdk-data-line={417}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{e}}%mdk-data-line={417}\n{}\nrooted at operator %mdk-data-line={418}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T.o$}%mdk-data-line={418}\n{}, whose children are the result of\nmapping the function %mdk-data-line={419}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{expr}}%mdk-data-line={419}\n{} over (%mdk-data-line={419}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdToken{Keyword,Cpp}{this}}%mdk-data-line={419}\n{}, %mdk-data-line={419}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{f1}}%mdk-data-line={419}\n{}, %mdk-data-line={419}\n{}{\\dots}%mdk-data-line={419}\n{}, %mdk-data-line={419}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{fk}}%mdk-data-line={419}\n{}). \nThe helper function %mdk-data-line={420}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{expr}\\mdToken{Delimiter,Parenthesis,Python,BracketOpen}{(}\\mdToken{Identifier,Python}{v}\\mdToken{Delimiter,Parenthesis,Python,BracketClose}{)}}%mdk-data-line={420}\n{}\nevaluates to an expression tree in the case that %mdk-data-line={421}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{v}}%mdk-data-line={421}\n{} is of %mdk-data-line={421}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdToken{Type,Identifier,Cpp}{Symbolic}}%mdk-data-line={421}\n{} type\n(representing a type in %mdk-data-line={422}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={422}\n{}) and evaluates to %mdk-data-line={422}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{v}}%mdk-data-line={422}\n{} itself, a concrete value\nof some type in %mdk-data-line={423}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U$}%mdk-data-line={423}\n{}, otherwise.\nFinally, having computed the values %mdk-data-line={424}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{c}}%mdk-data-line={424}\n{} and %mdk-data-line={424}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{e}}%mdk-data-line={424}\n{}, the instrumented operator \nreturns %mdk-data-line={425}\n{}\\mdCode[class={code,code2,language-cpp2,lang-cpp2,cpp2,highlighted},language={cpp2}]{\\mdSpan[class={code-escaped}]{\\mdSpan[class={math-inline},elem={math-inline}]{$R'$}}\\mdToken{Delimiter,Parenthesis,Cpp,BracketOpen}{(}\\mdToken{Identifier,Cpp}{c}\\mdToken{Delimiter,Cpp}{,}\\mdToken{Identifier,Cpp}{e}\\mdToken{Delimiter,Parenthesis,Cpp,BracketClose}{)}}%mdk-data-line={425}\n{}, where %mdk-data-line={425}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$R$}%mdk-data-line={425}\n{} is the return type of operator %mdk-data-line={425}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T.o$}%mdk-data-line={425}\n{},\nand %mdk-data-line={426}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$R'$}%mdk-data-line={426}\n{} is a subtype of %mdk-data-line={426}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$R$}%mdk-data-line={426}\n{} from universe %mdk-data-line={426}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={426}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={428}]%\n%mdk-data-line={428}\n{}Looked at another way, the universe %mdk-data-line={428}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={428}\n{} represents the %mdk-data-line={428}\n{}{\\textquotedblleft}tainting{\\textquotedblright}%mdk-data-line={428}\n{} of\ntypes from %mdk-data-line={429}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U$}%mdk-data-line={429}\n{}. Tainted values flow from program inputs to the \noperands of operators. If an operator has been redefined\n(as above) then the taint propagates from its inputs to its outputs.\nOn the other hand, if the operator has not been redefined, then it will\nnot propagate the taint. In the context of DSE, %mdk-data-line={433}\n{}{\\textquotedblleft}taint{\\textquotedblright}%mdk-data-line={433}\n{} means\nthat the instrumented semantics carries along a symbolic expression tree %mdk-data-line={434}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$e$}%mdk-data-line={434}\n{}\nalong with a concrete value %mdk-data-line={435}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={435}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={437}]%\n%mdk-data-line={437}\n{}The choice of types from the universe %mdk-data-line={437}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={437}\n{} determines how symbolic\nexpressions are constructed. For each %mdk-data-line={438}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T \\in U$}%mdk-data-line={438}\n{}, the %mdk-data-line={438}\n{}{\\textquotedblleft}most symbolic{\\textquotedblright}%mdk-data-line={438}\n{}\n(least concrete) choice is the %mdk-data-line={439}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'$}%mdk-data-line={439}\n{} that redefines every operator of %mdk-data-line={439}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={439}\n{}\n(as shown in Figure%mdk-data-line={440}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-subtype}{}{\\mdSpan[class={figure-label}]{5}}%mdk-data-line={440}\n{}).\nThe %mdk-data-line={441}\n{}{\\textquotedblleft}least symbolic{\\textquotedblright}%mdk-data-line={441}\n{} (most concrete) choice is %mdk-data-line={441}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T' = T$}%mdk-data-line={441}\n{} which\nredefines no operators.  Let %mdk-data-line={442}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$symbolic(T)$}%mdk-data-line={442}\n{} be the\nset of types in %mdk-data-line={443}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$U'$}%mdk-data-line={443}\n{} that are subtypes of %mdk-data-line={443}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={443}\n{}. The types\nin %mdk-data-line={444}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$symbolic(T)$}%mdk-data-line={444}\n{} are partially ordered by subset inclusion on the \nset of operators from %mdk-data-line={445}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T$}%mdk-data-line={445}\n{} they redefine.%\n\\end{mdP}%\n\n\\mdHxx[id=sec-sp2dse,label={[3]\\{.heading-label\\}},toc={},data-line={449},caption={[[3]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}From Strongest Postconditions to DSE},bookmark={3.{\\hspace{0.5em}}From Strongest Postconditions to DSE}]{%mdk-data-line={449}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{3}.{\\hspace{0.5em}}}%mdk-data-line={449}\n{}From Strongest Postconditions to DSE}\\begin{mdP}[data-line={451}]%\n%mdk-data-line={451}\n{}The previous section showed how symbolic expressions can be computed via a\nset of instrumented types, where the expressions are computed as a side-effect \nof the execution of program operations.\nThis section shows how these symbolic expressions can be used to form a \n%mdk-data-line={455}\n{}\\mdEm{path-condition}%mdk-data-line={455}\n{} (which then can be compiled into a logic formula and \npassed to an automated theorem prover to find new inputs to drive a \nprogram%mdk-data-line={457}\n{}{'}%mdk-data-line={457}\n{}s execution along new paths). We derive a %mdk-data-line={457}\n{}\\mdEm{path-condition}%mdk-data-line={457}\n{}\ndirectly from the %mdk-data-line={458}\n{}\\mdEm{strongest postcondition}%mdk-data-line={458}\n{} (symbolic) semantics of our\nprogramming language, refining it to model the basic operations\nof an interpreter.%\n\\end{mdP}%\n\\mdHxxx[id=sec-strongest-postconditions,label={[3.1]\\{.heading-label\\}},toc={},data-line={462},caption={[[3.1]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Strongest Postconditions},bookmark={3.1.{\\hspace{0.5em}}Strongest Postconditions}]{%mdk-data-line={462}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{3.1}.{\\hspace{0.5em}}}%mdk-data-line={462}\n{}Strongest Postconditions}\\begin{mdP}[class={para-continue},data-line={464}]%\n%mdk-data-line={464}\n{}The strongest postcondition transformer %mdk-data-line={464}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$SP$}%mdk-data-line={464}\n{}{\\mdNbsp}\\mdSpan[class={citations},target-element={bibitem}]{[\\mdA[class={bibref,localref},target-element={bibitem}]{dijkstra76}{}{\\mdSpan[class={bibitem-label}]{6}}]}%mdk-data-line={464}\n{} is defined over\na predicate %mdk-data-line={465}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={465}\n{} representing a set of \npre-states and a statement %mdk-data-line={466}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$S$}%mdk-data-line={466}\n{} from our language. The transformer %mdk-data-line={466}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$SP(P,S)$}%mdk-data-line={466}\n{}\nyields a predicate %mdk-data-line={467}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$Q$}%mdk-data-line={467}\n{} such that for any state %mdk-data-line={467}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$s$}%mdk-data-line={467}\n{} satisfying\npredicate %mdk-data-line={468}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={468}\n{}, the execution of statement %mdk-data-line={468}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$S$}%mdk-data-line={468}\n{} from state %mdk-data-line={468}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$s$}%mdk-data-line={468}\n{},\nif it does not go wrong or diverge, yields a state %mdk-data-line={469}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$s'$}%mdk-data-line={469}\n{} satisfying predicate %mdk-data-line={469}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$Q$}%mdk-data-line={469}\n{}. \nThe strongest postcondition for the statements in our language\nis defined by the following five rules:%\n\\end{mdP}%\n\\begin{mdDiv}[class={mathpre,para-block,input-mathpre},elem={mathpre},data-line={473}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={474}\n\\begin{mdMathprearray}%mdk\n1.\\mdMathspace{2}&\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mdMathspace{1}\\mathid{x}\\mdMathspace{1}:=\\mdMathspace{1}\\mathid{E})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{1}\\exists \\mathid{y}\\mdMathspace{1}.\\mdMathspace{1}(\\mathid{x}\\mdMathspace{1}=\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}[\\mdMathspace{1}\\mathid{x}\\mdMathspace{1}\\rightarrow \\mathid{y}\\mdMathspace{1}])\\mdMathspace{1}\\wedge \\mathid{P}\\mdMathspace{1}[\\mdMathspace{1}\\mathid{x}\\mdMathspace{1}\\rightarrow \\mathid{y}\\mdMathspace{1}]\\mdMathbr{}\n2.\\mdMathspace{2}&\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mdMathspace{1}\\mathkw{skip})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{1}\\mathid{P}\\mdMathbr{}\n3.\\mdMathspace{2}&\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mathid{S}_{1};\\mathid{S}_{2})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{1}\\mathid{SP}(\\mathid{SP}(\\mathid{P},\\mathid{S}_{1}),\\mdMathspace{1}\\mathid{S}_{2})\\mdMathbr{}\n4.\\mdMathspace{2}&\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mathkw{if}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{S}_1\\mdMathspace{1}\\mathkw{else}\\mdMathspace{1}\\mathid{S}_2\\mdMathspace{1}\\mathkw{end})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathbr{}\n\\mdMathindent{4}&\\mdMathspace{7}\\mathid{SP}(\\mathid{P}\\wedge \\mathid{E},\\mathid{S}_1)\\mdMathspace{1}\\vee  \\mathid{SP}(\\mathid{P}\\mdMathspace{1}\\wedge \\neg \\mathid{E},\\mathid{S}_2)\\mdMathbr{}\n5.\\mdMathspace{2}&\\mdMathspace{1}\\mathid{SP}(\\mathid{P},\\mathkw{while}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{do}\\mdMathspace{1}\\mathid{S}\\mdMathspace{1}\\mathkw{end})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathbr{}\n\\mdMathindent{4}&\\mdMathspace{6}\\mathid{SP}(\\mathid{P},\\mathkw{if}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{S};\\mdMathspace{1}\\mathkw{while}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{do}\\mdMathspace{1}\\mathid{S}\\mdMathspace{1}\\mathkw{end}\\mdMathspace{1}\\mathkw{else}\\mdMathspace{1}\\mathkw{skip}\\mdMathspace{1}\\mathkw{end})\n\\end{mdMathprearray}%mdk\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[data-line={483}]%\n%mdk-data-line={483}\n{}Rule (1) defines the strongest postcondition for \nthe assignment statement. The assignment is modeled\nlogically by the equality %mdk-data-line={485}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x = E$}%mdk-data-line={485}\n{} where any free occurrence of %mdk-data-line={485}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x$}%mdk-data-line={485}\n{} in %mdk-data-line={485}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={485}\n{}\nis replaced by the existentially quantified variable %mdk-data-line={486}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$y$}%mdk-data-line={486}\n{}, which represents\nthe value of %mdk-data-line={487}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x$}%mdk-data-line={487}\n{} in the pre-state. The same substitution (%mdk-data-line={487}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$[x \\rightarrow y ]$}%mdk-data-line={487}\n{})\nis applied to the pre-state predicate %mdk-data-line={488}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={488}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={490}]%\n%mdk-data-line={490}\n{}Rules (2)-(5) define the strongest postcondition for the four control-flow\nstatements. The rules for the %mdk-data-line={491}\n{}\\mdStrong{skip}%mdk-data-line={491}\n{} statement and sequencing (;) are\nstraightforward. \nOf particular interest, note that the rule for the %mdk-data-line={493}\n{}\\mdStrong{if-then-else}%mdk-data-line={493}\n{} \nstatement splits cases on the expression %mdk-data-line={494}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={494}\n{}. \nIt is here that DSE will choose one of the cases for us, as the concrete\nexecution will evaluate %mdk-data-line={496}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={496}\n{} either to be true or false. This gives rise\nto the path-condition (either %mdk-data-line={497}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P \\wedge E$}%mdk-data-line={497}\n{} or %mdk-data-line={497}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P \\wedge \\neg E$}%mdk-data-line={497}\n{}). The\nrecursive rule for the %mdk-data-line={498}\n{}\\mdStrong{while}%mdk-data-line={498}\n{} loop unfolds as many times as the\nexpression %mdk-data-line={499}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={499}\n{} evaluates true, adding to the path-condition.%\n\\end{mdP}%\n\\mdHxxx[id=sp-refined,label={[3.2]\\{.heading-label\\}},toc={},data-line={501},caption={[[3.2]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}From \\$SP\\$ to DSE},bookmark={3.2.{\\hspace{0.5em}}From \\$SP\\$ to DSE}]{%mdk-data-line={501}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{3.2}.{\\hspace{0.5em}}}%mdk-data-line={501}\n{}From %mdk-data-line={501}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$SP$}%mdk-data-line={501}\n{} to DSE}\\begin{mdP}[data-line={503}]%\n%mdk-data-line={503}\n{}Assume that an execution begins with the assignment of initial values %mdk-data-line={503}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c_1 \\ldots c_k$}%mdk-data-line={503}\n{} to the program %mdk-data-line={503}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={503}\n{}{'}%mdk-data-line={503}\n{}s inputs\n%mdk-data-line={504}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$V = \\{ v_1 : T_1 \\ldots v_k : T_k \\}$}%mdk-data-line={504}\n{}.  To seed symbolic execution, some of the types %mdk-data-line={504}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T_i$}%mdk-data-line={504}\n{} are\nreplaced by symbolic counterparts %mdk-data-line={505}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T'_i$}%mdk-data-line={505}\n{}, in which case\n%mdk-data-line={506}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$v_i$}%mdk-data-line={506}\n{} is initialized to the value %mdk-data-line={506}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$sc_i = T'_i (c_i,SC(v_i))$}%mdk-data-line={506}\n{} instead of the value %mdk-data-line={506}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c_i$}%mdk-data-line={506}\n{},\nwhere %mdk-data-line={507}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$SC(v_i)$}%mdk-data-line={507}\n{} is the symbolic constant representing the initial value of variable %mdk-data-line={507}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$v_i$}%mdk-data-line={507}\n{}.\nThe symbolic constant %mdk-data-line={508}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$SC(v_i)$}%mdk-data-line={508}\n{} can be thought of as representing any value of\ntype %mdk-data-line={509}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$T_i$}%mdk-data-line={509}\n{}, which includes the value %mdk-data-line={509}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c_i$}%mdk-data-line={509}\n{}.%mdk-data-line={509}\n{} %mdk-data-line={509}\n{}%\n\\end{mdP}%\n\\begin{mdP}[class={indent,para-continue},data-line={511}]%\n%mdk-data-line={511}\n{}Let %mdk-data-line={511}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$V_s$}%mdk-data-line={511}\n{} and %mdk-data-line={511}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$V_c$}%mdk-data-line={511}\n{} partition the variables of %mdk-data-line={511}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$V$}%mdk-data-line={511}\n{} into those variables\nthat are treated symbolically (%mdk-data-line={512}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$V_s$}%mdk-data-line={512}\n{}) and those that are treated concretely \n(%mdk-data-line={513}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$V_c$}%mdk-data-line={513}\n{}). The initial state of the program is characterized by the formula%\n\\end{mdP}%\n\\begin{mdDiv}[class={equation,para-block},label={[(1)]\\{.equation-label\\}},elem={equation},line-adjust={0},data-line={515}]%\n%mdk-data-line={515}\n{}\\mdSpan[class={equation-before}]{\\mdSpan[class={equation-label}]{(1)}}%mdk-data-line={515}\n{}\n\\begin{mdDiv}[class={mathdisplay,para-block,input-math},elem={mathdisplay},color={},math-needpdf={},line-adjust={0},data-line={516}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={516}\nInit = (\\bigwedge_{v_i \\in V_s} v_i = sc_i) ~\\wedge~ (~\\bigwedge_{v_i \\in V_c} v_i = c_i)\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[data-line={519}]%\n%mdk-data-line={519}\n{}Thus, we see that the initial value of every input variable is characterized by \na symbolic constant %mdk-data-line={520}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$sc_i$}%mdk-data-line={520}\n{} or constant %mdk-data-line={520}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c_i$}%mdk-data-line={520}\n{}.  We assume that every non-input\nvariable in the program is initialized before being used.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={523}]%\n%mdk-data-line={523}\n{}The strongest postcondition is formulated to deal with open programs,\nprograms in which some variables are used before being assigned to. \nThis surfaces in Rule (1) for assignment, which uses existential quantification\nto refer to the value of variable %mdk-data-line={526}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x$}%mdk-data-line={526}\n{} in the pre-state.%\n\\end{mdP}%\n\\begin{mdP}[class={indent,para-continue},data-line={528}]%\n%mdk-data-line={528}\n{}By construction,\nwe have that every variable is defined before being used.  This means\nthat the precondition %mdk-data-line={530}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P$}%mdk-data-line={530}\n{} can be reformulated as a pair %mdk-data-line={530}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$<\\sigma,P_c>$}%mdk-data-line={530}\n{},\nwhere %mdk-data-line={531}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma$}%mdk-data-line={531}\n{} is a store mapping variables to values and %mdk-data-line={531}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P_c$}%mdk-data-line={531}\n{} is\nthe path-condition, a list of symbolic expressions (predicates) \ncorresponding\nto the expressions %mdk-data-line={534}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={534}\n{} evaluated in the context of an %mdk-data-line={534}\n{}\\mdStrong{if-then-else}%mdk-data-line={534}\n{} \nstatement. Initially, we have that :%\n\\end{mdP}%\n\\begin{mdDiv}[class={equation,para-block},label={[(2)]\\{.equation-label\\}},elem={equation},line-adjust={0},data-line={537}]%\n%mdk-data-line={537}\n{}\\mdSpan[class={equation-before}]{\\mdSpan[class={equation-label}]{(2)}}%mdk-data-line={537}\n{}\n\\begin{mdDiv}[class={mathdisplay,para-block,input-math},elem={mathdisplay},color={},math-needpdf={},line-adjust={0},data-line={538}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={538}\n\\sigma = \\{ (v_i,sc_i) | v_i \\in V_s \\} \\cup \\{ (v_i,c_i) | v_i \\in V_c \\}\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[class={para-continue},data-line={541}]%\n%mdk-data-line={541}\n{}representing the initial condition %mdk-data-line={541}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$Init$}%mdk-data-line={541}\n{}, and %mdk-data-line={541}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P_c = []$}%mdk-data-line={541}\n{}, the empty list.\nWe use %mdk-data-line={542}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma'$}%mdk-data-line={542}\n{} to refer to the formula that the store %mdk-data-line={542}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma$}%mdk-data-line={542}\n{}\ninduces:%\n\\end{mdP}%\n\\begin{mdDiv}[class={equation,para-block},label={[(3)]\\{.equation-label\\}},elem={equation},line-adjust={0},data-line={545}]%\n%mdk-data-line={545}\n{}\\mdSpan[class={equation-before}]{\\mdSpan[class={equation-label}]{(3)}}%mdk-data-line={545}\n{}\n\\begin{mdDiv}[class={mathdisplay,para-block,input-math},elem={mathdisplay},color={},math-needpdf={},line-adjust={0},data-line={546}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={546}\n\\sigma ' =  \\bigwedge_{(v,V) \\in \\sigma} (v = V)\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[data-line={549}]%\n%mdk-data-line={549}\n{}Thus, the pair %mdk-data-line={549}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$<\\sigma,P_c>$}%mdk-data-line={549}\n{} represents the predicate \n%mdk-data-line={550}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P = \\sigma' \\wedge (\\bigwedge_{c \\in P_c} c)$}%mdk-data-line={550}\n{}.\nA store %mdk-data-line={551}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma$}%mdk-data-line={551}\n{} supports two operations: %mdk-data-line={551}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma[x]$}%mdk-data-line={551}\n{} which denotes the\nvalue that %mdk-data-line={552}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x$}%mdk-data-line={552}\n{} maps to under %mdk-data-line={552}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma$}%mdk-data-line={552}\n{}; %mdk-data-line={552}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma[x \\mapsto V]$}%mdk-data-line={552}\n{}, which \nproduces a new store in which %mdk-data-line={553}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x$}%mdk-data-line={553}\n{} maps to value %mdk-data-line={553}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$V$}%mdk-data-line={553}\n{} and is everywhere \nelse the same as %mdk-data-line={554}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma$}%mdk-data-line={554}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent,para-continue},data-line={556}]%\n%mdk-data-line={556}\n{}Now, we can redefine strongest postcondition for assignment to eliminate the\nuse of existential quantification and model the operation of an interpreter,\nby separating out the notion of the store:%\n\\end{mdP}%\n\\begin{mdDiv}[class={mathpre,para-block,input-mathpre},elem={mathpre},data-line={560}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={561}\n\\begin{mdMathprearray}%mdk\n1.\\mdMathspace{1}&\\mdMathspace{1}\\mathid{SP}(<\\sigma,\\mdMathspace{1}\\mathid{P}_\\mathid{c}>,\\mdMathspace{1}\\mathid{x}\\mdMathspace{1}:=\\mdMathspace{1}\\mathid{E})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{1}<\\sigma[\\mathid{x}\\mdMathspace{1}\\mapsto \\mathid{eval}(\\sigma,\\mathid{E})],\\mdMathspace{1}\\mathid{P}_\\mathid{c}>\\\\\n\\end{mdMathprearray}%mdk\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[data-line={564}]%\n%mdk-data-line={564}\n{}where %mdk-data-line={564}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$eval(\\sigma,E)$}%mdk-data-line={564}\n{} evaluates expression %mdk-data-line={564}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={564}\n{} under the store %mdk-data-line={564}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma$}%mdk-data-line={564}\n{}\n(where every occurrence of a free variable\n%mdk-data-line={566}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$v$}%mdk-data-line={566}\n{} in %mdk-data-line={566}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$E$}%mdk-data-line={566}\n{} is replaced by the value %mdk-data-line={566}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\sigma[v]$}%mdk-data-line={566}\n{}). \nThis is the standard substitution rule of a standard operational semantics.%\n\\end{mdP}%\n\\begin{mdP}[class={indent,para-continue},data-line={569}]%\n%mdk-data-line={569}\n{}We also redefine the rule for the %mdk-data-line={569}\n{}\\mdStrong{if-then-else}%mdk-data-line={569}\n{} statement so\nthat it chooses which branch to take and appends the appropriate\nsymbolic expression (predicate) to the path-condition %mdk-data-line={571}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$P_c$}%mdk-data-line={571}\n{}:%\n\\end{mdP}%\n\\begin{mdDiv}[class={mathpre,para-block,input-mathpre},elem={mathpre},data-line={573}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={574}\n\\begin{mdMathprearray}%mdk\n4.\\mdMathspace{1}&\\mdMathspace{1}\\mathid{SP}(<\\sigma,\\mdMathspace{1}\\mathid{P}_\\mathid{c}>,\\mdMathspace{1}\\mathkw{if}\\mdMathspace{1}\\mathid{E}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{S}_1\\mdMathspace{1}\\mathkw{else}\\mdMathspace{1}\\mathid{S}_2\\mdMathspace{1}\\mathkw{end})\\mdMathspace{1}{~\\buildrel\\triangle\\over=~}\\mdMathspace{2}\\mdMathbr{}\n\\mdMathindent{3}&\\mdMathspace{3}\\mathkw{let}\\mdMathspace{1}\\mathid{choice}\\mdMathspace{1}=\\mdMathspace{1}\\mathid{eval}(\\sigma,\\mathid{E})\\mdMathspace{1}\\mathkw{in}\\mdMathbr{}\n\\mdMathindent{3}&\\mdMathspace{3}\\mathkw{if}\\mdMathspace{1}\\mathid{choice}\\mdMathspace{1}\\mathkw{then}\\mdMathspace{1}\\mathid{SP}(<\\sigma,\\mdMathspace{1}\\mathid{P}_\\mathid{c}\\mdMathspace{1}::\\mdMathspace{1}\\mathid{expr}(\\mathid{choice})\\mdMathspace{1}>,\\mathid{S}_1)\\mdMathspace{1}\\mdMathbr{}\n\\mdMathindent{3}&\\mdMathspace{3}\\mathkw{else}\\mdMathspace{1}\\mathid{SP}(<\\sigma,\\mdMathspace{1}\\mathid{P}_\\mathid{c}\\mdMathspace{1}::\\mdMathspace{1}\\neg \\mathid{expr}(\\mathid{choice})\\mdMathspace{1}>,\\mathid{S}_2)\n\\end{mdMathprearray}%mdk\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[data-line={580}]%\n%mdk-data-line={580}\n{}The other strongest postcondition rules remain unchanged.%\n\\end{mdP}%\n\\mdHxxx[id=sec-summing-it-up,label={[3.3]\\{.heading-label\\}},toc={},data-line={582},caption={[[3.3]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Summing it up},bookmark={3.3.{\\hspace{0.5em}}Summing it up}]{%mdk-data-line={582}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{3.3}.{\\hspace{0.5em}}}%mdk-data-line={582}\n{}Summing it up}\\begin{mdP}[data-line={584}]%\n%mdk-data-line={584}\n{}We have shown how the symbolic predicate transformer %mdk-data-line={584}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$SP$}%mdk-data-line={584}\n{} can \nbe refined into a symbolic interpreter operating over the\nsymbolic types defined in the previous section.\nIn the case when every input variable is symbolic and every\noperator is redefined, the path-condition is equivalent to the\n%mdk-data-line={589}\n{}\\mdEm{strongest postcondition}%mdk-data-line={589}\n{} of the execution path %mdk-data-line={589}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={589}\n{}. \nThis guarantees that the path-condition for %mdk-data-line={590}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={590}\n{} is %mdk-data-line={590}\n{}\\mdEm{sound}%mdk-data-line={590}\n{}.\nIn the case where a subset of the input variables are symbolic\nand/or not all operators are redefined, the path-condition of %mdk-data-line={592}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p$}%mdk-data-line={592}\n{} \nis not guaranteed to be sound. We leave it as an exercise to the\nreader to establish sufficient conditions under which the \nuse of concrete values in place of symbolic expressions is \nguaranteed to result in sound path-conditions.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={598}]%\n%mdk-data-line={598}\n{}This section does not address the compilation of a symbolic\nexpression to the (logic) language of an underlying ATP, nor the\nlifting of a satisfying assignment to a formula back to the\nlevel of the source language. This is best done for a particular\nsource language and ATP, as detailed in the next section.%\n\\end{mdP}%\n\\mdHxx[id=sec-impl,label={[4]\\{.heading-label\\}},toc={},data-line={605},caption={[[4]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Architecture of PyExZ3},bookmark={4.{\\hspace{0.5em}}Architecture of PyExZ3}]{%mdk-data-line={605}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{4}.{\\hspace{0.5em}}}%mdk-data-line={605}\n{}Architecture of PyExZ3}\\begin{mdP}[data-line={607}]%\n%mdk-data-line={607}\n{}In this section we present the high-level architecture\nof a simple DSE tool for the Python language, written in Python, called%mdk-data-line={608}\n{}{\\mdNbsp}\\mdA[data-linkid={pyexz3}]{https://github.com/thomasjball/PyExZ3/}{}{PyExZ3}%mdk-data-line={608}\n{}. \nFigure%mdk-data-line={609}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={figure}]{fig-arch}{}{\\mdSpan[class={figure-label}]{6}}%mdk-data-line={609}\n{}\nshows the class diagram (dashed edges are %mdk-data-line={610}\n{}{\\textquotedblleft}has-a{\\textquotedblright}%mdk-data-line={610}\n{} relationships; solid edges\nare %mdk-data-line={611}\n{}{\\textquotedblleft}is-a{\\textquotedblright}%mdk-data-line={611}\n{} relationships) of the tool.%\n\\end{mdP}%\n\\begin{mdDiv}[class={figure,floating,align-center},id=fig-arch,label={[6]\\{.figure-label\\}},elem={figure},toc-line={[6]\\{.figure-label\\}. Classes in PyExZ3},toc={tof},float-env={figure},float-name={Figure},caption={Classes in PyExZ3},page-align={here},data-line={613}]%\n\\begin{mdP}[data-line={614}]%\n%mdk-data-line={614}\n{}\\mdImg[width={1.00\\linewidth},data-linkid={arch}]{arch.png}%mdk-data-line={614}\n{}%\n\\end{mdP}%\n\\mdHr[class={figureline,madoko},data-line={615}]{}\\begin{mdDiv}[data-line={616}]%\n%mdk-data-line={616}\n{}\\mdSpan[class={figure-caption}]{\\mdSpan[class={caption-before}]{\\mdStrong{Figure{\\mdNbsp}\\mdSpan[class={figure-label}]{6}.} }Classes in PyExZ3}%mdk-data-line={616}\n{}%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\mdHxxx[id=sec-loading-the-code-under-test,label={[4.1]\\{.heading-label\\}},toc={},data-line={619},caption={[[4.1]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Loading the code under test},bookmark={4.1.{\\hspace{0.5em}}Loading the code under test}]{%mdk-data-line={619}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{4.1}.{\\hspace{0.5em}}}%mdk-data-line={619}\n{}Loading the code under test}\\begin{mdP}[data-line={621}]%\n%mdk-data-line={621}\n{}The %mdk-data-line={621}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Loader}}%mdk-data-line={621}\n{} class takes as input the name of a Python file (e.g., %mdk-data-line={621}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}\\mdToken{Delimiter,Python}{.}\\mdToken{Identifier,Python}{py}}%mdk-data-line={621}\n{}) \nto import. The loader expects to find a function named\n%mdk-data-line={623}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}}%mdk-data-line={623}\n{} inside the file %mdk-data-line={623}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}\\mdToken{Delimiter,Python}{.}\\mdToken{Identifier,Python}{py}}%mdk-data-line={623}\n{}, which will serve as the starting point\nfor symbolic execution. The %mdk-data-line={624}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{FunctionInvocation}}%mdk-data-line={624}\n{} class\nwraps this starting point. By default, each parameter to %mdk-data-line={625}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}}%mdk-data-line={625}\n{} is\na %mdk-data-line={626}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicInteger}}%mdk-data-line={626}\n{} unless there is decorator %mdk-data-line={626}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Delimiter,Python}{@}\\mdToken{Identifier,Python}{symbolic}}%mdk-data-line={626}\n{} specifying\nthe type to use for a particular argument.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={629}]%\n%mdk-data-line={629}\n{}The loader provides the capability to reload the\nmodule %mdk-data-line={630}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}\\mdToken{Delimiter,Python}{.}\\mdToken{Identifier,Python}{py}}%mdk-data-line={630}\n{} so that the function %mdk-data-line={630}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}}%mdk-data-line={630}\n{} can be \nreexecuted within the same process from the same initial\nstate with different inputs (see the class %mdk-data-line={632}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{ExplorationEngine}}%mdk-data-line={632}\n{})\nvia the %mdk-data-line={633}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{FunctionInvocation}}%mdk-data-line={633}\n{} class.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={635}]%\n%mdk-data-line={635}\n{}Finally, the loader looks for specially named functions %mdk-data-line={635}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{expected\\_result}}%mdk-data-line={635}\n{}\n(%mdk-data-line={636}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{expected\\_result\\_set}}%mdk-data-line={636}\n{}) in file %mdk-data-line={636}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}\\mdToken{Delimiter,Python}{.}\\mdToken{Identifier,Python}{py}}%mdk-data-line={636}\n{} to use as a test oracle after\nthe path exploration (by %mdk-data-line={637}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{ExplorationEngine}}%mdk-data-line={637}\n{}) has completed. These\nfunctions are expected to return a list of values to check\nagainst the list of return values collected from the executions of \nthe %mdk-data-line={640}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{foo}}%mdk-data-line={640}\n{} function.\nThe presence of the function %mdk-data-line={641}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{expected\\_result}}%mdk-data-line={641}\n{} (%mdk-data-line={641}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{expected\\_result\\_set}}%mdk-data-line={641}\n{}) \nyields a comparison of the two lists as bags (sets). We use such weaker\ntests, rather than list equality, because the order in which paths\nare explored by the %mdk-data-line={644}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{ExplorationEngine}}%mdk-data-line={644}\n{} can easily change due to small\ndifferences in the input programs.%\n\\end{mdP}%\n\\mdHxxx[id=sec-symbolic-types,label={[4.2]\\{.heading-label\\}},toc={},data-line={647},caption={[[4.2]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Symbolic types},bookmark={4.2.{\\hspace{0.5em}}Symbolic types}]{%mdk-data-line={647}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{4.2}.{\\hspace{0.5em}}}%mdk-data-line={647}\n{}Symbolic types}\\begin{mdP}[data-line={649}]%\n%mdk-data-line={649}\n{}Python supports multiple inheritance and, more importantly,\nallows user-defined classes to inherit \nfrom its built-in types (such as %mdk-data-line={651}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{object}}%mdk-data-line={651}\n{} and %mdk-data-line={651}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{int}}%mdk-data-line={651}\n{}).\nWe use these two features two implement\nsymbolic versions of Python objects and integers, \nfollowing the instrumented type approach defined in Section%mdk-data-line={654}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-semantics}{}{\\mdSpan[class={heading-label}]{2}}%mdk-data-line={654}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={656}]%\n%mdk-data-line={656}\n{}The abstract class %mdk-data-line={656}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicType}}%mdk-data-line={656}\n{} contains the \nsymbolic expression tree and provides basic functions for constructing \nand accessing the tree.  This class does double duty, as it is used\nto represent the (typed) symbolic constants associated with the parameters to the \nfunction, as well as the expression trees (per Section%mdk-data-line={660}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-semantics}{}{\\mdSpan[class={heading-label}]{2}}%mdk-data-line={660}\n{}). Recall\nthat the symbolic constants only appear as leaves of expression trees.\nThis means that the expression tree stored in a %mdk-data-line={662}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicType}}%mdk-data-line={662}\n{} will\nhave instances of a %mdk-data-line={663}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicType}}%mdk-data-line={663}\n{} as some of its leaves, namely\nthose leaves representing the symbolic constants.\n%mdk-data-line={665}\n{}%mdk-data-line={665}\n{}\nThe abstract class provides an %mdk-data-line={666}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{unwrap}}%mdk-data-line={666}\n{} method which returns\nthe pair of concrete value and expression tree associated with\nthe %mdk-data-line={668}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicType}}%mdk-data-line={668}\n{}, as well as a %mdk-data-line={668}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{wrap}}%mdk-data-line={668}\n{} method that takes a \npair of concrete value and expression tree and creates a %mdk-data-line={669}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicType}}%mdk-data-line={669}\n{}\nencapsulating them.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={672}]%\n%mdk-data-line={672}\n{}The class %mdk-data-line={672}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicObject}}%mdk-data-line={672}\n{} inherits from both %mdk-data-line={672}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{object}}%mdk-data-line={672}\n{} and %mdk-data-line={672}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicType}}%mdk-data-line={672}\n{} and\noverrides the basic comparison operations (%mdk-data-line={673}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_eq\\_\\_}}%mdk-data-line={673}\n{}, %mdk-data-line={673}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_neq\\_\\_}}%mdk-data-line={673}\n{}, %mdk-data-line={673}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_lt\\_\\_}}%mdk-data-line={673}\n{}, %mdk-data-line={673}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_le\\_\\_}}%mdk-data-line={673}\n{},\n%mdk-data-line={674}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_gt\\_\\_}}%mdk-data-line={674}\n{}, and %mdk-data-line={674}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_ge\\_\\_}}%mdk-data-line={674}\n{}).\nThe class %mdk-data-line={675}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicInteger}}%mdk-data-line={675}\n{} inherits from both %mdk-data-line={675}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{int}}%mdk-data-line={675}\n{} and %mdk-data-line={675}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicObject}}%mdk-data-line={675}\n{}\nand overrides a number of %mdk-data-line={676}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{int}}%mdk-data-line={676}\n{}{'}%mdk-data-line={676}\n{}s arithmetic methods\n(%mdk-data-line={677}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_add\\_\\_}}%mdk-data-line={677}\n{}, %mdk-data-line={677}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_sub\\_\\_}}%mdk-data-line={677}\n{}, %mdk-data-line={677}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_mul\\_\\_}}%mdk-data-line={677}\n{}, %mdk-data-line={677}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_mod\\_\\_}}%mdk-data-line={677}\n{}, %mdk-data-line={677}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_floordiv\\_}}%mdk-data-line={677}\n{})\nand bitwise methods\n(%mdk-data-line={679}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_and\\_\\_}}%mdk-data-line={679}\n{}, %mdk-data-line={679}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_or\\_\\_}}%mdk-data-line={679}\n{}, %mdk-data-line={679}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_xor\\_\\_}}%mdk-data-line={679}\n{}, %mdk-data-line={679}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_lshift\\_\\_}}%mdk-data-line={679}\n{}, %mdk-data-line={679}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_rshift\\_\\_}}%mdk-data-line={679}\n{}).%\n\\end{mdP}%\n\\mdHxxx[id=sec-tracing-control-flow,label={[4.3]\\{.heading-label\\}},toc={},data-line={682},caption={[[4.3]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Tracing control-flow},bookmark={4.3.{\\hspace{0.5em}}Tracing control-flow}]{%mdk-data-line={682}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{4.3}.{\\hspace{0.5em}}}%mdk-data-line={682}\n{}Tracing control-flow}\\begin{mdP}[data-line={684}]%\n%mdk-data-line={684}\n{}As Python interprets a program, it will evaluate expressions, \nsubstituting the value of a variable in its place in an \nexpression, applying operators (methods) to \nparameter values and assigning the return values of methods\nto variables. Value of type %mdk-data-line={688}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicInteger}}%mdk-data-line={688}\n{} will simply flow\nthrough this interpretation, without necessitating any change \nto the program or the interpreter. This takes care\nof the case of the strongest-postcondition rule for assignment,\nas elaborated in Section%mdk-data-line={692}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h2}]{sp-refined}{}{\\mdSpan[class={heading-label}]{3.2}}%mdk-data-line={692}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={694}]%\n%mdk-data-line={694}\n{}The strong-postcondition rule for a conditional test requires\na little more work. In Python, any object can be tested in\nan %mdk-data-line={696}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Keyword,Python}{if}}%mdk-data-line={696}\n{} or %mdk-data-line={696}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Keyword,Python}{while}}%mdk-data-line={696}\n{} condition or as the operand of a Boolean operation\n(%mdk-data-line={697}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Keyword,Python}{and}}%mdk-data-line={697}\n{}, %mdk-data-line={697}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Keyword,Python}{or}}%mdk-data-line={697}\n{}, %mdk-data-line={697}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Keyword,Python}{not}}%mdk-data-line={697}\n{})\nThe Python base class %mdk-data-line={698}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{object}}%mdk-data-line={698}\n{} provides a method named %mdk-data-line={698}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_bool\\_\\_}}%mdk-data-line={698}\n{} that\nthe Python runtime calls whenever it needs to perform such a conditional test.\nThis hook provides us what we need to trace the conditional\ncontrol-flow of a Python execution.  We override this method in the class\n%mdk-data-line={702}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicObject}}%mdk-data-line={702}\n{} in order to inform the %mdk-data-line={702}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{PathToConstraint}}%mdk-data-line={702}\n{} object (defined\nlater) of the symbolic expression for the conditional (as captured by\nthe %mdk-data-line={704}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicInteger}}%mdk-data-line={704}\n{} subclass).%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={706}]%\n%mdk-data-line={706}\n{}Note that the use of this hook in combination with the\ntainted types will only trace those conditionals\nin a Python execution whose values inherit from %mdk-data-line={708}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicObject}}%mdk-data-line={708}\n{}; \nby definition, %mdk-data-line={709}\n{}{\\textquotedblleft}untainted{\\textquotedblright}%mdk-data-line={709}\n{} conditionals do not depend on symbolic \ninputs so there is no value in adding them to the path-condition.%\n\\end{mdP}%\n\\mdHxxx[id=sec-recording-path-conditions,label={[4.4]\\{.heading-label\\}},toc={},data-line={712},caption={[[4.4]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Recording path-conditions},bookmark={4.4.{\\hspace{0.5em}}Recording path-conditions}]{%mdk-data-line={712}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{4.4}.{\\hspace{0.5em}}}%mdk-data-line={712}\n{}Recording path-conditions}\\begin{mdP}[data-line={714}]%\n%mdk-data-line={714}\n{}A %mdk-data-line={714}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Predicate}}%mdk-data-line={714}\n{} records a conditional (more precisely the symbolic expression\nfound in %mdk-data-line={715}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicInteger}}%mdk-data-line={715}\n{}) and\nwhich way it evaluated in an execution.  A %mdk-data-line={716}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={716}\n{}\nhas a %mdk-data-line={717}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Predicate}}%mdk-data-line={717}\n{}, a parent %mdk-data-line={717}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={717}\n{} and a set\nof %mdk-data-line={718}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={718}\n{} children. %mdk-data-line={718}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraints}}%mdk-data-line={718}\n{} form a tree, where\neach path starting from the root of the tree represents\na path-condition. The tree represents all path-conditions that have\nbeen explored so far.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={723}]%\n%mdk-data-line={723}\n{}The class %mdk-data-line={723}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{PathToConstraint}}%mdk-data-line={723}\n{} has a reference to the root of \nthe tree of %mdk-data-line={724}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={724}\n{}s\nand is responsible for installing a new %mdk-data-line={725}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={725}\n{} in the tree\nwhen notified by the overridden %mdk-data-line={726}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Predefined,Python}{\\_\\_bool\\_\\_}}%mdk-data-line={726}\n{} method of %mdk-data-line={726}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicObject}}%mdk-data-line={726}\n{}.\n%mdk-data-line={727}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{PathToConstraint}}%mdk-data-line={727}\n{} also tracks whether or not the current execution\nis following an existing path in the tree and grows the\ntree as needed. In fact, it actually\ntracks whether or not the current execution follows a particular\n%mdk-data-line={731}\n{}\\mdEm{expected path}%mdk-data-line={731}\n{} in the tree.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={733}]%\n%mdk-data-line={733}\n{}The expected path is the result\nof the %mdk-data-line={734}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{ExplorationEngine}}%mdk-data-line={734}\n{} picking a constraint %mdk-data-line={734}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={734}\n{} in the tree,\nand asking the ATP if the path-condition consisting of the prefix\nof predicates up to but not including %mdk-data-line={736}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={736}\n{} in the tree, \nfollowed by the negation of %mdk-data-line={737}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={737}\n{}{'}%mdk-data-line={737}\n{}s predicate is satisfiable. If the \nATP returns %mdk-data-line={738}\n{}{\\textquotedblleft}satisfiable{\\textquotedblright}%mdk-data-line={738}\n{} (with a new input %mdk-data-line={738}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$i$}%mdk-data-line={738}\n{}), then the assumption\nis that path-condition prefix is sound (that is, the execution of\nthe program on input %mdk-data-line={740}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$i$}%mdk-data-line={740}\n{} will follow the prefix).%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={742}]%\n%mdk-data-line={742}\n{}However, it is possible \nfor the path-condition to be unsound and for \nthe executed path to diverge early \nfrom the expected path, due to the fact that not every operation\nhas a symbolic encoding.  The tool simply reports the divergence\nand continues to process the execution as usual (as a diverging\npath may lead to some other interesting part of the code).%\n\\end{mdP}%\n\\mdHxxx[id=sec-from-symbolic-types-to-z3,label={[4.5]\\{.heading-label\\}},toc={},data-line={750},caption={[[4.5]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}From symbolic types to Z3},bookmark={4.5.{\\hspace{0.5em}}From symbolic types to Z3}]{%mdk-data-line={750}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{4.5}.{\\hspace{0.5em}}}%mdk-data-line={750}\n{}From symbolic types to Z3}\\begin{mdP}[data-line={752}]%\n%mdk-data-line={752}\n{}As we have explained DSE, the symbolic expressions are \nrepresented at the level of the source language. As detailed later\nin Section%mdk-data-line={754}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-int2z3}{}{\\mdSpan[class={heading-label}]{5}}%mdk-data-line={754}\n{}, we must translate \nfrom the source language to the input language of an\nautomated theorem prover (ATP), in this case%mdk-data-line={756}\n{}{\\mdNbsp}\\mdA[data-linkid={z3}]{http://z3.codeplex.org/}{}{Z3}%mdk-data-line={756}\n{}.  This\nseparation of languages is quite useful, as we may have\nthe need to translate a given symbolic expression\nto the ATP%mdk-data-line={759}\n{}{'}%mdk-data-line={759}\n{}s language multiple times, to make use of different\nfeatures of the underlying ATP. \nFurthermore, this separation\nof concerns allows us to easily retarget the DSE tool to a\ndifferent ATP.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={765}]%\n%mdk-data-line={765}\n{}The base class %mdk-data-line={765}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3Expression}}%mdk-data-line={765}\n{} represents a Z3 formula. The two \nsubclasses %mdk-data-line={766}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3Integer}}%mdk-data-line={766}\n{} and %mdk-data-line={766}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3BitVector}}%mdk-data-line={766}\n{} represent different ways \nto model arithmetic reasoning about integers in Z3. We will describe\nthe details of these encodings in Section%mdk-data-line={768}\n{}{\\mdNbsp}\\mdA[class={localref},target-element={h1}]{sec-int2z3}{}{\\mdSpan[class={heading-label}]{5}}%mdk-data-line={768}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={770}]%\n%mdk-data-line={770}\n{}The class %mdk-data-line={770}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3Wrapper}}%mdk-data-line={770}\n{} is responsible for performing the\ntranslation from the source language (Python) to Z3%mdk-data-line={771}\n{}{'}%mdk-data-line={771}\n{}s input language, \ninvoking Z3, and lifting a Z3 answer back to the level of Python. \nThe %mdk-data-line={773}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{findCounterexample}}%mdk-data-line={773}\n{} method does all the work, taking as\ninput a list of %mdk-data-line={774}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Predicate}}%mdk-data-line={774}\n{}s (called %mdk-data-line={774}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{assertions}}%mdk-data-line={774}\n{}) \nas well as a single %mdk-data-line={775}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Predicate}}%mdk-data-line={775}\n{} (called\nthe %mdk-data-line={776}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{query}}%mdk-data-line={776}\n{}). The %mdk-data-line={776}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{assertions}}%mdk-data-line={776}\n{} represent a path-condition\nprefix derived from the %mdk-data-line={777}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={777}\n{} tree that we wish the next\nexecution to follow, while %mdk-data-line={778}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{query}}%mdk-data-line={778}\n{} represents the predicate\nfollowing the prefix in the tree that we will negate.%\n\\end{mdP}%\n\\begin{mdP}[class={indent,para-continue},data-line={781}]%\n%mdk-data-line={781}\n{}The method constructs the formula%\n\\end{mdP}%\n\\begin{mdDiv}[class={equation,para-block},label={[(4)]\\{.equation-label\\}},elem={equation},line-adjust={0},data-line={783}]%\n%mdk-data-line={783}\n{}\\mdSpan[class={equation-before}]{\\mdSpan[class={equation-label}]{(4)}}%mdk-data-line={783}\n{}\n\\begin{mdDiv}[class={mathdisplay,para-block,input-math},elem={mathdisplay},color={},math-needpdf={},line-adjust={0},data-line={784}]%\n\\begin{mdDiv}[class={math-display}]%\n\\[%mdk-data-line={784}\n(\\bigwedge_{a \\in asserts} a) \\wedge \\neg query\n\\]%\n\\end{mdDiv}%%\n\\end{mdDiv}%%\n\\end{mdDiv}%\n\\begin{mdP}[data-line={787}]%\n%mdk-data-line={787}\n{}and asks Z3 if it is satisfiable. The method performs\na standard syntactic %mdk-data-line={788}\n{}{\\textquotedblleft}cone of influence{\\textquotedblright}%mdk-data-line={788}\n{} (CIF)\nreduction on the %mdk-data-line={789}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{asserts}}%mdk-data-line={789}\n{} with respect to the \n%mdk-data-line={790}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{query}}%mdk-data-line={790}\n{} to shrink the size of the formula. For example,\nif %mdk-data-line={791}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{asserts}}%mdk-data-line={791}\n{} is the set of predicates %mdk-data-line={791}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\{ (x<a), (a<0), (y>0) \\}$}%mdk-data-line={791}\n{} and\nthe query is %mdk-data-line={792}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$(x=0)$}%mdk-data-line={792}\n{}, then the CIF yields the\nset  %mdk-data-line={793}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\{ (x<a), (a<0) \\}$}%mdk-data-line={793}\n{}, which does not include the predicate %mdk-data-line={793}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$(y>0)$}%mdk-data-line={793}\n{},\nas the variable %mdk-data-line={794}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$y$}%mdk-data-line={794}\n{} is not in the set of variables (transitively)\nrelated to variable %mdk-data-line={795}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$x$}%mdk-data-line={795}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={797}]%\n%mdk-data-line={797}\n{}If the formula is satisfiable a model is requested\nfrom Z3 and lifted back to Python%mdk-data-line={798}\n{}{'}%mdk-data-line={798}\n{}s type universe.  Note that\nbecause of the CIF reduction, the model may not mention certain\ninput variables, in which case we simply keep their values from\nthe execution from which the %mdk-data-line={801}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{asserts}}%mdk-data-line={801}\n{} and %mdk-data-line={801}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Identifier,Python}{query}}%mdk-data-line={801}\n{} were derived.%\n\\end{mdP}%\n\\mdHxxx[id=sec-putting-it-all-together,label={[4.6]\\{.heading-label\\}},toc={},data-line={803},caption={[[4.6]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Putting it all together},bookmark={4.6.{\\hspace{0.5em}}Putting it all together}]{%mdk-data-line={803}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{4.6}.{\\hspace{0.5em}}}%mdk-data-line={803}\n{}Putting it all together}\\begin{mdP}[data-line={805}]%\n%mdk-data-line={805}\n{}The class %mdk-data-line={805}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{ExplorationEngine}}%mdk-data-line={805}\n{} ties everything together. It kicks off\nan execution of the Python code under test using %mdk-data-line={806}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{FunctionInvocation}}%mdk-data-line={806}\n{}.\nAs the Python code executes, building symbolic expressions via %mdk-data-line={807}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{SymbolicType}}%mdk-data-line={807}\n{}\nand its subclasses, callbacks to %mdk-data-line={808}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{PathToConstraint}}%mdk-data-line={808}\n{} create\na path-condition, represented by %mdk-data-line={809}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={809}\n{} and %mdk-data-line={809}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Predicate}}%mdk-data-line={809}\n{}. \nNewly discovered %mdk-data-line={810}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraints}}%mdk-data-line={810}\n{} are added to the end of a deque maintained by\n%mdk-data-line={811}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{ExplorationEngine}}%mdk-data-line={811}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={813}]%\n%mdk-data-line={813}\n{}Given the first seed execution, %mdk-data-line={813}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{ExplorationEngine}}%mdk-data-line={813}\n{} starts the work of \nexploring paths in a breadth-first fashion. It removes a %mdk-data-line={814}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={814}\n{} %mdk-data-line={814}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={814}\n{}\nfrom the front of its deque and, if %mdk-data-line={815}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={815}\n{} has not been already %mdk-data-line={815}\n{}{\\textquotedblleft}processed{\\textquotedblright}%mdk-data-line={815}\n{}, \nuses %mdk-data-line={816}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3Wrapper}}%mdk-data-line={816}\n{} to find a new input (as discussed in the previous section)\nwhere %mdk-data-line={817}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={817}\n{} is the query (to be negated) and the path to %mdk-data-line={817}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={817}\n{} in the \n%mdk-data-line={818}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={818}\n{} tree forms the assertions.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={820}]%\n%mdk-data-line={820}\n{}A %mdk-data-line={820}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Constraint}}%mdk-data-line={820}\n{} %mdk-data-line={820}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={820}\n{} in the tree is considered %mdk-data-line={820}\n{}{\\textquotedblleft}processed{\\textquotedblright}%mdk-data-line={820}\n{} if an execution \nhas covered %mdk-data-line={821}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c'$}%mdk-data-line={821}\n{}, a sibling of %mdk-data-line={821}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={821}\n{} in the tree that represents\nthe negation of the predicate associated with %mdk-data-line={822}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={822}\n{}, or if constraint %mdk-data-line={822}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$c$}%mdk-data-line={822}\n{}\nhas been removed from the deque.%\n\\end{mdP}%\n\\mdHxx[id=sec-int2z3,label={[5]\\{.heading-label\\}},toc={},data-line={825},caption={[[5]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}From Python Integers to Z3 Arithmetic},bookmark={5.{\\hspace{0.5em}}From Python Integers to Z3 Arithmetic}]{%mdk-data-line={825}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{5}.{\\hspace{0.5em}}}%mdk-data-line={825}\n{}From Python Integers to Z3 Arithmetic}\\begin{mdP}[data-line={827}]%\n%mdk-data-line={827}\n{}In languages such as C and Java, integers are finite-precision,\ngenerally limited to the size of a machine word (32 or 64 bits, for example).\nFor such languages, satisfiability of finite-precision integer arithmetic\nis decidable and can be reduced to Z3%mdk-data-line={830}\n{}{'}%mdk-data-line={830}\n{}s theory of bit-vectors, where\neach arithmetic operation is encoded by a circuit. This translation permits \nreasoning about non-linear arithmetic problems, such as \n%mdk-data-line={833}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$\\exists x,y,z : x*z + y \\leq (z/y)+5$}%mdk-data-line={833}\n{}.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={835}]%\n%mdk-data-line={835}\n{}Python (3.0) integers, however, are not finite-precision. They are only\nlimited by the size of machine memory. This means, for example, that\nPython integers don%mdk-data-line={837}\n{}{'}%mdk-data-line={837}\n{}t overflow or underflow. It also means that\nwe can%mdk-data-line={838}\n{}{'}%mdk-data-line={838}\n{}t hope to decide algorithmically whether or not a given\nequation over integer variables has a solution in general. Hilbert%mdk-data-line={839}\n{}{'}%mdk-data-line={839}\n{}s famous\n10th problem and its solution by Matiyasevich tells us that it is\nundecidable whether or not a polynomial\nequation of the form %mdk-data-line={842}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$p(x_1, \\ldots, x_n) = 0$}%mdk-data-line={842}\n{} with integer coefficients\nhas an solution in the integers.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={845}]%\n%mdk-data-line={845}\n{}This means that we will resort to heuristic approaches in our use\nof the%mdk-data-line={846}\n{}{\\mdNbsp}\\mdA[data-linkid={z3}]{http://z3.codeplex.org/}{}{Z3}%mdk-data-line={846}\n{} ATP.  The special case of linear integer arithmetic (LIA)\nis decidable and supported by Z3. In order to deal with non-linear operations,\nwe use uninterpreted functions (UF). Thus, if Z3 returns\n%mdk-data-line={849}\n{}{\\textquotedblleft}unsatisfiable{\\textquotedblright}%mdk-data-line={849}\n{} we know that there is no solution, but if the\nZ3 %mdk-data-line={850}\n{}{\\textquotedblleft}satisfiable{\\textquotedblright}%mdk-data-line={850}\n{}, we must treat the answer as a %mdk-data-line={850}\n{}{\\textquotedblleft}don{'}t know{\\textquotedblright}%mdk-data-line={850}\n{}. \nThe class %mdk-data-line={851}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3Integer}}%mdk-data-line={851}\n{} is used to translate a symbolic expression\ninto the theory LIA+UF and check for unsatisfiability. We leave it as an\nimplementation exercise to check if a symbolic expression can\nbe converted to LIA (without the use of UF) in order to make\nuse of %mdk-data-line={855}\n{}{\\textquotedblleft}satisfiable{\\textquotedblright}%mdk-data-line={855}\n{} answers from the LIA solver.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={857}]%\n%mdk-data-line={857}\n{}If the translation to %mdk-data-line={857}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3Integer}}%mdk-data-line={857}\n{} does not return %mdk-data-line={857}\n{}{\\textquotedblleft}unsatisfiable{\\textquotedblright}%mdk-data-line={857}\n{}, we\nuse Z3%mdk-data-line={858}\n{}{'}%mdk-data-line={858}\n{}s bit-vector decision procedure (via the class\n%mdk-data-line={859}\n{}\\mdCode[class={code,code1,language-python,lang-python,python,highlighted},language={python}]{\\mdToken{Namespace,Identifier,Python}{Z3BitVector}}%mdk-data-line={859}\n{}) to heuristically\ntry to find satisfiable answers, even in the presence of non-linear \narithmetic. We start with bit-vectors of size %mdk-data-line={861}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$N=32$}%mdk-data-line={861}\n{} and %mdk-data-line={861}\n{}\\mdEm{bound}%mdk-data-line={861}\n{} the values\nof the symbolic constants to fit within 8 bits in order to find \nsatisfiable solutions\nwith small values. Also, because Python integers do not overflow/underflow, \nthe bound helps us reserve space in the bit-vector to allow the\nresults of operations to exceed the bound while not overflowing\nthe bit-vector. As long as Z3 returns %mdk-data-line={867}\n{}{\\textquotedblleft}unsatisfiable{\\textquotedblright}%mdk-data-line={867}\n{} we increase\nthe bound. If the bound reaches %mdk-data-line={868}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$N$}%mdk-data-line={868}\n{}, we increase %mdk-data-line={868}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$N$}%mdk-data-line={868}\n{} by 8 bits,\nleaving the bound where it is and continue.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={871}]%\n%mdk-data-line={871}\n{}If Z3 returns\n%mdk-data-line={872}\n{}{\\textquotedblleft}satisfiable{\\textquotedblright}%mdk-data-line={872}\n{}, it may be the case that Z3 found a solution\nthat involved overflow in the bit-vector world of arithmetic\n(modulo %mdk-data-line={874}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$2^N-1$}%mdk-data-line={874}\n{}). Therefore,\nthe solution is validated back in the\nPython world by evaluating the formula under that solution\nusing Python semantics. \nIf the formula does not evaluate to the same\nvalue in both worlds, then we increase %mdk-data-line={879}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$N$}%mdk-data-line={879}\n{} by 8 bits (to \ncreate a gap between the bound and %mdk-data-line={880}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$N$}%mdk-data-line={880}\n{}) and continue to search\nfor a solution.%\n\\end{mdP}%\n\\begin{mdP}[class={indent},data-line={883}]%\n%mdk-data-line={883}\n{}The process terminates when we find a valid satisfying solution \nor %mdk-data-line={884}\n{}\\mdSpan[class={math-inline},elem={math-inline}]{$N=64$}%mdk-data-line={884}\n{} and the bound reaches 64 (in which case, we return %mdk-data-line={884}\n{}{\\textquotedblleft}don{'}t know{\\textquotedblright}%mdk-data-line={884}\n{}).%\n\\end{mdP}%\n\\mdHxx[id=sec-extensions,label={[6]\\{.heading-label\\}},toc={},data-line={886},caption={[[6]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Extensions},bookmark={6.{\\hspace{0.5em}}Extensions}]{%mdk-data-line={886}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{6}.{\\hspace{0.5em}}}%mdk-data-line={886}\n{}Extensions}\\begin{mdP}[data-line={888}]%\n%mdk-data-line={888}\n{}We have presented the basics of dynamic symbolic execution \n(for Python).\nA more thorough treatment would deal with other data types besides\nintegers, such as Python dictionaries, strings and lists, each\nof which presents their own challenges for symbolic reasoning. \nThere are many other interesting challenges in DSE, such\nas dealing with user-defined classes (rather than built-in types\nas done here) and multi-threaded execution.%\n\\end{mdP}%\n\\mdHxx[id=sec-acknowledgements,label={[7]\\{.heading-label\\}},toc={},data-line={897},caption={[[7]\\{.heading-label\\}.{\\hspace{0.5em}}]\\{.heading-before\\}Acknowledgements},bookmark={7.{\\hspace{0.5em}}Acknowledgements}]{%mdk-data-line={897}\n{}\\mdSpan[class={heading-before}]{\\mdSpan[class={heading-label}]{7}.{\\hspace{0.5em}}}%mdk-data-line={897}\n{}Acknowledgements}\\begin{mdP}[data-line={899}]%\n%mdk-data-line={899}\n{}Many thanks to the students of the 2014 Marktoberdorf Summer School\non Dependable Software Systems Engineering\nfor their questions and feedback about the first author%mdk-data-line={901}\n{}{'}%mdk-data-line={901}\n{}s lectures on dynamic\nsymbolic execution. The following students of the summer school\nhelpfully provided tests for the%mdk-data-line={903}\n{}{\\mdNbsp}\\mdA[data-linkid={pyexz3}]{https://github.com/thomasjball/PyExZ3/}{}{PyExZ3}%mdk-data-line={903}\n{} tool: Daniel Darvas,\nDamien Rusinek, Christian Dehnert and Thomas Pani. Thanks also to Peter\nChapman for his contributions.%\n\\end{mdP}%\n\n\\mdHxx[id=sec-references,label={8},toc={},data-line={963},caption={References},bookmark={References}]{%mdk-data-line={963}\n{}References}\\begin{mdBibliography}[class={bibliography,bib-numeric},elem={bibliography},bibstyle={plainnat},bibdata={dse},caption={14},data-line={964;out{\\textbackslash}DSE-bib.bbl:2}]%\n\\begin{mdBibitem}[class={bibitem},id=cadare05,label={[1]\\{.bibitem-label\\}},elem={bibitem},cite-label={Cadar and Engler(2005)},caption={Cristian Cadar and Dawson{\\textbackslash} R. 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Engler.\n%mdk-data-line={964;out\\DSE-bib.bbl:7}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:7}\n{} Execution generated test cases: How to make systems code crash\n  itself.\n%mdk-data-line={964;out\\DSE-bib.bbl:9}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:9}\n{} In %mdk-data-line={964;out\\DSE-bib.bbl:9}\n{}\\mdEm{Proceedings of 12th International SPIN Workshop}%mdk-data-line={964;out\\DSE-bib.bbl:9}\n{}%mdk-data-line={964;out\\DSE-bib.bbl:9}\n{}, pages\n  2%mdk-data-line={964;out\\DSE-bib.bbl:10}\n{}{\\textendash}%mdk-data-line={964;out\\DSE-bib.bbl:10}\n{}23, 2005.%\n\\end{mdBibitem}%\n\\begin{mdBibitem}[class={bibitem},id=cadars13,label={[2]\\{.bibitem-label\\}},elem={bibitem},cite-label={Cadar and Sen(2013)},caption={Cristian Cadar and Koushik Sen. \\\\Symbolic execution for software testing: three decades later.},searchterm={Symbolic+execution+software+testing+three+decades+later++Cristian+Cadar+Koushik+},data-line={964;out{\\textbackslash}DSE-bib.bbl:13}]%\n%mdk-data-line={964;out\\DSE-bib.bbl:14}\n{}\\mdSpan[class={bibitem-before}]{[\\mdSpan[class={bibitem-label}]{2}]{\\mdNbsp}{\\mdNbsp}}%mdk-data-line={964;out\\DSE-bib.bbl:14}\n{}Cristian Cadar and Koushik Sen.\n%mdk-data-line={964;out\\DSE-bib.bbl:15}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:15}\n{} Symbolic execution for software testing: three decades later.\n%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{} %mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}\\mdEm{Communications of the ACM}%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}, 56%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}\\mdSpan[penalty={0}]{}%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{} (2):%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}\\mdSpan[penalty={0}]{}%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{} 82%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}{\\textendash}%mdk-data-line={964;out\\DSE-bib.bbl:16}\n{}90,\n  2013.%\n\\end{mdBibitem}%\n\\begin{mdBibitem}[class={bibitem},id=cadargpde06,label={[3]\\{.bibitem-label\\}},elem={bibitem},cite-label={Cadar et{\\textbackslash} al.(2006)Cadar, Ganesh, Pawlowski, Dill, and\\\\  Engler},caption={Cristian Cadar, Vijay Ganesh, Peter{\\textbackslash} M. 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Dill, and Dawson%mdk-data-line={964;out\\DSE-bib.bbl:21}\n{}{\\mdNbsp}%mdk-data-line={964;out\\DSE-bib.bbl:21}\n{}R.\n  Engler.\n%mdk-data-line={964;out\\DSE-bib.bbl:23}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:23}\n{} EXE: automatically generating inputs of death.\n%mdk-data-line={964;out\\DSE-bib.bbl:24}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:24}\n{} In %mdk-data-line={964;out\\DSE-bib.bbl:24}\n{}\\mdEm{Proceedings of the 13th ACM Conference on Computer and\n  Communications Security}%mdk-data-line={964;out\\DSE-bib.bbl:25}\n{}%mdk-data-line={964;out\\DSE-bib.bbl:25}\n{}, pages 322%mdk-data-line={964;out\\DSE-bib.bbl:25}\n{}{\\textendash}%mdk-data-line={964;out\\DSE-bib.bbl:25}\n{}335, 2006.%\n\\end{mdBibitem}%\n\\begin{mdBibitem}[class={bibitem},id=clarke76,label={[4]\\{.bibitem-label\\}},elem={bibitem},cite-label={Clarke(1976)},caption={Lori{\\textbackslash} A. Clarke. \\\\A system to generate test data and symbolically execute programs.},searchterm={+system+generate+test+data+symbolically+execute+programs++Lori+Clarke+},data-line={964;out{\\textbackslash}DSE-bib.bbl:28}]%\n%mdk-data-line={964;out\\DSE-bib.bbl:29}\n{}\\mdSpan[class={bibitem-before}]{[\\mdSpan[class={bibitem-label}]{4}]{\\mdNbsp}{\\mdNbsp}}%mdk-data-line={964;out\\DSE-bib.bbl:29}\n{}Lori%mdk-data-line={964;out\\DSE-bib.bbl:29}\n{}{\\mdNbsp}%mdk-data-line={964;out\\DSE-bib.bbl:29}\n{}A. Clarke.\n%mdk-data-line={964;out\\DSE-bib.bbl:30}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:30}\n{} A system to generate test data and symbolically execute programs.\n%mdk-data-line={964;out\\DSE-bib.bbl:31}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:31}\n{} %mdk-data-line={964;out\\DSE-bib.bbl:31}\n{}\\mdEm{IEEE Transactions on Software Engineering}%mdk-data-line={964;out\\DSE-bib.bbl:31}\n{}%mdk-data-line={964;out\\DSE-bib.bbl:31}\n{}, 2%mdk-data-line={964;out\\DSE-bib.bbl:31}\n{}\\mdSpan[penalty={0}]{}%mdk-data-line={964;out\\DSE-bib.bbl:31}\n{}\n  (3):%mdk-data-line={964;out\\DSE-bib.bbl:32}\n{}\\mdSpan[penalty={0}]{}%mdk-data-line={964;out\\DSE-bib.bbl:32}\n{} 215%mdk-data-line={964;out\\DSE-bib.bbl:32}\n{}{\\textendash}%mdk-data-line={964;out\\DSE-bib.bbl:32}\n{}222, 1976.%\n\\end{mdBibitem}%\n\\begin{mdBibitem}[class={bibitem},id=demourab08,label={[5]\\{.bibitem-label\\}},elem={bibitem},cite-label={de{\\textbackslash} Moura and Bj{\\o}rner(2008)},caption={Leonardo{\\textbackslash} Mendon{\\c{c}}a de{\\textbackslash} Moura and Nikolaj Bj{\\o}rner. \\\\Z3: an efficient SMT solver.},searchterm={+efficient+solver++Leonardo+Mendon+Moura+Nikolaj+rner+},data-line={964;out{\\textbackslash}DSE-bib.bbl:35}]%\n%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}\\mdSpan[class={bibitem-before}]{[\\mdSpan[class={bibitem-label}]{5}]{\\mdNbsp}{\\mdNbsp}}%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}Leonardo%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}{\\mdNbsp}%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}Mendon%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}{\\c{c}}%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}a de%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}{\\mdNbsp}%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}Moura and Nikolaj Bj%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}{\\o}%mdk-data-line={964;out\\DSE-bib.bbl:36}\n{}rner.\n%mdk-data-line={964;out\\DSE-bib.bbl:37}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:37}\n{} Z3: an efficient SMT solver.\n%mdk-data-line={964;out\\DSE-bib.bbl:38}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:38}\n{} In %mdk-data-line={964;out\\DSE-bib.bbl:38}\n{}\\mdEm{Proceedings of the 14th International Conference of Tools\n  and Algorithms for the Construction and Analysis of Systems}%mdk-data-line={964;out\\DSE-bib.bbl:39}\n{}%mdk-data-line={964;out\\DSE-bib.bbl:39}\n{}, pages 337%mdk-data-line={964;out\\DSE-bib.bbl:39}\n{}{\\textendash}%mdk-data-line={964;out\\DSE-bib.bbl:39}\n{}340,\n  2008.%\n\\end{mdBibitem}%\n\\begin{mdBibitem}[class={bibitem},id=dijkstra76,label={[6]\\{.bibitem-label\\}},elem={bibitem},cite-label={Dijkstra(1976)},caption={Edsger{\\textbackslash} W. 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Mathur, and Mary%mdk-data-line={964;out\\DSE-bib.bbl:71}\n{}{\\mdNbsp}%mdk-data-line={964;out\\DSE-bib.bbl:71}\n{}Lou Soffa.\n%mdk-data-line={964;out\\DSE-bib.bbl:72}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:72}\n{} Generating test data for branch coverage.\n%mdk-data-line={964;out\\DSE-bib.bbl:73}\n{}\\mdSpan[class={newblock}]{}%mdk-data-line={964;out\\DSE-bib.bbl:73}\n{} In %mdk-data-line={964;out\\DSE-bib.bbl:73}\n{}\\mdEm{Proceedings of the Automate Software Engineering\n  Conference}%mdk-data-line={964;out\\DSE-bib.bbl:74}\n{}%mdk-data-line={964;out\\DSE-bib.bbl:74}\n{}, pages 219%mdk-data-line={964;out\\DSE-bib.bbl:74}\n{}{\\textendash}%mdk-data-line={964;out\\DSE-bib.bbl:74}\n{}228, 2000.%\n\\end{mdBibitem}%\n\\begin{mdBibitem}[class={bibitem},id=king76,label={[11]\\{.bibitem-label\\}},elem={bibitem},cite-label={King(1976)},caption={James{\\textbackslash} C. 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\\noindent\\usebox\\@snippetBox%\n}\n\n\\newenvironment{mdDisplaySnippet}[1][\\arabic{snippets}]%\n  {\\stepcounter{snippets}\\edef\\@snippetName{#1}%\n   \\begin{lrbox}{\\@snippetBox}\\vbox\\bgroup}%\n  {\\egroup\\end{lrbox}\\@savedepth\\newpage}\n\n\\newenvironment{mdInlineSnippet}[1][\\arabic{snippets}]%\n  {\\stepcounter{snippets}\\def\\@snippetName{#1}%\n   \\begin{lrbox}{\\@snippetBox}}%\n  {\\end{lrbox}\\@savedepth\\newpage}\n\n\\newenvironment{mdSnippets}%\n    {\\pagestyle{empty}%\n     \\abovedisplayskip=0pt%\n     \\belowdisplayskip=0pt%\n     \\abovedisplayshortskip=0pt%\n     \\belowdisplayshortskip=0pt%\n     \\newwrite\\foo\\immediate\\openout\\foo=\\jobname.dim%\n     \\immediate\\write\\foo{\\%ordinal,(hash)name,width,(total) height,depth,image width,image height,dpi,base64 encoding}%\n    }%\n    {\\closeout\\foo}\n"
  },
  {
    "path": "marktoberdorf_slides/collatz.py",
    "content": "def collatz(n):\n    if n <= 1:\n        return 1\n    else:\n        if n % 2 == 0:\n            return collatz(n // 2)\n        else:\n            return collatz(3 * n + 1)\n\ndef max_iters():\n    return 10\n\ndef expected_result_set():\n    return [1]\n"
  },
  {
    "path": "marktoberdorf_slides/examples/adder.py",
    "content": "from z3 import *\n\ndef add1(x0,y0,c0):\n\treturn (simplify(Xor(Xor(x0,y0),c0)), \n                simplify(Or(And(x0,y0),And(x0,c0),And(y0,c0))))\n\ndef addN(xN,yN,cin):\n\tres = []\n\tcout = cin\n\tfor (x,y) in zip(xN,yN):\n\t\t(new_res,cout) = add1(x,y,cout)\n\t\tres.append(new_res)\n\treturn (res,cout)\n\n"
  },
  {
    "path": "marktoberdorf_slides/examples/automata.py",
    "content": "from z3 import *\n\nChar = BitVecSort(8)\nList = Datatype('List')\nList.declare('cons', ('head', Char), ('tail', List))\nList.declare('nil')\nList = List.create()\n\n# abbreviations\ncons = List.cons\nhead = List.head\ntail = List.tail\nnil = List.nil\nBool = BoolSort()\n\n# don't use model-based quantifier instantiation here\nset_param('smt.mbqi', False)\nsolver = SimpleSolver()\n\n# we will set up axioms to solve for the regular expression \n# [0-9]+|[a-z]\n\nq0 = Function('q0',List,Bool)\nq1 = Function('q1',List,Bool)\nq2 = Function('q2',List,Bool)\nq3 = Function('q3',List,Bool)\nq4 = Function('q4',List,Bool)\n\ny = Const('y',List)\n\n# qo -[e]-> q1, qo -[e]-> q3 \ndef q0axiom():\n\tbody = q0(y) == Or(q1(y),q3(y))\n\treturn ForAll(y, body, patterns = [q0(y)])\n\n# q1 -[0..9]-> q2\ndef q1axiom():\n\tbody = q1(y) == And(y!=nil,ULE(ord('0'),head(y)),ULE(head(y),ord('9')),\\\n                            q2(tail(y)))\n\treturn ForAll(y, body, patterns = [q1(y)])\n\n# q2 -[0..9] -> q2  (q2 is final) \ndef q2axiom():\n\tbody = q2(y) == Or(y==nil,And(y!=nil,ULE(ord('0'),head(y)),ULE(head(y),ord('9')),q2(tail(y))))\n\treturn ForAll(y, body, patterns = [q2(y)])\n\n# q3 -[a-z]-> q4\ndef q3axiom():\n\tbody = q3(y) == And(y!=nil,ULE(ord('a'),head(y)),ULE(head(y),ord('z')),q4(tail(y)))\n\treturn ForAll(y, body, patterns = [q3(y)])\n\n# (q4 is final)\ndef q4axiom():\n\tbody = q4(y) == (y==nil)\n\treturn ForAll(y, body, patterns = [q4(y)])\n\n#to generate longer results\ndef lengthGE(x,k):\n  axiom = BoolVal(True)\n  for i in range(k):\n    axiom = And(axiom,Not(x == nil))\n    x = tail(x)\n  return axiom\n\ndef test():\n  solver.add(q0axiom(),q1axiom(),q2axiom(),q3axiom(),q4axiom())\n  y = Const('y',List)\n  solver.add(lengthGE(y,4))\n  solver.add(q0(y))\n  print(solver.assertions())\n  ok = solver.check()\n  assert(ok != unsat)\n  print(solver.model())\n  yVal = solver.model().evaluate(y,True)\n  print(yVal)\n  \ntest()\n"
  },
  {
    "path": "marktoberdorf_slides/examples/check_adder.py",
    "content": "from adder import *\n\nN = 128\n\n# now prove that adder is equal to bitvector add \n\nxN = BoolVector(\"x\",N)\nyN = BoolVector(\"y\",N)\n\nx_bv = BitVec(\"x_bv\",N)\ny_bv = BitVec(\"y_bv\",N)\n\n# need to equate the inputs (Extract(lo,hi,bv)\n\ndef eq_bool(bv, i, b):\n\treturn (Extract(i,i,bv) == 1) == b\n\ndef eq_bitvec_boolvec(bitvec,boolvec):\n\treturn [ eq_bool(bitvec,i,boolvec[i]) for i in range(bitvec.size())  ]\n\ninputs_equal = eq_bitvec_boolvec(x_bv,xN) + eq_bitvec_boolvec(y_bv,yN)\n\nres,cout = addN(xN,yN,BoolVal(False))\n\nres_bv = x_bv + y_bv\n\noutputs_equal = eq_bitvec_boolvec(res_bv,res)\n\ns = Solver()\ns.add(And(inputs_equal))\ns.add(Not(And(outputs_equal)))\n#print(s.assertions())\nresult = s.check()\nif result == sat:\n\tprint(s.model())\nelse:\n\tprint(result)\n"
  },
  {
    "path": "marktoberdorf_slides/examples/check_mult.py",
    "content": "from mult import *\n\n# now prove that adder is equal to bitvector add \n\nN = 16\n\nxN = BoolVector(\"x\",N)\nyN = BoolVector(\"y\",N)\n\nx_bv = BitVec(\"x_bv\",N)\ny_bv = BitVec(\"y_bv\",N)\n\n# need to equate the inputs (Extract(lo,hi,bv)\n\ndef eq_bool(bv, i, b):\n\treturn (Extract(i,i,bv) == 1) == b\n\ndef eq_bitvec_boolvec(bitvec,boolvec):\n\treturn [ eq_bool(bitvec,i,boolvec[i]) for i in range(bitvec.size())  ]\n\ninputs_equal = eq_bitvec_boolvec(x_bv,xN) + eq_bitvec_boolvec(y_bv,yN)\n\nres,cout = multN(xN,yN)\n\nres_bv = x_bv * y_bv\n\noutputs_equal = eq_bitvec_boolvec(res_bv,res)\n\ns = Solver()\ns.add(And(inputs_equal))\ns.add(Not(And(outputs_equal)))\nresult = s.check()\nprint(result)\n"
  },
  {
    "path": "marktoberdorf_slides/examples/first.py",
    "content": "from z3 import *\n\nx = BitVec(\"x\",8)\ny = BitVec(\"y\",8)\nz = BitVec(\"z\",8)\n\ns = Solver()\ns.add(x > 0,y > 0,z==x+y)\ns.add(Not(z > 0))\n\nresult = s.check()\nif result == sat:\n\tprint(s.model())\nelse:\n\tprint(result)\n\n"
  },
  {
    "path": "marktoberdorf_slides/examples/hats.py",
    "content": "from z3 import *\n\nN = 7\n\n# set up the bit vectors for the hats and what each person will guess\nhat    = [ BitVec(\"Hat\"+str(i),10) for i in range(N) ]\nguess  = [ BitVec(\"Guess\"+str(i),10) for i in range(N) ]\n\n# compute the sum of each person i's guess and the hats numbers for all j != i\nsum    = [ sum([guess[i]]+[ hat[j] for j in range(N) if j!=i]) for i in range(N) ]\n\n# the program for each person i\nprog   = [ (sum[i] % N) == i for i in range(N) ]\n\ns = Solver()\n\ndef bounded(x):\n\treturn And(0<=x,x<N)\ns.add([ bounded(hat[i]) for i in range(N) ])\ns.add([ bounded(guess[i]) for i in range(N) ])\n\ns.add([ prog[i] for i in range(N) ])\n\n# at least one person's guess is their own hat number\ns.add(Not(Or([ guess[i] == hat[i] for i in range(N) ])))\n\nprint(s.check())\n"
  },
  {
    "path": "marktoberdorf_slides/examples/mult.py",
    "content": "from adder import *\n\n#  x2 x1 x0\n#  y2 y1 y0\n  \n#  And(x2,y0) And(x1,y0) And(x0,y0)\n#  And(x1,y1) And(x0,y1) False\n#  And(x0,y2) False      False\n\ndef multN(xN,yN):\n\ti = 0\n\tintermediate = []\n\tN = len(xN)\n\n\t# create N vectors\n\twhile(i<N):\n\t\tfalse_prefix = [ BoolVal(False) for j in range(i) ]\n\t\tsuffix = [ And(xN[i],yN[j-i]) for j in range(i,N) ]\n\t\tvec = false_prefix + suffix\n\t\tintermediate.append(vec)\n\t\ti = i + 1\n\n\t# now add them up\n\tres = intermediate[0]\n\tfor j in intermediate[1:]:\n\t\t(res,cout) = addN(res,j,BoolVal(False))\n\n\treturn (res,cout)"
  },
  {
    "path": "marktoberdorf_slides/examples/mult2.py",
    "content": "from z3 import *\n\nN = 13\nzerosN = [BoolVal(False)] * N\n\ndef multcell(a,b,c,s):\n\t# I have to admit it is based on http://www.xilinx.com/univ/teaching_materials/dsp_primer/sample/lecture_notes/FPGAArithmetic_mult.pdf , slide 3.\n\ty = And(a,b)\n\t(sout,cout) = add1(s,y,c)\n\treturn (cout,sout)\n\t\ndef multN(xN,yN):\n\t# to sum up the columns\n\tsums = zerosN\n\t\n\tfor row in range(N):\n\t\tb = yN[row] # yN is not shifted\n\t\t# initialise carry to 0\n\t\tc = BoolVal(False)\n\t\t\n\t\tfor col in range(row,N): # note that col<row cells are skipped!\n\t\t\t# collect the inputs\n\t\t\t\n\t\t\t# xN is shifted in each row\n\t\t\ta = xN[col-row]\n\t\t\t# sum up the columns\n\t\t\ts = sums[col]\n\t\t\t\n\t\t\t# make a multiplication cell\n\t\t\t(co,so) = multcell(a,b,c,s)\n\t\t\t\n\t\t\t# outputs\n\t\t\tc = co\n\t\t\tsums[col] = so\n\treturn sums\n\t\ndef add1(x0,y0,c0):\n\treturn (simplify(Xor(Xor(x0,y0),c0)) , simplify(Or(And(x0,y0),And(x0,c0),And(y0,c0))))"
  },
  {
    "path": "pyexz3.py",
    "content": "# Copyright: see copyright.txt\n\nimport os\nimport sys\nimport logging\nimport traceback\nfrom optparse import OptionParser\n\nfrom symbolic.loader import *\nfrom symbolic.explore import ExplorationEngine\n\nprint(\"PyExZ3 (Python Exploration with Z3)\")\n\nsys.path = [os.path.abspath(os.path.join(os.path.dirname(__file__)))] + sys.path\n\nusage = \"usage: %prog [options] <path to a *.py file>\"\nparser = OptionParser(usage=usage)\n\nparser.add_option(\"-l\", \"--log\", dest=\"logfile\", action=\"store\", help=\"Save log output to a file\", default=\"\")\nparser.add_option(\"-s\", \"--start\", dest=\"entry\", action=\"store\", help=\"Specify entry point\", default=\"\")\nparser.add_option(\"-g\", \"--graph\", dest=\"dot_graph\", action=\"store_true\", help=\"Generate a DOT graph of execution tree\")\nparser.add_option(\"-m\", \"--max-iters\", dest=\"max_iters\", type=\"int\", help=\"Run specified number of iterations\", default=0)\nparser.add_option(\"--cvc\", dest=\"cvc\", action=\"store_true\", help=\"Use the CVC SMT solver instead of Z3\", default=False)\nparser.add_option(\"--z3\", dest=\"cvc\", action=\"store_false\", help=\"Use the Z3 SMT solver\")\n\n(options, args) = parser.parse_args()\n\nif not (options.logfile == \"\"):\n\tlogging.basicConfig(filename=options.logfile,level=logging.DEBUG)\n\nif len(args) == 0 or not os.path.exists(args[0]):\n\tparser.error(\"Missing app to execute\")\n\tsys.exit(1)\n\nsolver = \"cvc\" if options.cvc else \"z3\"\n\nfilename = os.path.abspath(args[0])\n\t\n# Get the object describing the application\napp = loaderFactory(filename,options.entry)\nif app == None:\n\tsys.exit(1)\n\nprint (\"Exploring \" + app.getFile() + \".\" + app.getEntry())\n\nresult = None\ntry:\n\tengine = ExplorationEngine(app.createInvocation(), solver=solver)\n\tgeneratedInputs, returnVals, path = engine.explore(options.max_iters)\n\t# check the result\n\tresult = app.executionComplete(returnVals)\n\n\t# output DOT graph\n\tif (options.dot_graph):\n\t\tfile = open(filename+\".dot\",\"w\")\n\t\tfile.write(path.toDot())\t\n\t\tfile.close()\n\nexcept ImportError as e:\n\t# createInvocation can raise this\n\tlogging.error(e)\n\tsys.exit(1)\n\nif result == None or result == True:\n\tsys.exit(0);\nelse:\n\tsys.exit(1);\t\n"
  },
  {
    "path": "run_tests.py",
    "content": "import os\nimport re\nimport sys\nimport subprocess\nfrom optparse import OptionParser\nfrom sys import platform as _platform\n\nclass bcolors:\n    SUCCESS = '\\033[32m'\n    WARNING = '\\033[33m'\n    FAIL = '\\033[31m'\n    ENDC = '\\033[0m'\n\ndef myprint(color, s, *args):\n  if _platform != \"win32\" and sys.stdout.isatty():\n    print(color, s, bcolors.ENDC, *args)\n  else:\n    print(*args)\n\nusage = \"usage: %prog [options] <test directory>\"\nparser = OptionParser()\nparser.add_option(\"--cvc\", dest=\"cvc\", action=\"store_true\", help=\"Use the CVC SMT solver instead of Z3\", default=False)\nparser.add_option(\"--z3\", dest=\"cvc\", action=\"store_false\", help=\"Use the Z3 SMT solver\")\n(options, args) = parser.parse_args()\n\nif len(args) == 0 or not os.path.exists(args[0]):\n    parser.error(\"Please supply directory of tests\")\n    sys.exit(1)\n    \ntest_dir = os.path.abspath(args[0])\n\nif not os.path.isdir(test_dir):\n    print(\"Please provide a directory of test scripts.\")\n    sys.exit(1)\n\nfiles = [ f for f in os.listdir(test_dir) if re.search(\".py$\",f) ]\n\nfailed = []\nfor f in files:\n\t# execute the python runner for this test\n        full = os.path.join(test_dir, f)\n        with open(os.devnull, 'w') as devnull:\n            solver = \"--cvc\" if options.cvc else \"--z3\"\n            ret = subprocess.call([sys.executable, \"pyexz3.py\", \"--m=25\", solver, full], stdout=devnull)\n        if (ret == 0):\n            myprint(bcolors.SUCCESS, \"✓\", \"Test \" + f + \" passed.\")\n        else:\n            failed.append(f)\n            myprint(bcolors.FAIL, \"✗\", \"Test \" + f + \" failed.\")\n\nif failed != []:\n\tprint(\"RUN FAILED\")\n\tprint(failed)\n\tsys.exit(1)\nelse:\n\tsys.exit(0)"
  },
  {
    "path": "setup.bat",
    "content": "set Z3HOME=c:\\z3\\bin\nset PYTHONHOME=c:\\Python34\nset GRAPHVIZHOME=\"c:\\Program Files (x86)\\Graphviz2.38\\bin\"\nset PATH=%PATH%;%PYTHONHOME%;%Z3HOME%;%GRAPHVIZHOME%\nset PYTHONPATH=%PYTHONPATH%;%Z3HOME%\n\n"
  },
  {
    "path": "setup.sh",
    "content": "#!/bin/bash\n\n# Homebrew default locations (both for python and z3)\nexport PYTHONMODS=\"usr/local/lib/python2.7/site-packages:/Library/Python/2.7/site-packages:~/Library/Python/2.7/lib/python/site-packages\" \nexport Z3HOME=\"/usr/local/Cellar/z3/4.3.1/lib/python2.7/site-packages/\"\nexport Z3BIN=\"/usr/local/Cellar/z3/4.3.1/bin\"\n\n\n# Python setup\nexport PYTHONPATH=$PYTHONMODS:$Z3HOME\n\n# Path setup\nexport PATH=$PATH:$Z3BIN\n\n\n"
  },
  {
    "path": "symbolic/__init__.py",
    "content": ""
  },
  {
    "path": "symbolic/args.py",
    "content": "def symbolic(**arg_types):\n\tdef decorator(f):\n\t\tf.symbolic_args = arg_types\n\t\treturn f\n\treturn decorator\n\ndef concrete(**arg_types):\n\tdef decorator(f):\n\t\tf.concrete_args = arg_types\n\t\treturn f\n\treturn decorator\n"
  },
  {
    "path": "symbolic/constraint.py",
    "content": "# Copyright: see copyright.txt\n\nimport logging\n\nlog = logging.getLogger(\"se.constraint\")\n\nclass Constraint:\n\tcnt = 0\n\t\"\"\"A constraint is a list of predicates leading to some specific\n\t   position in the code.\"\"\"\n\tdef __init__(self, parent, last_predicate):\n\t\tself.inputs = None\n\t\tself.predicate = last_predicate\n\t\tself.processed = False\n\t\tself.parent = parent\n\t\tself.children = []\n\t\tself.id = self.__class__.cnt\n\t\tself.__class__.cnt += 1\n\n\tdef __eq__(self, other):\n\t\t\"\"\"Two Constraints are equal iff they have the same chain of predicates\"\"\"\n\t\tif isinstance(other, Constraint):\n\t\t\tif not self.predicate == other.predicate:\n\t\t\t\treturn False\n\t\t\treturn self.parent is other.parent\n\t\telse:\n\t\t\treturn False\n\n\tdef getAssertsAndQuery(self):\n\t\tself.processed = True\n\n\t\t# collect the assertions\n\t\tasserts = []\n\t\ttmp = self.parent\n\t\twhile tmp.predicate is not None:\n\t\t\tasserts.append(tmp.predicate)\n\t\t\ttmp = tmp.parent\n\n\t\treturn asserts, self.predicate\t       \n\n\tdef getLength(self):\n\t\tif self.parent == None:\n\t\t\treturn 0\n\t\treturn 1 + self.parent.getLength()\n\n\tdef __str__(self):\n\t\treturn str(self.predicate) + \"  (processed: %s, path_len: %d)\" % (self.processed,self.getLength())\n\n\tdef __repr__(self):\n\t\ts = repr(self.predicate) + \" (processed: %s)\" % (self.processed)\n\t\tif self.parent is not None:\n\t\t\ts += \"\\n  path: %s\" % repr(self.parent)\n\t\treturn s\n\n\tdef findChild(self, predicate):\n\t\tfor c in self.children:\n\t\t\tif predicate == c.predicate:\n\t\t\t\treturn c\n\t\treturn None\n\n\tdef addChild(self, predicate):\n\t\tassert(self.findChild(predicate) is None)\n\t\tc = Constraint(self, predicate)\n\t\tself.children.append(c)\n\t\treturn c\n\n"
  },
  {
    "path": "symbolic/cvc_expr/__init__.py",
    "content": ""
  },
  {
    "path": "symbolic/cvc_expr/exprbuilder.py",
    "content": "import logging\n\nfrom symbolic.cvc_expr.integer import CVCInteger\nfrom symbolic.cvc_expr.string import CVCString\nfrom symbolic.symbolic_types import SymbolicInteger, SymbolicStr\nfrom symbolic.symbolic_types.symbolic_type import SymbolicObject\nimport utils\n\nlog = logging.getLogger(\"se.cvc_expr.exprbuilder\")\n\n\nclass ExprBuilder(object):\n    def __init__(self, asserts, query, solver):\n        self.solver = solver\n        self.solver.guards = []\n        self.em = self.solver.getExprManager()\n        self.cvc_vars = {}\n        self.query = self._toCVC(asserts, query)\n\n    def _toCVC(self, asserts, query):\n        smt_query = self._predToCVC(query).not_op()\n        for p in asserts:\n            smt_query &= self._predToCVC(p)\n        for guard in self.solver.guards:\n            smt_query &= guard\n        return smt_query\n\n    def _predToCVC(self, pred, env=None):\n        sym_expr = self._astToCVCExpr(pred.symtype, env)\n        if env is None:\n            if not sym_expr.cvc_expr.getType().isBoolean():\n                sym_expr = (sym_expr == CVCInteger.constant(0, self.solver)).not_op()\n            if not pred.result:\n                sym_expr = sym_expr.not_op()\n        else:\n            if not pred.result:\n                sym_expr = sym_expr.not_op()\n        return sym_expr\n\n    def _getVariable(self, symbolic_var):\n        name = symbolic_var.name\n        if name in self.cvc_vars:\n            return self.cvc_vars[name]\n        variable = None\n        if isinstance(symbolic_var, SymbolicInteger):\n            variable = CVCInteger.variable(name, self.solver)\n        elif isinstance(symbolic_var, SymbolicStr):\n            variable = CVCString.variable(name, self.solver)\n        self.cvc_vars[name] = variable\n        return variable\n\n    def _wrapIf(self, expr, env):\n        if env is None:\n            return expr.ite(CVCInteger.constant(1, self.solver), CVCInteger.constant(0, self.solver))\n        else:\n            return expr\n\n    def _astToCVCExpr(self, expr, env=None):\n        if isinstance(expr, list):\n            op = expr[0]\n            args = [self._astToCVCExpr(a, env) for a in expr[1:]]\n            cvc_l = args[0]\n            cvc_r = args[1] if len(args) > 1 else None\n            cvc_3 = args[2] if len(args) > 2 else None\n\n            # arithmetical operations\n            if op == \"+\":\n                return cvc_l + cvc_r\n            elif op == \"-\":\n                return cvc_l - cvc_r\n            elif op == \"*\":\n                return cvc_l * cvc_r\n            elif op == \"//\":\n                return cvc_l / cvc_r\n            elif op == \"%\":\n                return cvc_l % cvc_r\n\n            # bitwise\n            elif op == \"<<\":\n                return cvc_l << cvc_r\n            elif op == \">>\":\n                return cvc_l >> cvc_r\n            elif op == \"^\":\n                return cvc_l ^ cvc_r\n            elif op == \"|\":\n                return cvc_l | cvc_l\n            elif op == \"&\":\n                return cvc_l & cvc_r\n\n            # string\n            elif op == \"str.len\":\n                return cvc_l.len()\n            elif op == \"str.find\":\n                return cvc_l.find(cvc_r, cvc_3)\n            elif op == \"str.replace\":\n                return cvc_l.replace(cvc_r, cvc_3)\n            elif op == \"str.startswith\":\n                return self._wrapIf(cvc_l.startswith(cvc_r), env)\n\n            # collection operators\n            elif op == \"getitem\":\n                return cvc_l[cvc_r]\n            elif op == \"slice\":\n                return cvc_l[cvc_r:cvc_3]\n            # equality gets coerced to integer\n            elif op == \"==\":\n                if cvc_l is None or cvc_r is None:\n                    # forces false condition no model contains None\n                    return self._wrapIf(self._astToCVCExpr(0, env) !=\n                                        self._astToCVCExpr(0, env), env)\n                else:\n                    return self._wrapIf((cvc_l == cvc_r), env)\n            elif op == \"!=\":\n                if cvc_l is None or cvc_r is None:\n                    return self._wrapIf(self._astToCVCExpr(0, env) == self._astToCVCExpr(0, env), env)\n                else:\n                    return self._wrapIf((cvc_l != cvc_r), env)\n            elif op == \"<\":\n                return self._wrapIf((cvc_l < cvc_r), env)\n            elif op == \">\":\n                return self._wrapIf((cvc_l > cvc_r), env)\n            elif op == \"<=\":\n                return self._wrapIf((cvc_l <= cvc_r), env)\n            elif op == \">=\":\n                return self._wrapIf((cvc_l >= cvc_r), env)\n            elif op == \"in\":\n                return self._wrapIf((cvc_l.__contains__(cvc_r)), env)\n            else:\n                utils.crash(\"Unknown BinOp during conversion from ast to CVC (expressions): %s\" % op)\n\n        elif isinstance(expr, SymbolicObject):\n            if expr.isVariable():\n                if env is None:\n                    variable = self._getVariable(expr)\n                    return variable\n                else:\n                    return env[expr.name]\n            else:\n                return self._astToCVCExpr(expr.expr, env)\n\n        elif isinstance(expr, int) | isinstance(expr, str):\n            if env is None:\n                if isinstance(expr, int):\n                    return CVCInteger.constant(expr, self.solver)\n                elif isinstance(expr, str):\n                    return CVCString.constant(expr, self.solver)\n            else:\n                return expr\n        elif expr is None:\n            return None\n        else:\n            utils.crash(\"Unknown node during conversion from ast to CVC (expressions): %s\" % expr)\n    \n"
  },
  {
    "path": "symbolic/cvc_expr/expression.py",
    "content": "import logging\n\nimport CVC4\n\n\nlog = logging.getLogger(\"se.cvc_expr.expr\")\n\n\nclass CVCExpression(object):\n    CVC_TYPE = 'Bool'\n\n    def __init__(self, cvc_expr, solver):\n        self.cvc_expr = cvc_expr\n        self.solver = solver\n        self.em = self.solver.getExprManager()\n\n    @classmethod\n    def variable(cls, name, solver):\n        raise NotImplementedError\n\n    @classmethod\n    def constant(cls, v, solver):\n        raise NotImplementedError\n\n    def getvalue(self):\n        raise NotImplementedError\n\n    def ite(self, th, el):\n        assert th.cvc_expr.getType().toString() == el.cvc_expr.getType().toString()\n        return th.__class__(self.em.mkExpr(CVC4.ITE, self.cvc_expr,\n                                           th.cvc_expr, el.cvc_expr), self.solver)\n\n    def __and__(self, other):\n        return CVCExpression(self.em.mkExpr(CVC4.AND, self.cvc_expr, other.cvc_expr), self.solver)\n\n    def __xor__(self, other):\n        return CVCExpression(self.em.mkExpr(CVC4.XOR, self.cvc_expr, other.cvc_expr), self.solver)\n\n    def __or__(self, other):\n        return CVCExpression(self.em.mkExpr(CVC4.OR, self.cvc_expr, other.cvc_expr), self.solver)\n\n    def not_op(self):\n        return CVCExpression(self.em.mkExpr(CVC4.NOT, self.cvc_expr), self.solver)\n\n    def __ne__(self, other):\n        return CVCExpression(self.em.mkExpr(CVC4.NOT, self.em.mkExpr(CVC4.EQUAL, self.cvc_expr, other.cvc_expr)),\n                             self.solver)\n\n    def __eq__(self, other):\n        return CVCExpression(self.em.mkExpr(CVC4.EQUAL, self.cvc_expr, other.cvc_expr), self.solver)\n\n    def __lt__(self, other):\n        return CVCExpression(self.em.mkExpr(CVC4.LT, self.cvc_expr, other.cvc_expr), self.solver)\n\n    def __gt__(self, other):\n        return CVCExpression(self.em.mkExpr(CVC4.GT, self.cvc_expr, other.cvc_expr), self.solver)\n\n    def __ge__(self, other):\n        return CVCExpression(self.em.mkExpr(CVC4.GEQ, self.cvc_expr, other.cvc_expr), self.solver)\n\n    def __le__(self, other):\n        return CVCExpression(self.em.mkExpr(CVC4.LEQ, self.cvc_expr, other.cvc_expr), self.solver)\n\n    def __str__(self):\n        return self.cvc_expr.toString()"
  },
  {
    "path": "symbolic/cvc_expr/integer.py",
    "content": "import logging\n\nimport CVC4\nfrom CVC4 import Rational, Integer\n\nfrom .expression import CVCExpression\n\nlog = logging.getLogger(\"se.cvc.integer\")\n\n\nclass CVCInteger(CVCExpression):\n    \"\"\"Python numbers are represented as integers in the CVC path expression. For bitwise operations, integers are\n    transformed into bit vectors of size _bv_size and then converted back to a natural number using CVC's\n    BITVECTOR_TO_NAT operator. In order to maintain sound reasoning of the behavior of generated inputs, all inputs to\n    bit vector operations are asserted positive through _assert_bvsanity as well as the outputs through\n    _assert_bvbounds. These assumptions restrict the symbolic execution from finding valid solutions to path formulas\n    in order to avoid generating path expressions with solutions that do not match program behavior. For example, x = -1\n    is a valid solution to x != CVC4.BITVECTOR_TO_NAT(CVC4.INT_TO_BITVECTOR(x)) since the output of the right-hand side\n    of the equation will be positive (natural numbers are >= 0).\n\n    Possible improvements:\n\n    1) _bv_size is currently fixed at a low number. The Z3 integration starts with small bit vectors and gradually\n    increases the size until a solution is found. Match that functionality in CVCInteger.\n\n    2) Create an alternative implementation of CVCInteger that uses bit vectors for all operations. In the presence of\n    bitwise operations, the conversion between bit vectors and integers is expensive.\n\n    3) Encode in the formula a BITVECTOR_TO_INT conversion that performs two's complement arithmetic.\"\"\"\n\n    CVC_TYPE = 'Int'\n\n    _bv_size = 8\n\n    @classmethod\n    def variable(cls, name, solver):\n        em = solver.getExprManager()\n        expr = em.mkVar(name, em.integerType())\n        return cls(expr, solver)\n\n    @classmethod\n    def constant(cls, v, solver):\n        em = solver.getExprManager()\n        return cls(em.mkConst(Rational(Integer(str(v)))), solver)\n\n    def getvalue(self):\n        \"\"\"In order to mitigate the limited accuracy of the C-type values returned by the CVC getters, strings are parsed\n        into Python numbers. This fix was added to pass the PyExZ3/test/bignum.py test case.\"\"\"\n        ce = self.solver.getValue(self.cvc_expr)\n        rational = ce.getConstRational()\n        numerator = int(rational.getNumerator().toString())\n        denominator = int(rational.getDenominator().toString())\n        if rational.isIntegral():\n            return numerator // denominator\n        else:\n            return numerator / denominator\n\n    def __add__(self, other):\n        return CVCInteger(self.em.mkExpr(CVC4.PLUS, self.cvc_expr, other.cvc_expr), self.solver)\n\n    def __sub__(self, other):\n        return CVCInteger(self.em.mkExpr(CVC4.MINUS, self.cvc_expr, other.cvc_expr), self.solver)\n\n    def __mul__(self, other):\n        return CVCInteger(self.em.mkExpr(CVC4.MULT, self.cvc_expr, other.cvc_expr), self.solver)\n\n    def __truediv__(self, other):\n        return CVCInteger(self.em.mkExpr(CVC4.DIVISION, self.cvc_expr, other.cvc_expr), self.solver)\n\n    def __mod__(self, other):\n        return CVCInteger(self.em.mkExpr(CVC4.INTS_MODULUS, self.cvc_expr, other.cvc_expr), self.solver)\n\n    def __or__(self, other):\n        return self._bvhelper(other, CVC4.BITVECTOR_OR)\n\n    def __and__(self, other):\n        return self._bvhelper(other, CVC4.BITVECTOR_AND)\n\n    def __xor__(self, other):\n        return self._bvhelper(other, CVC4.BITVECTOR_XOR)\n\n    def __lshift__(self, other):\n        return self._bvhelper(other, CVC4.BITVECTOR_SHL)\n\n    def __rshift__(self, other):\n        return self._bvhelper(other, CVC4.BITVECTOR_ASHR)\n\n    def tobv(self):\n        bvconversion = self.em.mkConst(CVC4.IntToBitVector(self._bv_size))\n        return self.em.mkExpr(bvconversion, self.cvc_expr)\n\n    def bvsanity(self):\n        return self == CVCExpression(self.em.mkExpr(CVC4.BITVECTOR_TO_NAT, self.tobv()), self.solver)\n\n    def _bvhelper(self, other, op):\n        calculation = self.em.mkExpr(op, self.tobv(), other.tobv())\n        self.solver.guards.append(self.bvsanity() & other.bvsanity())\n        self._assert_bvbounds(calculation)\n        return CVCInteger(self.em.mkExpr(CVC4.BITVECTOR_TO_NAT, calculation), self.solver)\n\n    def _assert_bvbounds(self, bvexpr):\n        bitextract = self.em.mkConst(CVC4.BitVectorExtract(0, 0))\n        self.solver.guards.append((CVCExpression(\n            self.em.mkExpr(CVC4.EQUAL, self.em.mkExpr(bitextract, bvexpr),\n                           self.em.mkConst(CVC4.BitVector(1, 0))),\n            self.solver)))\n"
  },
  {
    "path": "symbolic/cvc_expr/string.py",
    "content": "import logging\n\nimport CVC4\n\nfrom .expression import CVCExpression\nfrom .integer import CVCInteger\n\nlog = logging.getLogger(\"se.cvc.string\")\n\n\nclass CVCString(CVCExpression):\n    CVC_TYPE = 'String'\n\n    @classmethod\n    def variable(cls, name, solver):\n        em = solver.getExprManager()\n        expr = em.mkVar(name, em.stringType())\n        return cls(expr, solver)\n\n    @classmethod\n    def constant(cls, v, solver):\n        em = solver.getExprManager()\n\n        chararray = [CVC4.CVC4String_convertCharToUnsignedInt(c)\n                     for c in bytes(v, 'UTF-8')]\n        cvcstr = CVC4.CVC4String(chararray)\n        assert cvcstr.size() == len(v)\n        return cls(em.mkConst(cvcstr), solver)\n\n    def getvalue(self):\n        ce = self.solver.getValue(self.cvc_expr)\n        chararray = [CVC4.CVC4String_convertUnsignedIntToChar(c)\n                     for c in ce.getConstString().getVec()]\n        v = bytes(chararray).decode()\n        assert len(v) == ce.getConstString().size()\n        return v\n\n    def len(self):\n        return CVCInteger(self.em.mkExpr(CVC4.STRING_LENGTH, self.cvc_expr), self.solver)\n\n    def __add__(self, other):\n        return CVCString(self.em.mkExpr(CVC4.STRING_CONCAT,\n            self.cvc_expr, other.cvc_expr), self.solver)\n\n    def __contains__(self, item):\n        return CVCExpression(self.em.mkExpr(CVC4.STRING_STRCTN,\n            self.cvc_expr, item.cvc_expr), self.solver)\n\n    def __getitem__(self, item):\n        if isinstance(item, slice):\n            offset = item.stop - item.start\n            self.solver.guards.append(item.start\n                                      >= CVCInteger.constant(0, self.solver))\n            self.solver.guards.append(offset\n                                      >= CVCInteger.constant(0, self.solver))\n            # Adding these forms of guards globally\n            # can limit the number of solutions generated\n            # but restructuring to avoid reducing the\n            # solution space is a significant undertaking.\n            self.solver.guards.append(self.len() > item.start)\n            self.solver.guards.append(self.len() >= item.stop)\n            return CVCString(self.em.mkExpr(CVC4.STRING_SUBSTR, self.cvc_expr,\n                                            item.start.cvc_expr, offset.cvc_expr),\n                             self.solver)\n        return CVCString(self.em.mkExpr(CVC4.STRING_CHARAT, self.cvc_expr,\n                                        item.cvc_expr), self.solver)\n\n    def find(self, findstr, beg):\n        \"\"\"CVC4's String IndexOf functionality is capable of specifying\n        an index to begin the search. However, the current\n        implementation searches from the beginning of the string.\"\"\"\n        return CVCInteger(\n            self.em.mkExpr(CVC4.STRING_STRIDOF, self.cvc_expr,\n                           findstr.cvc_expr, beg.cvc_expr),\n            self.solver)\n\n    def replace(self, old, new):\n        return CVCString(self.em.mkExpr(CVC4.STRING_STRREPL, self.cvc_expr, old.cvc_expr, new.cvc_expr), self.solver)\n\n    def startswith(self, prefix):\n        return CVCExpression(self.em.mkExpr(CVC4.STRING_PREFIX,\n                                            prefix.cvc_expr,\n                                            self.cvc_expr), self.solver)\n"
  },
  {
    "path": "symbolic/cvc_wrap.py",
    "content": "import logging\n\nimport utils\n\nimport CVC4\nfrom CVC4 import ExprManager, SmtEngine, SExpr\n\nfrom symbolic.cvc_expr.exprbuilder import ExprBuilder\n\nlog = logging.getLogger(\"se.cvc\")\n\n\nclass CVCWrapper(object):\n    options = {'produce-models': 'true',\n               # Enable experimental string support\n               'strings-exp': 'true',\n               # Enable modular arithmetic with constant modulus\n               'rewrite-divk': 'true',\n               # Per Query timeout of 5 seconds\n               'tlimit-per': 5000,\n               'output-language': 'smt2',\n               'input-language': 'smt2'}\n    logic = 'ALL_SUPPORTED'\n\n    def __init__(self):\n        self.asserts = None\n        self.query = None\n        self.em = None\n        self.solver = None\n\n    def findCounterexample(self, asserts, query):\n        \"\"\"Tries to find a counterexample to the query while\n           asserts remains valid.\"\"\"\n        self.em = ExprManager()\n        self.solver = SmtEngine(self.em)\n        for name, value in CVCWrapper.options.items():\n            self.solver.setOption(name, SExpr(str(value)))\n        self.solver.setLogic(CVCWrapper.logic)\n        self.query = query\n        self.asserts = self._coneOfInfluence(asserts, query)\n        result = self._findModel()\n        log.debug(\"Query -- %s\" % self.query)\n        log.debug(\"Asserts -- %s\" % asserts)\n        log.debug(\"Cone -- %s\" % self.asserts)\n        log.debug(\"Result -- %s\" % result)\n        return result\n\n    def _findModel(self):\n        self.solver.push()\n        exprbuilder = ExprBuilder(self.asserts, self.query, self.solver)\n        self.solver.assertFormula(exprbuilder.query.cvc_expr)\n        try:\n            result = self.solver.checkSat()\n            log.debug(\"Solver returned %s\" % result.toString())\n            if not result.isSat():\n                ret = None\n            elif result.isUnknown():\n                ret = None\n            elif result.isSat():\n                ret = self._getModel(exprbuilder.cvc_vars)\n            else:\n                raise Exception(\"Unexpected SMT result\")\n        except RuntimeError as r:\n            log.debug(\"CVC exception %s\" % r)\n            ret = None\n        self.solver.pop()\n        return ret\n\n    @staticmethod\n    def _getModel(variables):\n        \"\"\"Retrieve the model generated for the path expression.\"\"\"\n        return {name: cvc_var.getvalue() for (name, cvc_var) in variables.items()}\n\n    @staticmethod\n    def _coneOfInfluence(asserts, query):\n        cone = []\n        cone_vars = set(query.getVars())\n        ws = [a for a in asserts if len(set(a.getVars()) & cone_vars) > 0]\n        remaining = [a for a in asserts if a not in ws]\n        while len(ws) > 0:\n            a = ws.pop()\n            a_vars = set(a.getVars())\n            cone_vars = cone_vars.union(a_vars)\n            cone.append(a)\n            new_ws = [a for a in remaining if len(set(a.getVars()) & cone_vars) > 0]\n            remaining = [a for a in remaining if a not in new_ws]\n            ws = ws + new_ws\n        return cone\n"
  },
  {
    "path": "symbolic/explore.py",
    "content": "# Copyright: see copyright.txt\n\nfrom collections import deque\nimport logging\nimport os\n\nfrom .z3_wrap import Z3Wrapper\nfrom .path_to_constraint import PathToConstraint\nfrom .invocation import FunctionInvocation\nfrom .symbolic_types import symbolic_type, SymbolicType\n\nlog = logging.getLogger(\"se.conc\")\n\nclass ExplorationEngine:\n\tdef __init__(self, funcinv, solver=\"z3\"):\n\t\tself.invocation = funcinv\n\t\t# the input to the function\n\t\tself.symbolic_inputs = {}  # string -> SymbolicType\n\t\t# initialize\n\t\tfor n in funcinv.getNames():\n\t\t\tself.symbolic_inputs[n] = funcinv.createArgumentValue(n)\n\n\t\tself.constraints_to_solve = deque([])\n\t\tself.num_processed_constraints = 0\n\n\t\tself.path = PathToConstraint(lambda c : self.addConstraint(c))\n\t\t# link up SymbolicObject to PathToConstraint in order to intercept control-flow\n\t\tsymbolic_type.SymbolicObject.SI = self.path\n\n\t\tif solver == \"z3\":\n\t\t\tself.solver = Z3Wrapper()\n\t\telif solver == \"cvc\":\n\t\t\tfrom .cvc_wrap import CVCWrapper\n\t\t\tself.solver = CVCWrapper()\n\t\telse:\n\t\t\traise Exception(\"Unknown solver %s\" % solver)\n\n\t\t# outputs\n\t\tself.generated_inputs = []\n\t\tself.execution_return_values = []\n\n\tdef addConstraint(self, constraint):\n\t\tself.constraints_to_solve.append(constraint)\n\t\t# make sure to remember the input that led to this constraint\n\t\tconstraint.inputs = self._getInputs()\n\n\tdef explore(self, max_iterations=0):\n\t\tself._oneExecution()\n\t\t\n\t\titerations = 1\n\t\tif max_iterations != 0 and iterations >= max_iterations:\n\t\t\tlog.debug(\"Maximum number of iterations reached, terminating\")\n\t\t\treturn self.execution_return_values\n\n\t\twhile not self._isExplorationComplete():\n\t\t\tselected = self.constraints_to_solve.popleft()\n\t\t\tif selected.processed:\n\t\t\t\tcontinue\n\t\t\tself._setInputs(selected.inputs)\t\t\t\n\n\t\t\tlog.info(\"Selected constraint %s\" % selected)\n\t\t\tasserts, query = selected.getAssertsAndQuery()\n\t\t\tmodel = self.solver.findCounterexample(asserts, query)\n\n\t\t\tif model == None:\n\t\t\t\tcontinue\n\t\t\telse:\n\t\t\t\tfor name in model.keys():\n\t\t\t\t\tself._updateSymbolicParameter(name,model[name])\n\n\t\t\tself._oneExecution(selected)\n\n\t\t\titerations += 1\t\t\t\n\t\t\tself.num_processed_constraints += 1\n\n\t\t\tif max_iterations != 0 and iterations >= max_iterations:\n\t\t\t\tlog.info(\"Maximum number of iterations reached, terminating\")\n\t\t\t\tbreak\n\n\t\treturn self.generated_inputs, self.execution_return_values, self.path\n\n\t# private\n\n\tdef _updateSymbolicParameter(self, name, val):\n\t\tself.symbolic_inputs[name] = self.invocation.createArgumentValue(name,val)\n\n\tdef _getInputs(self):\n\t\treturn self.symbolic_inputs.copy()\n\n\tdef _setInputs(self,d):\n\t\tself.symbolic_inputs = d\n\n\tdef _isExplorationComplete(self):\n\t\tnum_constr = len(self.constraints_to_solve)\n\t\tif num_constr == 0:\n\t\t\tlog.info(\"Exploration complete\")\n\t\t\treturn True\n\t\telse:\n\t\t\tlog.info(\"%d constraints yet to solve (total: %d, already solved: %d)\" % (num_constr, self.num_processed_constraints + num_constr, self.num_processed_constraints))\n\t\t\treturn False\n\n\tdef _getConcrValue(self,v):\n\t\tif isinstance(v,SymbolicType):\n\t\t\treturn v.getConcrValue()\n\t\telse:\n\t\t\treturn v\n\n\tdef _recordInputs(self):\n\t\targs = self.symbolic_inputs\n\t\tinputs = [ (k,self._getConcrValue(args[k])) for k in args ]\n\t\tself.generated_inputs.append(inputs)\n\t\tprint(inputs)\n\t\t\n\tdef _oneExecution(self,expected_path=None):\n\t\tself._recordInputs()\n\t\tself.path.reset(expected_path)\n\t\tret = self.invocation.callFunction(self.symbolic_inputs)\n\t\tprint(ret)\n\t\tself.execution_return_values.append(ret)\n\n"
  },
  {
    "path": "symbolic/invocation.py",
    "content": "# Copyright: see copyright.txt\n\nclass FunctionInvocation:\n\tdef __init__(self, function, reset):\n\t\tself.function = function\n\t\tself.reset = reset\n\t\tself.arg_constructor = {}\n\t\tself.initial_value = {}\n\n\tdef callFunction(self,args):\n\t\tself.reset()\n\t\treturn self.function(**args)\n\n\tdef addArgumentConstructor(self, name, init, constructor):\n\t\tself.initial_value[name] = init\n\t\tself.arg_constructor[name] = constructor\n\n\tdef getNames(self):\n\t\treturn self.arg_constructor.keys()\n\n\tdef createArgumentValue(self,name,val=None):\n\t\tif val == None:\n\t\t\tval = self.initial_value[name]\n\t\treturn self.arg_constructor[name](name,val)\n\n\t\n\n"
  },
  {
    "path": "symbolic/loader.py",
    "content": "# Copyright: copyright.txt\n\nimport inspect\nimport re\nimport os\nimport sys\nfrom .invocation import FunctionInvocation\nfrom .symbolic_types import SymbolicInteger, getSymbolic\n\n# The built-in definition of len wraps the return value in an int() constructor, destroying any symbolic types.\n# By redefining len here we can preserve symbolic integer types.\nimport builtins\nbuiltins.len = (lambda x : x.__len__())\n\nclass Loader:\n\tdef __init__(self, filename, entry):\n\t\tself._fileName = os.path.basename(filename)\n\t\tself._fileName = self._fileName[:-3]\n\t\tif (entry == \"\"):\n\t\t\tself._entryPoint = self._fileName\n\t\telse:\n\t\t\tself._entryPoint = entry;\n\t\tself._resetCallback(True)\n\n\tdef getFile(self):\n\t\treturn self._fileName\n\n\tdef getEntry(self):\n\t\treturn self._entryPoint\n\t\n\tdef createInvocation(self):\n\t\tinv = FunctionInvocation(self._execute,self._resetCallback)\n\t\tfunc = self.app.__dict__[self._entryPoint]\n\t\targspec = inspect.getargspec(func)\n\t\t# check to see if user specified initial values of arguments\n\t\tif \"concrete_args\" in func.__dict__:\n\t\t\tfor (f,v) in func.concrete_args.items():\n\t\t\t\tif not f in argspec.args:\n\t\t\t\t\tprint(\"Error in @concrete: \" +  self._entryPoint + \" has no argument named \" + f)\n\t\t\t\t\traise ImportError()\n\t\t\t\telse:\n\t\t\t\t\tLoader._initializeArgumentConcrete(inv,f,v)\n\t\tif \"symbolic_args\" in func.__dict__:\n\t\t\tfor (f,v) in func.symbolic_args.items():\n\t\t\t\tif not f in argspec.args:\n\t\t\t\t\tprint(\"Error (@symbolic): \" +  self._entryPoint + \" has no argument named \" + f)\n\t\t\t\t\traise ImportError()\n\t\t\t\telif f in inv.getNames():\n\t\t\t\t\tprint(\"Argument \" + f + \" defined in both @concrete and @symbolic\")\n\t\t\t\t\traise ImportError()\n\t\t\t\telse:\n\t\t\t\t\ts = getSymbolic(v)\n\t\t\t\t\tif (s == None):\n\t\t\t\t\t\tprint(\"Error at argument \" + f + \" of entry point \" + self._entryPoint + \" : no corresponding symbolic type found for type \" + str(type(v)))\n\t\t\t\t\t\traise ImportError()\n\t\t\t\t\tLoader._initializeArgumentSymbolic(inv, f, v, s)\n\t\tfor a in argspec.args:\n\t\t\tif not a in inv.getNames():\n\t\t\t\tLoader._initializeArgumentSymbolic(inv, a, 0, SymbolicInteger)\n\t\treturn inv\n\n\t# need these here (rather than inline above) to correctly capture values in lambda\n\tdef _initializeArgumentConcrete(inv,f,val):\n\t\tinv.addArgumentConstructor(f, val, lambda n,v: val)\n\n\tdef _initializeArgumentSymbolic(inv,f,val,st):\n\t\tinv.addArgumentConstructor(f, val, lambda n,v: st(n,v))\n\n\tdef executionComplete(self, return_vals):\n\t\tif \"expected_result\" in self.app.__dict__:\n\t\t\treturn self._check(return_vals, self.app.__dict__[\"expected_result\"]())\n\t\tif \"expected_result_set\" in self.app.__dict__:\n\t\t\treturn self._check(return_vals, self.app.__dict__[\"expected_result_set\"](),False)\n\t\telse:\n\t\t\tprint(self._fileName + \".py contains no expected_result function\")\n\t\t\treturn None\n\n\t# -- private\n\n\tdef _resetCallback(self,firstpass=False):\n\t\tself.app = None\n\t\tif firstpass and self._fileName in sys.modules:\n\t\t\tprint(\"There already is a module loaded named \" + self._fileName)\n\t\t\traise ImportError()\n\t\ttry:\n\t\t\tif (not firstpass and self._fileName in sys.modules):\n\t\t\t\tdel(sys.modules[self._fileName])\n\t\t\tself.app =__import__(self._fileName)\n\t\t\tif not self._entryPoint in self.app.__dict__ or not callable(self.app.__dict__[self._entryPoint]):\n\t\t\t\tprint(\"File \" +  self._fileName + \".py doesn't contain a function named \" + self._entryPoint)\n\t\t\t\traise ImportError()\n\t\texcept Exception as arg:\n\t\t\tprint(\"Couldn't import \" + self._fileName)\n\t\t\tprint(arg)\n\t\t\traise ImportError()\n\n\tdef _execute(self, **args):\n\t\treturn self.app.__dict__[self._entryPoint](**args)\n\n\tdef _toBag(self,l):\n\t\tbag = {}\n\t\tfor i in l:\n\t\t\tif i in bag:\n\t\t\t\tbag[i] += 1\n\t\t\telse:\n\t\t\t\tbag[i] = 1\n\t\treturn bag\n\n\tdef _check(self, computed, expected, as_bag=True):\n\t\tb_c = self._toBag(computed)\n\t\tb_e = self._toBag(expected)\n\t\tif as_bag and b_c != b_e or not as_bag and set(computed) != set(expected):\n\t\t\tprint(\"-------------------> %s test failed <---------------------\" % self._fileName)\n\t\t\tprint(\"Expected: %s, found: %s\" % (b_e, b_c))\n\t\t\treturn False\n\t\telse:\n\t\t\tprint(\"%s test passed <---\" % self._fileName)\n\t\t\treturn True\n\t\ndef loaderFactory(filename,entry):\n\tif not os.path.isfile(filename) or not re.search(\".py$\",filename):\n\t\tprint(\"Please provide a Python file to load\")\n\t\treturn None\n\ttry: \n\t\tdir = os.path.dirname(filename)\n\t\tsys.path = [ dir ] + sys.path\n\t\tret = Loader(filename,entry)\n\t\treturn ret\n\texcept ImportError:\n\t\tsys.path = sys.path[1:]\n\t\treturn None\n\n\n"
  },
  {
    "path": "symbolic/path_to_constraint.py",
    "content": "# Copyright: see copyright.txt\n\nimport logging\n\nfrom .predicate import Predicate\nfrom .constraint import Constraint\n\nlog = logging.getLogger(\"se.pathconstraint\")\n\nclass PathToConstraint:\n\tdef __init__(self, add):\n\t\tself.constraints = {}\n\t\tself.add = add\n\t\tself.root_constraint = Constraint(None, None)\n\t\tself.current_constraint = self.root_constraint\n\t\tself.expected_path = None\n\n\tdef reset(self,expected):\n\t\tself.current_constraint = self.root_constraint\n\t\tif expected==None:\n\t\t\tself.expected_path = None\n\t\telse:\n\t\t\tself.expected_path = []\n\t\t\ttmp = expected\n\t\t\twhile tmp.predicate is not None:\n\t\t\t\tself.expected_path.append(tmp.predicate)\n\t\t\t\ttmp = tmp.parent\n\n\tdef whichBranch(self, branch, symbolic_type):\n\t\t\"\"\" This function acts as instrumentation.\n\t\tBranch can be either True or False.\"\"\"\n\n\t\t# add both possible predicate outcomes to constraint (tree)\n\t\tp = Predicate(symbolic_type, branch)\n\t\tp.negate()\n\t\tcneg = self.current_constraint.findChild(p)\n\t\tp.negate()\n\t\tc = self.current_constraint.findChild(p)\n\n\t\tif c is None:\n\t\t\tc = self.current_constraint.addChild(p)\n\n\t\t\t# we add the new constraint to the queue of the engine for later processing\n\t\t\tlog.debug(\"New constraint: %s\" % c)\n\t\t\tself.add(c)\n\t\t\t\n\t\t# check for path mismatch\n\t\t# IMPORTANT: note that we don't actually check the predicate is the\n\t\t# same one, just that the direction taken is the same\n\t\tif self.expected_path != None and self.expected_path != []:\n\t\t\texpected = self.expected_path.pop()\n\t\t\t# while not at the end of the path, we expect the same predicate result\n\t\t\t# at the end of the path, we expect a different predicate result\n\t\t\tdone = self.expected_path == []\n\t\t\tif ( not done and expected.result != c.predicate.result or \\\n\t\t\t\tdone and expected.result == c.predicate.result ):\n\t\t\t\tprint(\"Replay mismatch (done=\",done,\")\")\n\t\t\t\tprint(expected)\n\t\t\t\tprint(c.predicate)\n\n\t\tif cneg is not None:\n\t\t\t# We've already processed both\n\t\t\tcneg.processed = True\n\t\t\tc.processed = True\n\t\t\tlog.debug(\"Processed constraint: %s\" % c)\n\n\t\tself.current_constraint = c\n\n\tdef toDot(self):\n\t\t# print the thing into DOT format\n\t\theader = \"digraph {\\n\"\n\t\tfooter = \"\\n}\\n\"\n\t\treturn header + self._toDot(self.root_constraint) + footer\n\n\tdef _toDot(self,c):\n\t\tif (c.parent == None):\n\t\t\tlabel = \"root\"\n\t\telse:\n\t\t\tlabel = c.predicate.symtype.toString()\n\t\t\tif not c.predicate.result:\n\t\t\t\tlabel = \"Not(\"+label+\")\"\n\t\tnode = \"C\" + str(c.id) + \" [ label=\\\"\" + label + \"\\\" ];\\n\"\n\t\tedges = [ \"C\" + str(c.id) + \" -> \" + \"C\" + str(child.id) + \";\\n\" for child in c.children ]\n\t\treturn node + \"\".join(edges) + \"\".join([ self._toDot(child) for child in c.children ])\n\t\t\n"
  },
  {
    "path": "symbolic/predicate.py",
    "content": "# Copyright - see copyright.txt\n\nclass Predicate:\n\t\"\"\"Predicate is one specific ``if'' encountered during the program execution.\n\t   \"\"\"\n\tdef __init__(self, st, result):\n\t\tself.symtype = st\n\t\tself.result = result\n\n\tdef getVars(self):\n\t\treturn self.symtype.getVars()\n\n\tdef __eq__(self, other):\n\t\tif isinstance(other, Predicate):\n\t\t\tres = self.result == other.result and self.symtype.symbolicEq(other.symtype)\n\t\t\treturn res\n\t\telse:\n\t\t\treturn False\n\n\tdef __hash__(self):\n\t\treturn hash(self.symtype)\n\n\tdef __str__(self):\n\t\treturn self.symtype.toString() + \" (%s)\" % (self.result)\n\n\tdef __repr__(self):\n\t\treturn self.__str__()\n\n\tdef negate(self):\n\t\t\"\"\"Negates the current predicate\"\"\"\n\t\tassert(self.result is not None)\n\t\tself.result = not self.result\n\n"
  },
  {
    "path": "symbolic/symbolic_types/__init__.py",
    "content": "# Copyright: see copyright.txt\n\nfrom .symbolic_int import SymbolicInteger as SymInt\nfrom .symbolic_int import SymbolicObject as SymObj\nfrom .symbolic_dict import SymbolicDict as SymD\nfrom .symbolic_str import SymbolicStr as SymS\nfrom .symbolic_type import SymbolicType as SymType\n\nSymObj.wrap = lambda conc, sym : SymbolicInteger(\"se\",conc,sym)\nSymbolicInteger = SymInt\nSymbolicDict = SymD\nSymbolicStr = SymS\nSymbolicType = SymType\n\ndef getSymbolic(v):\n\texported = [(int,SymbolicInteger),(dict,SymbolicDict),(str,SymbolicStr)]\n\tfor (t,s) in exported:\n\t\tif isinstance(v,t):\n\t\t\treturn s\n\treturn None\n\n\n\n"
  },
  {
    "path": "symbolic/symbolic_types/symbolic_dict.py",
    "content": "import ast\nimport sys\nfrom . symbolic_type import SymbolicObject\n\n# SymbolicDict: the key and values will both be SymbolicType for full generality\n\n# keys of dictionary must be immutable\n# values in dictionary may be mutable\n\n\n# TODO: big simplification: can only initialize with\n# an empty dictionary\nclass SymbolicDict(SymbolicObject,dict):\n\tdef __new__(cls, name, *args, **kwargs):\n\t\tself = dict.__new__(cls,args,kwargs)\n\t\treturn self\n\n\tdef __init__(self, name, kwargs):\n\t\tSymbolicObject.__init__(self,name,None)\n\t\tdict.__init__(self,kwargs)\n\n\tdef getConcrValue(self):\n\t\treturn self\n\t\t\n\tdef __bool__(self):\n\t\treturn bool(len(self))\n\n#\tdef wrap(conc,sym):\n#\t\tpass # TODO\n\n#\tdef __getitem__(self,key):\n#\t\tval = super.__getitem__(key)\n#\t\tif isinstance(val,SymbolicType):\n#\t\t\twrap = val.wrap\n#\t\telse:\n#\t\t\twrap = lambda c,s : c\n#\t\treturn self._do_bin_op(key, lambda d, k: val, ast.Index, wrap)\n\n#\tdef __setitem__(self,key,value):\n#\t\t# update the expression (this is a triple - not binary)\n#\t\tconcrete, symbolic =\\\n#\t\t\tself._do_sexpr([self,key,value], lambda d, k, v : d.super.__setitem__(k,v), ast.Store,\\\n#\t\t\t\t\tlambda c, s: c, s)\n#\t\t# note that we do an in place update of \n#                self.expr = symbolic\n\n#\tdef __contains__(self,key):\n#\t\tfor k in self.keys():\n#\t\t\tif k == key:\n#\t\t\t\treturn True\n#\t\treturn False\n\n#\tdef __delitem__(self,key)\n#\t\tif dict.__contains(self,key):\n#\t\t\tpass\n#\t\t\t# self.expr = Delete(self.expr,key)\n#\t\tdict.__delitem__(self,key)\n\n"
  },
  {
    "path": "symbolic/symbolic_types/symbolic_int.py",
    "content": "# Copyright: copyright.txt\n\nfrom . symbolic_type import SymbolicObject\n\n# we use multiple inheritance to achieve concrete execution for any\n# operation for which we don't have a symbolic representation. As\n# we can see a SymbolicInteger is both symbolic (SymbolicObject) and \n# concrete (int)\n\nclass SymbolicInteger(SymbolicObject,int):\n\t# since we are inheriting from int, we need to use new\n\t# to perform construction correctly\n\tdef __new__(cls, name, v, expr=None):\n\t\treturn int.__new__(cls, v)\n\n\tdef __init__(self, name, v, expr=None):\n\t\tSymbolicObject.__init__(self, name, expr)\n\t\tself.val = v\n\n\tdef getConcrValue(self):\n\t\treturn self.val\n\n\tdef wrap(conc,sym):\n\t\treturn SymbolicInteger(\"se\",conc,sym)\n\n\tdef __hash__(self):\n\t\treturn hash(self.val)\n\n\tdef _op_worker(self,args,fun,op):\n\t\treturn self._do_sexpr(args, fun, op, SymbolicInteger.wrap)\n\n# now update the SymbolicInteger class for operations we\n# will build symbolic terms for\n\nops =  [(\"add\",    \"+\"  ),\\\n\t(\"sub\",    \"-\"  ),\\\n\t(\"mul\",    \"*\"  ),\\\n\t(\"mod\",    \"%\"  ),\\\n\t(\"floordiv\", \"//\" ),\\\n\t(\"and\",    \"&\"  ),\\\n\t(\"or\",     \"|\"  ),\\\n\t(\"xor\",    \"^\"  ),\\\n\t(\"lshift\", \"<<\" ),\\\n\t(\"rshift\", \">>\" ) ]\n\ndef make_method(method,op,a):\n\tcode  = \"def %s(self,other):\\n\" % method\n\tcode += \"   return self._op_worker(%s,lambda x,y : x %s y, \\\"%s\\\")\" % (a,op,op)\n\tlocals_dict = {}\n\texec(code, globals(), locals_dict)\n\tsetattr(SymbolicInteger,method,locals_dict[method])\n\nfor (name,op) in ops:\n\tmethod  = \"__%s__\" % name\n\tmake_method(method,op,\"[self,other]\")\n\trmethod  = \"__r%s__\" % name\n\tmake_method(rmethod,op,\"[other,self]\")\n\n"
  },
  {
    "path": "symbolic/symbolic_types/symbolic_str.py",
    "content": "from . symbolic_type import SymbolicObject\nfrom symbolic.symbolic_types.symbolic_int import SymbolicInteger\nfrom string import whitespace\n\nclass SymbolicStr(SymbolicObject, str):\n\n    def __new__(cls, name, v, expr=None):\n        return str.__new__(cls, v)\n\n    def __init__(self, name, v, expr=None):\n        SymbolicObject.__init__(self, name, expr)\n        self.val = v\n\n    def getConcrValue(self):\n        return self.val\n\n    def wrap(conc, sym):\n        return SymbolicStr(\"se\", conc, sym)\n\n    def __hash__(self):\n        return hash(self.val)\n\n    def _op_worker(self, args, fun, op):\n        return self._do_sexpr(args, fun, op, SymbolicStr.wrap)\n\n    def __bool__(self):\n        return SymbolicObject.__bool__(self.__len__() != 0)\n\n    def __len__(self):\n        return self._do_sexpr([self], lambda x: len(x),\n                                \"str.len\", SymbolicInteger.wrap)\n\n    def __contains__(self, item):\n        return self._do_sexpr([self, item], lambda x, y: str.__contains__(x, y),\n                                \"in\", SymbolicInteger.wrap)\n\n    def __getitem__(self, key):\n        \"\"\"Negative indexes, out of bound slices, and slice skips are not currently supported.\"\"\"\n        if isinstance(key, slice):\n            start = key.start if key.start is not None else 0\n            stop = key.stop if key.stop is not None else self.__len__()\n            return self._do_sexpr([self, start, stop],\n                                  lambda x, y, z: str.__getitem__(x, slice(y, z)), \"slice\", SymbolicStr.wrap)\n        return self._do_sexpr([self, key], lambda x, y: str.__getitem__(x, y),\n                              \"getitem\", SymbolicStr.wrap)\n\n    def find(self, findstr, beg=0):\n        return self._do_sexpr([self, findstr, beg],\n                              lambda x, y, z: str.find(x, y, z),\n                              \"str.find\", SymbolicInteger.wrap)\n\n    def startswith(self, prefix):\n        return self._do_sexpr([self, prefix],\n                              lambda x, y: str.startswith(x, y),\n                              \"str.startswith\", SymbolicInteger.wrap)\n\n    def split(self, sep=None, maxsplit=None):\n        if sep is None:\n            sep = \" \"\n        if len(self) == 0:\n            return []\n        elif maxsplit == 0 or sep not in self:\n            return [self]\n        else:\n            sep_idx = self.find(sep)\n            maxsplit = None if maxsplit is None else maxsplit - 1\n            return [self[0:sep_idx]] + \\\n                   self[sep_idx + 1:].split(sep, maxsplit)\n\n    def count(self, sub):\n        \"\"\"String count is not a native function of the SMT solver. Instead, we implement count as a recursive series of\n        find operations. Note that not all of the functionality of count is supported at this time, such as the start\n        index.\"\"\"\n        if sub not in self:\n            ret = 0\n        elif sub == \"\":\n            ret = self.__len__() + 1\n        else:\n            find_idx = self.find(sub)\n            reststr = self[find_idx + sub.__len__():]\n            ret = reststr.count(sub) + 1\n        assert int(ret) == str.count(str(self), str(sub))\n        return ret\n\n    def _replace(self, old, new):\n        return self._do_sexpr([self, old, new], lambda x, y, z: str.replace(x, y, z),\n                              \"str.replace\", SymbolicStr.wrap)\n\n    def replace(self, old, new, maxreplace=-1):\n        \"\"\"CVC only replaces the first occurrence of old with new\n        (maxreplace=1). For this reason, SymbolicStr's replace is implemented\n        as a recurrence of single replaces.\"\"\"\n        if maxreplace == 0 or old not in self:\n            ret = self\n        else:\n            pivot_point = self.find(old) + old.__len__()\n            first_half = self[:pivot_point]\n            first_half = first_half._replace(old, new)\n            second_half = self[pivot_point:]\n            ret = first_half + second_half.replace(old, new, maxreplace-1)\n        assert str(ret) == str.replace(str(self), str(old), str(new), int(maxreplace))\n        return ret\n\n    def strip(self, chars=None):\n        if chars is None:\n            chars = whitespace\n        if self.__len__() == 0:\n            return self\n        for char in chars:\n            if self[0] == char:\n                return self[1:].strip(chars)\n        for char in chars:\n            if self[self.__len__() - 1] == char:\n                return self[:self.__len__() - 1].strip(chars)\n        return self\n\n# Currently only a subset of string operations are supported.\nops = [(\"add\", \"+\")]\n\ndef make_method(method,op,a):\n    code  = \"def %s(self,other):\\n\" % method\n    code += \"   return self._op_worker(%s,lambda x,y : x %s y, \\\"%s\\\")\" % (a,op,op)\n    locals_dict = {}\n    exec(code, globals(), locals_dict)\n    setattr(SymbolicStr, method, locals_dict[method])\n\nfor (name,op) in ops:\n    method  = \"__%s__\" % name\n    make_method(method,op,\"[self,other]\")\n    rmethod  = \"__r%s__\" % name\n    make_method(rmethod,op,\"[other,self]\")\n\n"
  },
  {
    "path": "symbolic/symbolic_types/symbolic_type.py",
    "content": "# Copyright: see copyright.txt\n\nimport utils\nimport inspect\nimport functools\n\n# the ABSTRACT base class for representing any expression that depends on a symbolic input\n# it also tracks the corresponding concrete value for the expression (aka concolic execution)\n\nclass SymbolicType(object):\n\tdef __init__(self, name, expr=None):\n\t\tself.name = name\n\t\tself.expr = expr\n\n\t# to be provided by subclass\n\n\tdef getConcrValue(self):\n\t\traise NotImplemented()\n\n\tdef wrap(conc,sym):\n\t\traise NotImplemented()\n\n\t# public funs\n\n\tdef isVariable(self):\n\t\treturn self.expr == None\n\n\tdef unwrap(self):\n\t\tif self.isVariable():\n\t\t\treturn (self.getConcrValue(),self)\n\t\telse:\n\t\t\treturn (self.getConcrValue(),self.expr)\n\n\tdef getVars(self):\n\t\tif self.isVariable():\n\t\t\treturn [self.name]\n\t\telif isinstance(self.expr,list):\n\t\t\treturn self._getVarsLeaves(self.expr)\n\t\telse:\n\t\t\treturn []\n\n\tdef _getVarsLeaves(self,l):\n\t\tif isinstance(l,list):\n\t\t\treturn functools.reduce(lambda a, x: self._getVarsLeaves(x) + a,l,[])\n\t\telif isinstance(l,SymbolicType):\n\t\t\treturn [l.name]\n\t\telse:\n\t\t\treturn []\n\n\t# creating the expression tree\n\tdef _do_sexpr(self,args,fun,op,wrap):\n\t\tunwrapped = [ (a.unwrap() if isinstance(a,SymbolicType) else (a,a)) for a in args ]\n\t\targs = zip(inspect.getargspec(fun).args, [ c for (c,s) in unwrapped ])\n\t\tconcrete = fun(**dict([a for a in args]))\n\t\tsymbolic = [ op ] + [ s for c,s in unwrapped ]\n\t\treturn wrap(concrete,symbolic)\n\n\tdef symbolicEq(self, other):\n\t\tif not isinstance(other,SymbolicType):\n\t\t\treturn False\n\t\tif self.isVariable() or other.isVariable():\n\t\t\treturn self.name == other.name\n\t\treturn self._eq_worker(self.expr,other.expr)\n\n\tdef _eq_worker(self, expr1, expr2):\n\t\tif type(expr1) != type(expr2):\n\t\t\treturn False\n\t\tif isinstance(expr1, list):\n\t\t\treturn len(expr1) == len(expr2) and\\\n\t\t\t       type(expr1[0]) == type(expr2[0]) and\\\n                               all([ self._eq_worker(x,y) for x,y in zip(expr1[1:],expr2[1:]) ])\n\t\telif isinstance(expr1, SymbolicType):\n\t\t\treturn expr1.name == expr2.name\n\t\telse:\n\t\t\treturn expr1 == expr2\n\n\tdef toString(self):\n\t\tif self.isVariable():\n\t\t\treturn self.name + \"#\" + str(self.getConcrValue())\n\t\telse:\n\t\t\treturn self._toString(self.expr)\n\n\tdef _toString(self,expr):\n\t\tif isinstance(expr,list):\n\t\t\treturn \"(\" + expr[0] + \" \" + \", \".join([ self._toString(a) for a in expr[1:] ]) + \")\"\n\t\telif isinstance(expr,SymbolicType):\n\t\t\treturn expr.toString()\n\t\telse:\n\t\t\treturn str(expr)\n\n# this class is also ABSTRACT although __init__.py does\n# initialize wrap to return SymbolicInteger for the \n# relational comparison operators\n\nclass SymbolicObject(SymbolicType,object): \n\tdef __init__(self, name, expr=None):\n\t\tSymbolicType.__init__(self,name,expr)\n\n\tSI = None    # this is set up by ConcolicEngine to link __bool__ to PathConstraint\n\n\tdef wrap(conc,sym):\n\t\t# see __init__.py\n\t\traise NotImplemented()\n\n\t# this is a critical interception point: the __bool__\n\t# method is called whenever a predicate is evaluated in\n\t# Python execution (if, while, and, or). This allows us\n\t# to capture the path condition\n\n\tdef __bool__(self):\n\t\tret = bool(self.getConcrValue())\n\t\tif SymbolicObject.SI != None:\n\t\t\tSymbolicObject.SI.whichBranch(ret,self)\n\t\treturn ret\n\n\t# compute both the symbolic and concrete image of operator\n\tdef _do_bin_op(self, other, fun, op, wrap):\n\t\treturn self._do_sexpr([self,other], fun, op, wrap)\n\n\tdef __eq__(self, other):\n\t\t# TODO: what it self is not symbolic and other is???\n\t\treturn self._do_bin_op(other, lambda x, y: x == y, \"==\", SymbolicObject.wrap)\n\n\tdef __ne__(self, other):\n\t\treturn self._do_bin_op(other, lambda x, y: x != y, \"!=\", SymbolicObject.wrap)\n\n\tdef __lt__(self, other):\n\t\treturn self._do_bin_op(other, lambda x, y: x < y, \"<\", SymbolicObject.wrap)\n\n\tdef __le__(self, other):\n\t\treturn self._do_bin_op(other, lambda x, y: x <= y, \"<=\", SymbolicObject.wrap)\n\n\tdef __gt__(self, other):\n\t\treturn self._do_bin_op(other, lambda x, y: x > y, \">\", SymbolicObject.wrap)\n\n\tdef __ge__(self, other):\n\t\treturn self._do_bin_op(other, lambda x, y: x >= y, \">=\", SymbolicObject.wrap)\n\n\n"
  },
  {
    "path": "symbolic/z3_expr/__init__.py",
    "content": ""
  },
  {
    "path": "symbolic/z3_expr/bitvector.py",
    "content": "from z3 import *\nfrom .expression import Z3Expression\n\nclass Z3BitVector(Z3Expression):\n\tdef __init__(self,N):\n\t\tZ3Expression.__init__(self)\n\t\tself.N = N\n\n\tdef _isIntVar(self,v):\n\t\treturn isinstance(v,BitVecRef)\n\n\tdef _variable(self,name,solver):\n\t\treturn BitVec(name,self.N,solver.ctx)\n\n\tdef _constant(self,v,solver):\n\t\treturn BitVecVal(v,self.N,solver.ctx)\n"
  },
  {
    "path": "symbolic/z3_expr/expression.py",
    "content": "import utils\n\nfrom symbolic.symbolic_types.symbolic_int import SymbolicInteger\nfrom symbolic.symbolic_types.symbolic_type import SymbolicType\nfrom z3 import *\n\nclass Z3Expression(object):\n\tdef __init__(self):\n\t\tself.z3_vars = {}\n\n\tdef toZ3(self,solver,asserts,query):\n\t\tself.z3_vars = {}\n\t\tsolver.assert_exprs([self.predToZ3(p,solver) for p in asserts])\n\t\tsolver.assert_exprs(Not(self.predToZ3(query,solver)))\n\n\tdef predToZ3(self,pred,solver,env=None):\n\t\tsym_expr = self._astToZ3Expr(pred.symtype,solver,env)\n\t\tif env == None:\n\t\t\tif not is_bool(sym_expr):\n\t\t\t\tsym_expr = sym_expr != self._constant(0,solver)\n\t\t\tif not pred.result:\n\t\t\t\tsym_expr = Not(sym_expr)\n\t\telse:\n\t\t\tif not pred.result:\n\t\t\t\tsym_expr = not sym_expr\n\t\treturn sym_expr\n\n\tdef getIntVars(self):\n\t\treturn [ v[1] for v in self.z3_vars.items() if self._isIntVar(v[1]) ]\n\n\t# ----------- private ---------------\n\n\tdef _isIntVar(self, v):\n\t\traise NotImplementedException\n\n\tdef _getIntegerVariable(self,name,solver):\n\t\tif name not in self.z3_vars:\n\t\t\tself.z3_vars[name] = self._variable(name,solver)\n\t\treturn self.z3_vars[name]\n\n\tdef _variable(self,name,solver):\n\t\traise NotImplementedException\n\n\tdef _constant(self,v,solver):\n\t\traise NotImplementedException\n\n\tdef _wrapIf(self,e,solver,env):\n\t\tif env == None:\n\t\t\treturn If(e,self._constant(1,solver),self._constant(0,solver))\n\t\telse:\n\t\t\treturn e\n\n\t# add concrete evaluation to this, to check\n\tdef _astToZ3Expr(self,expr,solver,env=None):\n\t\tif isinstance(expr, list):\n\t\t\top = expr[0]\n\t\t\targs = [ self._astToZ3Expr(a,solver,env) for a in expr[1:] ]\n\t\t\tz3_l,z3_r = args[0],args[1]\n\n\t\t\t# arithmetical operations\n\t\t\tif op == \"+\":\n\t\t\t\treturn self._add(z3_l, z3_r, solver)\n\t\t\telif op == \"-\":\n\t\t\t\treturn self._sub(z3_l, z3_r, solver)\n\t\t\telif op == \"*\":\n\t\t\t\treturn self._mul(z3_l, z3_r, solver)\n\t\t\telif op == \"//\":\n\t\t\t\treturn self._div(z3_l, z3_r, solver)\n\t\t\telif op == \"%\":\n\t\t\t\treturn self._mod(z3_l, z3_r, solver)\n\n\t\t\t# bitwise\n\t\t\telif op == \"<<\":\n\t\t\t\treturn self._lsh(z3_l, z3_r, solver)\n\t\t\telif op == \">>\":\n\t\t\t\treturn self._rsh(z3_l, z3_r, solver)\n\t\t\telif op == \"^\":\n\t\t\t\treturn self._xor(z3_l, z3_r, solver)\n\t\t\telif op == \"|\":\n\t\t\t\treturn self._or(z3_l, z3_r, solver)\n\t\t\telif op == \"&\":\n\t\t\t\treturn self._and(z3_l, z3_r, solver)\n\n\t\t\t# equality gets coerced to integer\n\t\t\telif op == \"==\":\n\t\t\t\treturn self._wrapIf(z3_l == z3_r,solver,env)\n\t\t\telif op == \"!=\":\n\t\t\t\treturn self._wrapIf(z3_l != z3_r,solver,env)\n\t\t\telif op == \"<\":\n\t\t\t\treturn self._wrapIf(z3_l < z3_r,solver,env)\n\t\t\telif op == \">\":\n\t\t\t\treturn self._wrapIf(z3_l > z3_r,solver,env)\n\t\t\telif op == \"<=\":\n\t\t\t\treturn self._wrapIf(z3_l <= z3_r,solver,env)\n\t\t\telif op == \">=\":\n\t\t\t\treturn self._wrapIf(z3_l >= z3_r,solver,env)\n\t\t\telse:\n\t\t\t\tutils.crash(\"Unknown BinOp during conversion from ast to Z3 (expressions): %s\" % op)\n\n\t\telif isinstance(expr, SymbolicInteger):\n\t\t\tif expr.isVariable():\n\t\t\t\tif env == None:\n\t\t\t\t\treturn self._getIntegerVariable(expr.name,solver)\n\t\t\t\telse:\n\t\t\t\t\treturn env[expr.name]\n\t\t\telse:\n\t\t\t\treturn self._astToZ3Expr(expr.expr,solver,env)\n\n\t\telif isinstance(expr, SymbolicType):\n\t\t\tutils.crash(\"{} is an unsupported SymbolicType of {}\".\n\t\t\t\t\t\tformat(expr, type(expr)))\n\n\t\telif isinstance(expr, int):\n\t\t\tif env == None:\n\t\t\t\treturn self._constant(expr,solver)\n\t\t\telse:\n\t\t\t\treturn expr\n\t\telse:\n\t\t\tutils.crash(\"Unknown node during conversion from ast to Z3 (expressions): %s\" % expr)\n\n\tdef _add(self, l, r, solver):\n\t\treturn l + r\n\n\tdef _sub(self, l, r, solver):\n\t\treturn l - r\n\n\tdef _mul(self, l, r, solver):\n\t\treturn l * r\n\n\tdef _div(self, l, r, solver):\n\t\treturn l / r\n\n\tdef _mod(self, l, r, solver):\n\t\treturn l % r\n\n\tdef _lsh(self, l, r, solver):\n\t\treturn l << r\n\n\tdef _rsh(self, l, r, solver):\n\t\treturn l >> r\n\n\tdef _xor(self, l, r, solver):\n\t\treturn l ^ r\n\n\tdef _or(self, l, r, solver):\n\t\treturn l | r\n\n\tdef _and(self, l, r, solver):\n\t\treturn l & r\n"
  },
  {
    "path": "symbolic/z3_expr/integer.py",
    "content": "from z3 import *\nfrom .expression import Z3Expression\n\nclass Z3Integer(Z3Expression):\n\tdef _isIntVar(self,v):\n\t\treturn isinstance(v,IntRef)\n\n\tdef _variable(self,name,solver):\n\t\treturn Int(name,solver.ctx)\n\n\tdef _constant(self,v,solver):\n\t\treturn IntVal(v,solver.ctx)\n\n\tdef _mod(self, l, r, solver):\n\t\tmod_fun = Function('int_mod', IntSort(), IntSort(), IntSort())\n\t\treturn mod_fun(l, r)\n\n\tdef _lsh(self, l, r, solver):\n\t\tlsh_fun = Function('int_lsh', IntSort(), IntSort(), IntSort())\n\t\treturn lsh_fun(l, r)\n\n\tdef _rsh(self, l, r, solver):\n\t\trsh_fun = Function('int_rsh', IntSort(), IntSort(), IntSort())\n\t\treturn rsh_fun(l, r)\n\n\tdef _xor(self, l, r, solver):\n\t\txor_fun = Function('int_xor', IntSort(), IntSort(), IntSort())\n\t\treturn xor_fun(l, r)\n\n\tdef _or(self, l, r, solver):\n\t\tor_fun = Function('int_or', IntSort(), IntSort(), IntSort())\n\t\treturn or_fun(l, r)\n\n\tdef _and(self, l, r, solver):\n\t\tand_fun = Function('int_and', IntSort(), IntSort(), IntSort())\n\t\treturn and_fun(l, r)\n"
  },
  {
    "path": "symbolic/z3_wrap.py",
    "content": "# Copyright: see copyright.txt\n\nimport sys\nimport ast\nimport logging\n\nfrom z3 import *\nfrom .z3_expr.integer import Z3Integer\nfrom .z3_expr.bitvector import Z3BitVector\n\nlog = logging.getLogger(\"se.z3\")\n\nclass Z3Wrapper(object):\n\tdef __init__(self):\n\t\tself.N = 32\n\t\tself.asserts = None\n\t\tself.query = None\n\t\tself.use_lia = True\n\t\tself.z3_expr = None\n\n\tdef findCounterexample(self, asserts, query):\n\t\t\"\"\"Tries to find a counterexample to the query while\n\t  \t asserts remains valid.\"\"\"\n\t\tself.solver = Solver()\n\t\tself.query = query\n\t\tself.asserts = self._coneOfInfluence(asserts,query)\n\t\tres = self._findModel()\n\t\tlog.debug(\"Query -- %s\" % self.query)\n\t\tlog.debug(\"Asserts -- %s\" % asserts)\n\t\tlog.debug(\"Cone -- %s\" % self.asserts)\n\t\tlog.debug(\"Result -- %s\" % res)\n\t\treturn res\n\n\t# private\n\n\t# this is very inefficient\n\tdef _coneOfInfluence(self,asserts,query):\n\t\tcone = []\n\t\tcone_vars = set(query.getVars())\n\t\tws = [ a for a in asserts if len(set(a.getVars()) & cone_vars) > 0 ]\n\t\tremaining = [ a for a in asserts if a not in ws ]\n\t\twhile len(ws) > 0:\n\t\t\ta = ws.pop()\n\t\t\ta_vars = set(a.getVars())\n\t\t\tcone_vars = cone_vars.union(a_vars)\n\t\t\tcone.append(a)\n\t\t\tnew_ws = [ a for a in remaining if len(set(a.getVars()) & cone_vars) > 0 ]\n\t\t\tremaining = [ a for a in remaining if a not in new_ws ]\n\t\t\tws = ws + new_ws\n\t\treturn cone\n\n\tdef _findModel(self):\n\t\t# Try QF_LIA first (as it may fairly easily recognize unsat instances)\n\t\tif self.use_lia:\n\t\t\tself.solver.push()\n\t\t\tself.z3_expr = Z3Integer()\n\t\t\tself.z3_expr.toZ3(self.solver,self.asserts,self.query)\n\t\t\tres = self.solver.check()\n\t\t\t#print(self.solver.assertions)\n\t\t\tself.solver.pop()\n\t\t\tif res == unsat:\n\t\t\t\treturn None\n\n\t\t# now, go for SAT with bounds\n\t\tself.N = 32\n\t\tself.bound = (1 << 4) - 1\n\t\twhile self.N <= 64:\n\t\t\tself.solver.push()\n\t\t\t(ret,mismatch) = self._findModel2()\n\t\t\tif (not mismatch):\n\t\t\t\tbreak\n\t\t\tself.solver.pop()\n\t\t\tself.N = self.N+8\n\t\t\tif self.N <= 64: print(\"expanded bit width to \"+str(self.N)) \n\t\t#print(\"Assertions\")\n\t\t#print(self.solver.assertions())\n\t\tif ret == unsat:\n\t\t\tres = None\n\t\telif ret == unknown:\n\t\t\tres = None\n\t\telif not mismatch:\n\t\t\tres = self._getModel()\n\t\telse:\n\t\t\tres = None\n\t\tif self.N<=64: self.solver.pop()\n\t\treturn res\n\n\tdef _setAssertsQuery(self):\n\t\tself.z3_expr = Z3BitVector(self.N)\n\t\tself.z3_expr.toZ3(self.solver,self.asserts,self.query)\n\n\tdef _findModel2(self):\n\t\tself._setAssertsQuery()\n\t\tint_vars = self.z3_expr.getIntVars()\n\t\tres = unsat\n\t\twhile res == unsat and self.bound <= (1 << (self.N-1))-1:\n\t\t\tself.solver.push()\n\t\t\tconstraints = self._boundIntegers(int_vars,self.bound)\n\t\t\tself.solver.assert_exprs(constraints)\n\t\t\tres = self.solver.check()\n\t\t\tif res == unsat:\n\t\t\t\tself.bound = (self.bound << 1)+1\n\t\t\t\tself.solver.pop()\n\t\tif res == sat:\n\t\t\t# Does concolic agree with Z3? If not, it may be due to overflow\n\t\t\tmodel = self._getModel()\n\t\t\t#print(\"Match?\")\n\t\t\t#print(self.solver.assertions)\n\t\t\tself.solver.pop()\n\t\t\tmismatch = False\n\t\t\tfor a in self.asserts:\n\t\t\t\teval = self.z3_expr.predToZ3(a,self.solver,model)\n\t\t\t\tif (not eval):\n\t\t\t\t\tmismatch = True\n\t\t\t\t\tbreak\n\t\t\tif (not mismatch):\n\t\t\t\tmismatch = not (not self.z3_expr.predToZ3(self.query,self.solver,model))\n\t\t\t#print(mismatch)\n\t\t\treturn (res,mismatch)\n\t\telif res == unknown:\n\t\t\tself.solver.pop()\n\t\treturn (res,False)\n\n\tdef _getModel(self):\n\t\tres = {}\n\t\tmodel = self.solver.model()\n\t\tfor name in self.z3_expr.z3_vars.keys():\n\t\t\ttry:\n\t\t\t\tce = model.eval(self.z3_expr.z3_vars[name])\n\t\t\t\tres[name] = ce.as_signed_long()\n\t\t\texcept:\n\t\t\t\tpass\n\t\treturn res\n\t\n\tdef _boundIntegers(self,vars,val):\n\t\tbval = BitVecVal(val,self.N,self.solver.ctx)\n\t\tbval_neg = BitVecVal(-val-1,self.N,self.solver.ctx)\n\t\treturn And([ v <= bval for v in vars]+[ bval_neg <= v for v in vars])\n\n"
  },
  {
    "path": "test/abs_test.py",
    "content": "def abs_test(a,b):\n\tif (a < 0):\n\t\tif (abs(a) == b):\n\t\t\treturn 0\n\t\treturn 1\n\treturn 2\n\ndef expected_result():\n\treturn [0,1,2]\n\n"
  },
  {
    "path": "test/andor.py",
    "content": "def andor(x,y):\n\tif(x or y):\n\t\treturn 1\n\telse:\n\t\treturn 2\n\ndef expected_result():\n\treturn [1,1,2]\n"
  },
  {
    "path": "test/arrayindex2.py",
    "content": "A = [0, 1, 0, 0, 1, 0, 1]\n\ndef arrayindex2(i):\n\n if i in [ j for j in range(len(A)) if A[j] ]:\n   return i\n else:\n   return \"OTHER\"\n\ndef expected_result():\n  return [ 1,4,6, \"OTHER\" ]\n"
  },
  {
    "path": "test/bad_eq.py",
    "content": "def bad_eq(i):\n\tif (0 == i):\n\t\treturn 0\n\treturn 1\n\ndef expected_result():\n\treturn [0,1]\n"
  },
  {
    "path": "test/bignum.py",
    "content": "import sys\n\ndef bignum(a):\n  if a == sys.maxsize:\n    return \"bv\"\n  if a == sys.maxsize+1:\n    return \"bignum\"\n  return \"other\"\n\ndef expected_result():\n  return [ \"other\", \"bv\", \"bignum\" ]\n"
  },
  {
    "path": "test/binary_search.py",
    "content": "from lib.bsearch import *\n\narray = [ 0, 4, 6, 95, 430, 4944, 119101 ]\n\ndef binary_search(k):\n\ti = bsearch(array,k)\n\tif(i>=0):\n\t\tif (not array[i]==k):\n\t\t\treturn \"ERROR\"\n\t\telse:\n\t\t\treturn str(k)\n\telse:\n\t\tif (k in array):\n\t\t\treturn \"ERROR\"\n\t\telse:\n\t\t\treturn \"NOT_FOUND\"\n\ndef expected_result(): \n\treturn [str(i) for i in array] + [ \"NOT_FOUND\" for i in range(len(array)+1) ]\n\n"
  },
  {
    "path": "test/bitwidth.py",
    "content": "def bitwidth(a):\n\tif (a + 1 < a):\n\t\treturn 0\n\telse:\n\t\treturn 1\n\ndef expected_result():\n\treturn [1]\n"
  },
  {
    "path": "test/complex.py",
    "content": "def complex(x,y):\n  if (y >= 1<<32):\n    h = hash(y)\n    print(\"hash(\",y,\") =\",h)\n    if (x == h):\n      if (y == (1<<32) + 203):\n        return 0\n      else:\n        return 1\n    else:\n      return 2\n\ndef expected_result_set():\n\treturn [0,1,2,None]\n\n\n"
  },
  {
    "path": "test/cseppento1.py",
    "content": "def cseppento1(x,y):\n    # based on B2a_IfElse by Lajos Cseppento\n    # see: L. Cseppento: Comparison of Symbolic Execution Based Test Generation Tools, B.Sc. Thesis, Budapest University of Technology and Economics, 2013.\n    if (x > 0 and y > 0):\n        return 1\n    elif (x < 0 and y > 0):\n        return 2\n    elif (x < 0 and y < 0):\n        return 3\n    elif (x > 0 and y < 0):\n        return 4\n    elif (x > 0 and y < 0):\n        # impossible branch , because the previous is the same\n        return -1\n    elif (x == 0 or y == 0):\n        return 0\n    else:\n        # impossible branch\n        return -2\n\ndef expected_result():\n    return [0,0,0,1,2,3,4] # should it be [0,1,2,3,4] instead?"
  },
  {
    "path": "test/cseppento2.py",
    "content": "def cseppento2(a,b):\n    # based on B2c_NonLinear by Lajos Cseppento\n    # see: L. Cseppento: Comparison of Symbolic Execution Based Test Generation Tools, B.Sc. Thesis, Budapest University of Technology and Economics, 2013.\n    if (2 * a * a - 5 * a + 3 == 0 and 2 * b * b - 5 * b + 3 == 0 and a != b):\n        return 1\n    else:\n        return 2\n\ndef expected_result():\n    return [2,2,2] # should it be [2] ?"
  },
  {
    "path": "test/cseppento3.py",
    "content": "def cseppento3(x):\n    # based on B3c_DoWhile by Lajos Cseppento\n    # see: L. Cseppento: Comparison of Symbolic Execution Based Test Generation Tools, B.Sc. Thesis, Budapest University of Technology and Economics, 2013.\n   \n# Sum of the positive integers <= min (x, 100)\n# 1+2+4+5+7+8+...[+98+100]\n    i = 1\n    sum = 0\n    while (i<=x):\n        i = i + 1\n        if (i % 3 == 0):\n            continue\n        if (i > 100):\n            break\n        sum = sum + i\n    return sum\n\n#def expected_result():\n#    return []\n    "
  },
  {
    "path": "test/cvc/effectivebool.py",
    "content": "\"\"\"Tests strings and integers used as a branch condition. The\ninterpreter calls the bool constructor with the contents of the branch\ncondition passed in as the argument. If a __bool__ function does not\nexist, __len__ is called and compared to zero.\"\"\"\nfrom symbolic.args import symbolic\n\n\n@symbolic(string=\"foo\", num=1)\ndef effectivebool(string, num):\n    if string:\n        return 0\n    elif num:\n        return 1\n    else:\n        return 2\n\n\ndef expected_result():\n    return [0, 1, 2]\n"
  },
  {
    "path": "test/cvc/emptystr.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"foo\")\ndef emptystr(s):\n    if s != '':\n        return 0\n    else:\n        return 1\n\n\ndef expected_result():\n    return [0, 1]\n"
  },
  {
    "path": "test/cvc/escape.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(string=\"foo\")\ndef escape(string):\n    if string and '\\\\' not in string and string.find(':') > 0:\n        return 0\n    else:\n        return 1\n\n\ndef expected_result_set():\n    return {0, 1}\n"
  },
  {
    "path": "test/cvc/none.py",
    "content": "from symbolic.args import *\n\n\n@symbolic(c=3)\ndef none(c):\n    if c == None:\n        return 1\n    elif c != None:\n        return 0\n\n\ndef expected_result_set():\n    return {0}"
  },
  {
    "path": "test/cvc/strcontains.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"foo\")\ndef strcontains(s):\n    if \"bar\" in s:\n        return 0\n    else:\n        return 1\n\n\ndef expected_result():\n    return [0, 1]\n"
  },
  {
    "path": "test/cvc/strcount.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"foo\")\ndef strcount(s):\n    if s.count(\"x\") == 2:\n        return 1\n    elif s.count(\"xy\") == 1:\n        return 2\n    elif s.count(\"\") == 3:\n        return 3\n    else:\n        return 0\n\n\ndef expected_result_set():\n    return {0, 1, 2, 3}\n"
  },
  {
    "path": "test/cvc/strfind.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"foo\")\ndef strfind(s):\n    find_idx = s.find(\"bar\")\n    if find_idx == 3:\n        return 0\n    elif find_idx == -1:\n        return 1\n    else:\n        return 2\n\n\ndef expected_result():\n    return [0, 1, 2]\n"
  },
  {
    "path": "test/cvc/strfindbeg.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"foo\")\ndef strfindbeg(s):\n    find_idx = s.find(\"bar\", 1)\n    if find_idx == 3:\n        return 0\n    elif find_idx == -1:\n        return 1\n    else:\n        return 2\n\n\ndef expected_result():\n    return [0, 1, 2]\n"
  },
  {
    "path": "test/cvc/strindex.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"foobar\")\ndef strindex(s):\n    \"\"\"Test case does not currently test negative indexes.\n    It is also currently unclear how we want to handle concrete\n    executions that raise errors. Currently the error stops\n    execution and prevents a branch predicate from forming.\"\"\"\n    if s[4] == 'Q':\n        return 0\n    else:\n        return 1\n\n\ndef expected_result():\n    return [0, 1]\n"
  },
  {
    "path": "test/cvc/stringadd.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"foo\")\ndef stringadd(s):\n    x = s + \"bar\"\n    if x == \"nobar\":\n        return 0\n    return 1\n\n\ndef expected_result():\n    return [0, 1]\n"
  },
  {
    "path": "test/cvc/stringtest.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"foo\")\ndef stringtest(s):\n    if (s == \"bar\"):\n        return 0\n    elif (s == '\\\\'):\n        return 2\n    else:\n        return 1\n\n\ndef expected_result_set():\n    return {0, 1, 2}\n"
  },
  {
    "path": "test/cvc/strmiddle.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"x\")\ndef strmiddle(s):\n    x = \"A\"+s+\"C\"\n    if \"B\" in x:\n        return 0\n    else:\n        return 1\n\n\ndef expected_result():\n    return [0, 1]\n"
  },
  {
    "path": "test/cvc/strreplace.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"bar\")\ndef strreplace(s):\n    if \"faa\" == s.replace(\"o\", \"a\"):\n        return 0\n    else:\n        return 1\n\n\ndef expected_result_set():\n    return {0, 1}\n"
  },
  {
    "path": "test/cvc/strslice.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"foo\")\ndef strslice(s):\n    if '\\\\' not in s and s[0:2] == \"//\":\n        return 0\n    return 1\n\n\ndef expected_result_set():\n    return {0, 1}\n"
  },
  {
    "path": "test/cvc/strsplit.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"foo\")\ndef strsplit(s):\n    if ['a', 'b'] == s.split(\"&\"):\n        return 0\n    return 1\n\n\ndef expected_result_set():\n    return {0, 1}\n"
  },
  {
    "path": "test/cvc/strstartswith.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"foo\")\ndef strstartswith(s):\n    if s.startswith('abc'):\n        return 0\n    return 1\n\n\ndef expected_result_set():\n    return {0, 1}\n"
  },
  {
    "path": "test/cvc/strstrip.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"foo\")\ndef strstrip(s):\n    if \" \" in s and \"abc\" == s.strip():\n        return 0\n    return 1\n\n\ndef expected_result_set():\n    return {0, 1}\n"
  },
  {
    "path": "test/cvc/strsubstring.py",
    "content": "from symbolic.args import symbolic\n\n\n@symbolic(s=\"foo\")\ndef strsubstring(s):\n    \"\"\"Test case for Python slicing, negative\n    indices and steps are not currently tested.\"\"\"\n    if s[2:] == \"obar\":\n        return 0\n    elif s[:2] == \"bb\":\n        return 1\n    elif s[1:3] == \"bb\":\n        return 2\n    else:\n        return 3\n\n\ndef expected_result():\n    return [0, 1, 2, 3]\n"
  },
  {
    "path": "test/decorator.py",
    "content": "from symbolic.args import *\n\n@concrete(a=1,b=2)\n@symbolic(c=3)\ndef decorator(a,b,c):\n\tif a+b+c == 6:\n\t\treturn 0\n\telse:\n\t\treturn 1\n\ndef expected_result():\n\treturn [0,1]"
  },
  {
    "path": "test/decorator_dict.py",
    "content": "from symbolic.args import *\n\n@symbolic(d=dict([(42,6)]))\ndef decorator_dict(d):\n\tif d[42] == 6:\n\t\treturn 0\n\telse:\n\t\treturn 1\n\ndef expected_result():\n\treturn [0]"
  },
  {
    "path": "test/diamond.py",
    "content": "def diamond(x,y,z):\n\tret = 0\n\tif (x):\n\t\tret = ret + 1\n\tif (y):\n\t\tret = ret + 1\n\tif (z):\n\t\tret = ret + 1\n\treturn ret\n\ndef expected_result():\n\treturn [ 0, 1, 1, 1, 2, 2, 2, 3]"
  },
  {
    "path": "test/dict.py",
    "content": "D =  dict({ (101,2), (1,3), (4,9) })\n\ndef dict(x):\n    # if x in D.keys():\n    if x in [ j for j in D.keys() ]:\n       return D[x]\n    else:\n       return \"NONE\"\n\ndef expected_result():\n    return [2,3,9,\"NONE\"]\n"
  },
  {
    "path": "test/dictionary.py",
    "content": "# Copyright: see copyright.txt\n\n# Test if engine explores all paths\n\nfrom lib.se_dict import SymbolicDictionary\n\ndef dictionary(in1):\n    d = SymbolicDictionary({})\n    d[3] = 10\n\n    if d.has_key(in1):\n        return 1\n    else:\n        return 2\n\ndef expected_result():\n\treturn [1,2]"
  },
  {
    "path": "test/elseif.py",
    "content": "# Copyright: see copyright.txt\n\n# Test if engine explores all paths\n\ndef elseif(in1):\n    if in1 ==  0:\n        return 0\n    elif in1 == 1:\n        return 1\n    elif in1 == 2:\n        return 2\n    elif in1 == 3:\n        return 3\n    elif in1 == 4:\n        return 4\n    elif in1 == 5:\n        return 5\n    elif in1 == 6:\n        return 6\n    elif in1 == 7:\n        return 7\n    elif in1 == 8:\n        return 8\n    else:\n        return 9\n    return 10\n\ndef expected_result():\n\treturn [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]"
  },
  {
    "path": "test/expand.py",
    "content": "def expand(in1, in2):\n    if (in1 + in2 >= 1 << 32): \n        return 0\n    else:\n        return 1\n\ndef expected_result():\n\treturn [0,1]\n"
  },
  {
    "path": "test/expressions.py",
    "content": "# Copyright: see copyright.txt\n\ndef expressions(in1, in2):\n    a = in1\n    b = in2 + 47\n    c = a * b\n    # only solution should be a==1, b==6 (assuming Python actually uses 32-bit integers)\n    if c==53: # a > 0 and a < 20 and b < 100 and c == 53:\n        d = -1\n    else:\n        d = 0\n\n    return d\n\ndef expected_result():\n\treturn [-1,0]\n"
  },
  {
    "path": "test/filesys.py",
    "content": "import os\nimport sys\n\ndef filesys(x):\n\t# if file doesn't exist, create it and \n\tg = 1\n\tif (os.path.exists(\"tmp.txt\")):\n\t\tfile = open(\"tmp.txt\",\"r\")\n\t\tg = int(file.readline())\n\t\tg = g + 3\n\t\tfile.close()\n\n\tfile = open(\"tmp.txt\",\"w\")\n\tfile.write(str(g))\n\tfile.close()\n\n\tif (x == g):\n\t\treturn 0\n\telif (x == g+1):\n\t\treturn 1\t\n\telse:\n\t\treturn 2\t\t\n\ndef expected_result_set():\n\treturn [2]\n"
  },
  {
    "path": "test/fp.py",
    "content": "def fp(a):\n\tif a % 2 == 0:\n\t\t# The next condition basically checks whether a is zero (in a weird way, though).\n\t\tif a == int(a / 2):\n\t\t\treturn 0\n\t\telse:\n\t\t\treturn 1\n\telse:\n\t\treturn 2\n\ndef expected_result():\n    return [0, 1, 2]"
  },
  {
    "path": "test/gcd.py",
    "content": "def rec_gcd(u,v):\n  if (u == v):\n    return u\n\n  if (u == 0):\n    return v\n\n  if (v == 0):\n    return u\n \n  if not (u & 1):  # u is even\n    if v & 1:      #v is odd\n      return rec_gcd(u >> 1, v)\n    else:            # both u and v are even\n      return rec_gcd(u >> 1, v >> 1) << 1\n  else:\n    if not (v & 1):  # u is odd, v is even\n      return rec_gcd(u, v >> 1)\n \n    # reduce larger argument\n    if (u > v):\n      return rec_gcd((u - v) >> 1, v)\n\n    return rec_gcd((v - u) >> 1, u)\n\ndef iter_gcd(u,v):\n  # GCD(0,v) == v; GCD(u,0) == u, GCD(0,0) == 0\n  if (u == 0): \n    return v\n  if (v == 0):\n    return u\n \n  # Let shift := lg K, where K is the greatest power of 2\n  # dividing both u and v.\n  shift = 0\n  while ((u | v) & 1) == 0:\n    u = u >> 1\n    v = v >> 1\n    shift = shift + 1\n\n  while ((u & 1) == 0):\n    u = u >> 1\n \n  # From here on, u is always odd.\n  while (True):\n    # remove all factors of 2 in v -- they are not common\n    # note: v is not zero, so while will terminate\n    while ((v & 1) == 0):\n      v = v >> 1\n \n    # Now u and v are both odd. Swap if necessary so u <= v,\n    # then set v = v - u (which is even). For bignums, the\n    # swapping is just pointer movement, and the subtraction\n    # can be done in-place.\n    if (u > v):\n       u, v = v, u\n    v = v - u     # Here v >= u.\n    if (v == 0):\n      break\n \n  # restore common factors of 2\n  return u << shift\n\ndef gcd(x,y):\n  if (x>=0 and y>=0):\n     rec_result = rec_gcd(x,y)\n     iter_result = iter_gcd(x,y)\n     if (rec_result != iter_result):\n       return \"ERROR\"\n     else:\n       return \"OK\"\n  else:\n     return \"PRE\"\n\ndef expected_result_set(): \n  return [\"PRE\",\"OK\"]\n\n\n"
  },
  {
    "path": "test/hashval.py",
    "content": "# Created by Thomas Ball (2014)\n\ndef compute(x):\n\tres = x\n\tres = res + (res << 10)\n\tres = res ^ (res >> 6)\n\treturn res\n\t\ndef hashval(key):\n\thv = compute(key)\n\tif (hv == key + 1):\n\t\treturn 0\n\telse:\n\t\treturn 1\n\ndef expected_result():\n\treturn [1]\n\n"
  },
  {
    "path": "test/len_test.py",
    "content": "from symbolic.args import *\n\nclass Foo():\n    def __init__(self, var):\n        self.var = var\n\n    def __len__(self):\n        return self.var\n\n@symbolic(a=0)\ndef len_test(a):\n    f = Foo(a)\n    if len(f) == 2:\n        return 1\n    else:\n        return 0\n\ndef expected_result():\n    return [0, 1]\n\n"
  },
  {
    "path": "test/lib/__init__.py",
    "content": "# hello there1\n"
  },
  {
    "path": "test/lib/bsearch.py",
    "content": "\ndef bsearch(a,k):\n\tlo, hi = 0, len(a)-1\n\twhile(lo <= hi):\n\t\tmid = (lo+hi) >> 1\n\t\tif (a[mid] == k):\n\t\t\treturn mid\n\t\telif (a[mid] < k):\n\t\t\tlo = mid+1\n\t\telse:\n\t\t\thi = mid-1\n\treturn -1\n"
  },
  {
    "path": "test/lib/se_dict.py",
    "content": "# Copyright: see copyright.txt\n\nclass SymbolicDictionary(dict):\n\tdef __init__(self, *args, **kwargs):\n\t\tdict.__init__(self)\n\t\tself.update(*args, **kwargs)\n\n\tdef has_key(self, key):\n\t\tfor k in self.keys():\n\t\t\tif k == key:\n\t\t\t\treturn True\n\t\treturn False\n\n"
  },
  {
    "path": "test/list.py",
    "content": "def list(a,b):\n   if ([a,b] == [1,2]):\n      return 1\n   else:\n      return 2\n\ndef expected_result_set():\n   return [1,2]"
  },
  {
    "path": "test/logical_op.py",
    "content": "# Copyright: see copyright.txt\n\ndef logical_op(in1, in2):\n    if (in1 & in2) == 1:\n        if (in1 & in2) == 7:\n            return 1\n        else:\n            return 2\n    else:\n        return 3\n\n    return 0\n\ndef expected_result():\n\treturn [2,3]"
  },
  {
    "path": "test/loop.py",
    "content": "def loop(in1,in2):\n    if in1 > in2:\n        i = 0\n        while i < in1 and False:\n            i = i+1\n        return 1\n    else:\n        return 0\n        \ndef expected_result():\n    return [0,1,1]"
  },
  {
    "path": "test/many_branches.py",
    "content": "# Copyright: see copyright.txt\n\n# Test if engine explores all paths\n\ndef many_branches(in1, in2, in3):\n    if in1 == 0:\n        if in2 == 5:\n            if in3 == 10:\n                return 1\n            else:\n                return 2\n        else:\n            if in3 <= 3:\n                return 3\n            else:\n                return 4\n    else:\n        if in1 == 1:\n            if in2 == 7:\n                return 5\n            else:\n                return 6\n        else:\n            if in2 == 9:\n                return 7\n            else:\n                return 8\n\n    return 0\n\ndef expected_result():\n\treturn [1, 2, 3, 4, 5, 6, 7, 8]"
  },
  {
    "path": "test/maxtest.py",
    "content": "# Created by Thomas Ball (2014)\n#\n\ndef max2(s,t):\n\tif (s < t):\n\t\treturn t\n\telse:\n\t\treturn s\n\ndef max4(x,y,x2,y2):\n\treturn max2(max2(x,y),max2(x2,y2))\n\ndef maxtest(a,b,c,d):\n\tm = max4(a,b,c,d)\n\tif (m < a or m < b or m < c or m < d):\n\t\treturn \"ERROR\"\n\telse:\n\t\treturn \"OK\"\n\ndef expected_result():\n\treturn [ \"OK\" for i in range(8) ]"
  },
  {
    "path": "test/mod.py",
    "content": "def mod(x,y):\n\tif 0 < y < 10 and x % (y+1) == 3:\n\t\treturn 0\n\telse:\n\t\treturn 1\n\ndef expected_result():\n\treturn [0,1,1,1]\n\n"
  },
  {
    "path": "test/modulo.py",
    "content": "def modulo(in1):\n    if in1 % 3 != 0:\n        return 1\n    elif in1 % 5 != 0:\n        return 2\n    return 0\n    \ndef expected_result():\n    return [0,1,2]"
  },
  {
    "path": "test/modulo2.py",
    "content": "def modulo2(in1):\n    if (in1 <= 0):\n        return -1\n\n    if in1 % 3 != 0:\n        return 1\n    elif in1 % 5 != 0:\n        return 2\n    return 0\n    \ndef expected_result():\n    return [-1,0,1,2]"
  },
  {
    "path": "test/mult_assmt.py",
    "content": "\ndef mult_assmt(in1,in2,in3):\n    v = in1\n    if v == in2:\n        if in3 > 0:\n            v = v + 1\n        \n        if v != in2:\n            return 1\n        else:\n            return 2\n    \n    return 0\n\ndef expected_result():\n    return [0,1,2]"
  },
  {
    "path": "test/polyspace.py",
    "content": "def where_are_the_errors(input):\n\tk = input // 100\n\tx = 2\n\ty = k + 5\n\twhile x < 10:\n\t\tx = x + 1\n\t\ty = y + 3\n\tif (3*k + 100) > 43:\n\t\ty = y + 1\n\t\t# x == 10, y == k+30\n\t\tif x - y == 0:\n\t\t\t# unreachable\n\t\t\treturn \"DIVZERO\"\n\t\tx = x // (x - y)\n\treturn x\n\ndef polyspace(i):\n\tret = where_are_the_errors(i)\n\treturn ret\n\t\ndef expected_result():\n\treturn [-1,10]"
  },
  {
    "path": "test/power.py",
    "content": "# currently, we don't handle the power operator symbolically,\n# so this test is not expected to cover the else branch. The\n# initial (concrete) value of x is zero, so we expect only to\n# cover the then branch.\n\ndef power(x):\n\tif (x+2) ** 2 == 4:\n\t\treturn 0\n\telse:\n\t\treturn 1\n\ndef expected_result():\n\treturn [0]\n\n"
  },
  {
    "path": "test/power2.py",
    "content": "def power2(x):\n\tb = x * x;\n\tif b > 0:\n\t\treturn 0\n\telif b == 0:\n\t\treturn 1\n\telse:\n\t\t# This path is not possible if it is guaranteed that overflows do not occur.\n\t\treturn 2\n\ndef expected_result():\n\treturn [0, 1]\n\n"
  },
  {
    "path": "test/powtest.py",
    "content": "def powtest(in1):\n    if in1*in1 == 0:\n        return 1\n    elif in1*in1 > 0:\n        return 2\n    return 0\n\ndef expected_result():\n    return [1,2]"
  },
  {
    "path": "test/reverse.py",
    "content": "# check that sub and rsub are modelled properly\n\ndef reverse(x):\n\tif (x - 5 == 0):\n\t\treturn 0\n\treturn 1\n\ndef expected_result():\n\treturn [0,1]\n\t"
  },
  {
    "path": "test/set.py",
    "content": "S =  { 1, 3, 9, 12, 15, 19 }\n\ndef set(x):\n    if x in [ j for j in S]:\n       return x\n    else:\n       return \"NONE\"\n\ndef expected_result():\n    return [1,3,9,12,15,19,\"NONE\"]\n"
  },
  {
    "path": "test/shallow_branches.py",
    "content": "# Copyright: see copyright.txt\n\n# Test if engine explores all paths\n\ndef shallow_branches(in1, in2, in3, in4, in5):\n    if in1 ==  0:\n        return 1\n\n    if in2 ==  0:\n        return 3\n\n    if in3 ==  0:\n        return 5\n\n    if in4 ==  0:\n        return 7\n\n    if in5 ==  0:\n        return 9\n\n    return 0\n\ndef expected_result():\n\treturn [0, 1, 3, 5, 7, 9]"
  },
  {
    "path": "test/simple.py",
    "content": "def simple(x):\n\tif (x+1 > 10):\n\t\treturn 42\n\telse:\n\t\treturn 43\n\ndef expected_result():\n\treturn [ 42, 43]"
  },
  {
    "path": "test/swap.py",
    "content": "def swap(in1,in2):\n    if in1 > in2:\n        # swap in1 and in2\n        in1 = in1 ^ in2;\n        in2 = in1 ^ in2;\n        in1 = in1 ^ in2;\n        \n        if in1 > in2:\n            # impossible\n            return 1\n        else:\n            return 2\n    else:\n        return 0\n        \ndef expected_result():\n    return [0,2]"
  },
  {
    "path": "test/tmp",
    "content": ""
  },
  {
    "path": "test/tuplecmp.py",
    "content": "def tuplelt(a, b):\n  a0, a1 = a\n  b0, b1 = b\n\n  if a0 < b0:\n    return True\n  elif a1 < b1:\n    return True\n  return False\n\ndef tuplecmp(a0, a1, b0, b1):\n  return tuplelt((a0, a1), (b0, b1))\n\ndef expected_result():\n  return [True, True, False]\n"
  },
  {
    "path": "test/unnecessary_condition.py",
    "content": "def unnecessary_condition(in1,in2):\n    if in1 > in2:\n        unnec = in1 < in2 and False\n        if unnec:\n            return 2\n        return 1\n    else:\n        return 0\n        \ndef expected_result():\n    return [0,1]"
  },
  {
    "path": "test/unnecessary_condition2.py",
    "content": "def unnecessary_condition2(in1,in2):\n    if in1 > in2:\n        # unnec always false\n        unnec = in1 < 5 and False # this is a branching point because of (in1 < 5)\n        if unnec: # this is not a real branching point\n            return 2\n        return 1\n    else:\n        return 0\n        \ndef expected_result():\n    return [0,1,1]"
  },
  {
    "path": "test/unnecessary_condition3.py",
    "content": "def unnecessary_condition3(in1):\n    if in1 > 0:\n        # in1 > 0\n        return op(in1) + 10\n    else:\n        return op(in1) + 20\n        \ndef op(in1):\n    if in1 < -10:\n        return 1\n    if in1 < -5:\n        return 2\n    if in1 < -3:\n        return 3\n    if in1 < 0:\n        return 4\n        \n    return 0\n    \ndef expected_result():\n    return [10,20,21,22,23,24]"
  },
  {
    "path": "test/unnecessary_condition4.py",
    "content": "def unnecessary_condition4(in1):\n    v = op(in1)\n\n    if in1 > 0:\n        # in1 > 0\n        return v + 10\n    else:\n        return v + 20\n        \ndef op(in1):\n    if in1 < -10:\n        return 1\n    if in1 < -5:\n        return 2\n    if in1 < -3:\n        return 3\n    if in1 < 0:\n        return 4\n        \n    return 0\n    \ndef expected_result():\n    return [10,20,21,22,23,24]"
  },
  {
    "path": "test/weird.py",
    "content": "def weird(x,y):\n\tif (x < y) + x == 1:\n\t\treturn 0\n\telse:\n\t\treturn 1\n\ndef expected_result():\n\treturn [0,1]\n\n#print weird(0,1)\n#print weird(2,3)\n"
  },
  {
    "path": "test/whileloop.py",
    "content": "def whileloop(x):\n\ty = 1\n\twhile (0 <= x < 10 and y<=x):\n\t\ty = y + 1\n\n"
  },
  {
    "path": "tools/symbolic_int_subtype.py",
    "content": "# Check if SymbolicInteger is a subtype of `int' modulo concolic execution.\n#\n# To run:\n# $ python sym_exec.py examples/symbolic_int_subtype.py\n\nfrom inspect import signature\n\nfrom symbolic.symbolic_types.symbolic_int import SymbolicInteger\n\n# Collect integer methods to test.\nINT_FUNCS = []\nfor fname in dir(int):\n  # TODO for some functions signature lookup fails, maybe hardcode arity\n  try:\n    if fname in ('__delattr__', '__getattribute__', '__setattr__',\n                 '__new__', '__init__'):\n      continue\n    INT_FUNCS.append((fname, len(signature(getattr(int, fname)).parameters)))\n  except:\n    pass\n\n# NOTE Uncomment this to have a restricted set of (correctly modelled) functions\n# INT_FUNCS = [ ('__add__', 2), ('__sub__', 2), ('__and__', 2), ('__or__', 2) ]\n\ndef symbolic_int_subtype(func_index, a, b, c):\n  # make concolic execution branch on elements of INT_FUNCS\n  if func_index in [ i for i, fname in enumerate(INT_FUNCS) if INT_FUNCS[i] ]:\n    name, arity = INT_FUNCS[func_index]\n    print(\"%s/%d\" % (name, arity))\n\n    assert(0 <= arity <= 3)\n\n    int_params = [ int(a), int(b), int(c) ]\n    sym_params = [ a, b, c ]\n\n    int_func = getattr(int, name)\n    sym_func = getattr(SymbolicInteger, name)\n\n    try:\n      int_return = int_func(*int_params[:arity])\n    except ValueError as e:\n      int_return = e\n    except ZeroDivisionError as e:\n      int_return = e\n\n    try:\n      sym_return = sym_func(*sym_params[:arity])\n    except ValueError as e:\n      sym_return = e\n    except ZeroDivisionError as e:\n      sym_return = e\n\n    same = ((isinstance(int_return, Exception) and\n             isinstance(sym_return, Exception) and\n             type(int_return) == type(sym_return)\n            ) or int_return == sym_return)\n\n    if not same:\n      s = (\"\\nint %s(%s): %r,\\nsym %s(%s): %r\" %\n           (name, ','.join(str(i) for i in int_params), int_return,\n           name, ','.join(str(i) for i in sym_params), sym_return))\n      raise Exception(s)\n\n    return (name, same)\n\n  else:\n    return \"OUTSIDE\"\n\n# TODO Implement oracle.\n#      How to predict the number of paths for a given int method?\n"
  },
  {
    "path": "utils.py",
    "content": "# Copyright: see copyright.txt\n\nimport sys\nimport traceback\n\ndef traceback():\n\tstack = traceback.format_stack()\n\trest = stack[:-2]\n\treturn \"\\n\"+\"\".join(rest)\n\ndef crash(msg):\n\t#stack = _traceback()\n\tprint(msg)\n\tsys.exit(-1)\n"
  },
  {
    "path": "vagrant.sh",
    "content": "# Configuration\nINSTALLDIR=/home/vagrant/pyex\n\n# System Maintenance \napt-get update\napt-get -y upgrade\n\n# Dependencies\napt-get -y install python3\napt-get -y install graphviz graphviz-dev\n\n## Z3\napt-get -y install g++\napt-get -y install python3-examples\ncd /tmp\ngit clone https://github.com/Z3Prover/z3.git\ncd z3\ngit checkout -b unstable origin/unstable\npython scripts/mk_make.py\ncd build\nmake\nmake install\ncp libz3.so /usr/lib/python3/dist-packages\npython3 /usr/share/doc/python3.2/examples/scripts/reindent.py z3*.py\ncp z3*.py /usr/lib/python3/dist-packages\ncp z3*.pyc /usr/lib/python3/dist-packages\ncd\n\n## CVC4\napt-get install -y libgmp-dev\napt-get install -y libboost-all-dev\napt-get install -y openjdk-7-jre openjdk-7-jdk\napt-get install -y swig\napt-get install -y python3-dev\ncd /tmp\ngit clone https://github.com/CVC4/CVC4.git\ncd CVC4\n./autogen.sh\ncontrib/get-antlr-3.4\nexport PYTHON_CONFIG=/usr/bin/python3.2-config\n./configure --enable-optimized --with-antlr-dir=/tmp/CVC4/antlr-3.4 ANTLR=/tmp/CVC4/antlr-3.4/bin/antlr3 --enable-language-bindings=python\necho \"python_cpp_SWIGFLAGS = -py3\" >> src/bindings/Makefile.am\nautoreconf\nmake\nmake doc\nmake install\necho \"/usr/local/lib\" > /etc/ld.so.conf.d/cvc4.conf\n/sbin/ldconfig\ncp builds/src/bindings/python/CVC4.py /usr/lib/python3/dist-packages/CVC4.py\ncp builds/src/bindings/python/.libs/CVC4.so /usr/lib/python3/dist-packages/_CVC4.so\ncd\n\n# Installation\nln -s /vagrant $INSTALLDIR\nln -s $INSTALLDIR/symbolic /usr/lib/python3/dist-packages/\ncat > /usr/bin/pyex <<EOF\n#/bin/sh\nPYTHONPATH=\\$PYTHONPATH:\"\\$(pwd)\" python3 $INSTALLDIR/pyexz3.py \\$*\nEOF\nchmod a+x /usr/bin/pyex\n\n# Tests\ncd $INSTALLDIR\npython3 run_tests.py test\n"
  }
]