Repository: srush/Transformer-Puzzles
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gitextract_c6so92vp/
├── .gitignore
├── LICENSE
├── README.md
└── TransformerPuzzlers.ipynb
================================================
FILE CONTENTS
================================================
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FILE: .gitignore
================================================
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================================================
FILE: LICENSE
================================================
MIT License
Copyright (c) 2023 Sasha Rush
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
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IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
================================================
FILE: README.md
================================================
# Transformer Puzzles
This notebook is a collection of short coding puzzles based on the internals of the Transformer. The puzzles are written in Python and can be done in this notebook. After completing these you will have a much better intutive sense of how a Transformer can compute certain logical operations.
These puzzles are based on [Thinking Like Transformers](https://arxiv.org/pdf/2106.06981.pdf) by Gail Weiss, Yoav Goldberg, Eran Yahav and derived from this [blog post](https://srush.github.io/raspy/).
## Goal
**Can we produce a Transformer that does basic elementary school addition?**
i.e. given a string "19492+23919" can we produce the correct output?
## Rules
Each exercise consists of a function with a argument `seq` and output `seq`. Like a transformer we cannot change length. Operations need to act on the entire sequence in parallel. There is a global `indices` which tells use the position in the sequence. If we want to do something different on certain positions we can use `where` like in Numpy or PyTorch. To run the seq we need to give it an initial input.
```python colab={"base_uri": "https://localhost:8080/", "height": 96} id="1b28dc98" outputId="f1ac1157-3db8-40c0-dbb2-7d9bad8943a0"
def even_vals(seq=tokens):
"Keep even positions, set odd positions to -1"
x = indices % 2
# Note that all operations broadcast so you can use scalars.
return where(x == 0, seq, -1)
seq = even_vals()
# Give the initial input tokens
seq.input([0,1,2,3,4])
```
The main operation you can use is "attention". You do this by defining a selector which forms a matrix based on `key` and `query`.
```python colab={"base_uri": "https://localhost:8080/", "height": 176} id="e2ee0ff8" outputId="a61ac19c-2550-4f3c-d653-50c323cdfd59"
before = key(indices) < query(indices)
before
```
We can combine selectors with logical operations.
```python colab={"base_uri": "https://localhost:8080/", "height": 201} id="c315ba6d" outputId="270d50fa-649c-438b-8606-d3d078478162"
before_or_same = before | (key(indices) == query(indices))
before_or_same
```
Once you have a selector, you can apply "attention" to sum over the grey positions. For example to compute cumulative such we run the following function.
```python colab={"base_uri": "https://localhost:8080/", "height": 326} id="e79c8c8b" outputId="44db7f90-502d-497c-c5ba-4062c09f0a9a"
def cumsum(seq=tokens):
return before_or_same.value(seq)
seq = cumsum()
seq.input([0, 1, 2, 3, 4])
```
Good luck!
================================================
FILE: TransformerPuzzlers.ipynb
================================================
{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"id": "view-in-github",
"colab_type": "text"
},
"source": [
""
]
},
{
"cell_type": "markdown",
"id": "f8117f57",
"metadata": {
"id": "f8117f57"
},
"source": [
"# Transformer Puzzles"
]
},
{
"cell_type": "markdown",
"id": "e9e822cb",
"metadata": {
"id": "e9e822cb"
},
"source": [
"This notebook is a collection of short coding puzzles based on the internals of the Transformer. The puzzles are written in Python and can be done in this notebook. After completing these you will have a much better intutive sense of how a Transformer can compute certain logical operations. \n",
"\n",
"These puzzles are based on [Thinking Like Transformers](https://arxiv.org/pdf/2106.06981.pdf) by Gail Weiss, Yoav Goldberg, Eran Yahav and derived from this [blog post](https://srush.github.io/raspy/)."
]
},
{
"cell_type": "markdown",
"id": "8e962052",
"metadata": {
"id": "8e962052"
},
"source": [
"## Goal\n",
"\n",
"**Can we produce a Transformer that does basic addition?**\n",
"\n",
"i.e. given a string \"19492+23919\" can we produce the correct output? "
]
},
{
"cell_type": "markdown",
"id": "d332140b",
"metadata": {
"id": "d332140b"
},
"source": [
"## Rules\n",
"\n",
"Each exercise consists of a function with a argument `seq` and output `seq`. Like a transformer we cannot change length. Operations need to act on the entire sequence in parallel. There is a global `indices` which tells use the position in the sequence. If we want to do something different on certain positions we can use `where` like in Numpy or PyTorch. To run the seq we need to give it an initial input. "
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "6c5c885a",
"metadata": {
"id": "6c5c885a"
},
"outputs": [],
"source": [
"%%capture\n",
"!pip install -qqq git+https://github.com/chalk-diagrams/chalk git+https://github.com/srush/RASPy "
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "51724e11",
"metadata": {
"id": "51724e11"
},
"outputs": [],
"source": [
"from IPython.display import display, HTML\n",
"from raspy import key, query, tokens, indices, where, draw\n",
"import random"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1b28dc98",
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 96
},
"id": "1b28dc98",
"outputId": "f1ac1157-3db8-40c0-dbb2-7d9bad8943a0"
},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
"[0, -1, 2, -1, 4]"
],
"image/svg+xml": "\n"
},
"metadata": {},
"execution_count": 3
}
],
"source": [
"def even_vals(seq=tokens):\n",
" \"Keep even positions, set odd positions to -1\"\n",
" x = indices % 2\n",
" # Note that all operations broadcast so you can use scalars.\n",
" return where(x == 0, seq, -1)\n",
"seq = even_vals()\n",
"\n",
"# Give the initial input tokens\n",
"seq.input([0,1,2,3,4])"
]
},
{
"cell_type": "markdown",
"id": "9dc23f88",
"metadata": {
"id": "9dc23f88"
},
"source": [
"The main operation you can use is \"attention\". You do this by defining a selector which forms a matrix based on `key` and `query`."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e2ee0ff8",
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 176
},
"id": "e2ee0ff8",
"outputId": "a61ac19c-2550-4f3c-d653-50c323cdfd59"
},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
"Selector Layer 0\n",
" 0 1 2 3 4\n",
" |h e l l o|\n",
" 0 1 2 3 4\n",
" |---------|\n",
"0 h 0 | 1 1 1 1\n",
"1 e 1 | 1 1 1\n",
"2 l 2 | 1 1\n",
"3 l 3 | 1\n",
"4 o 4 | "
],
"image/svg+xml": "\n"
},
"metadata": {},
"execution_count": 4
}
],
"source": [
"before = key(indices) < query(indices)\n",
"before"
]
},
{
"cell_type": "markdown",
"id": "a4de0a14",
"metadata": {
"id": "a4de0a14"
},
"source": [
"We can combine selectors with logical operations."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c315ba6d",
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 201
},
"id": "c315ba6d",
"outputId": "270d50fa-649c-438b-8606-d3d078478162"
},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
"Selector Layer 0\n",
" 0 1 2 3 4\n",
" |h e l l o|\n",
" 0 1 2 3 4\n",
" 0 1 2 3 4\n",
" |---------|\n",
"0 h 0 | 1 1 1 1 1\n",
"1 e 1 | 1 1 1 1\n",
"2 l 2 | 1 1 1\n",
"3 l 3 | 1 1\n",
"4 o 4 | 1"
],
"image/svg+xml": "\n"
},
"metadata": {},
"execution_count": 5
}
],
"source": [
"before_or_same = before | (key(indices) == query(indices))\n",
"before_or_same"
]
},
{
"cell_type": "markdown",
"id": "00bc66a3",
"metadata": {
"id": "00bc66a3"
},
"source": [
"Once you have a selector, you can apply \"attention\" to sum over the grey positions. For example to compute cumulative such we run the following function. "
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e79c8c8b",
"metadata": {
"lines_to_next_cell": 1,
"colab": {
"base_uri": "https://localhost:8080/",
"height": 326
},
"id": "e79c8c8b",
"outputId": "44db7f90-502d-497c-c5ba-4062c09f0a9a"
},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
"[0, 1, 3, 6, 10]"
],
"image/svg+xml": "\n"
},
"metadata": {},
"execution_count": 6
}
],
"source": [
"def cumsum(seq=tokens):\n",
" return before_or_same.value(seq)\n",
"seq = cumsum()\n",
"seq.input([0, 1, 2, 3, 4])"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "9ae57559",
"metadata": {
"cellView": "form",
"id": "9ae57559"
},
"outputs": [],
"source": [
"#@title Test Code (Collapse)\n",
"def atoi(seq=tokens):\n",
" return seq.map(lambda x: ord(x) - ord('0'))\n",
"\n",
"def test_output(user, spec, token_sets):\n",
" for ex_num, token_set in enumerate(token_sets): \n",
" out1 = user(*token_set[:-1])((token_set[-1]))\n",
" out2 = spec(*token_set)\n",
" print(f\"Example {ex_num}. Args:\", token_set, \"Expected:\", out2)\n",
" display(out1)\n",
" out1 = out1.toseq()\n",
" for i, o in enumerate(out2):\n",
" assert out1[i] == o, f\"Output: {out1} Expected: {out2}\"\n",
"\n",
" pups = [\n",
" \"2m78jPG\",\n",
" \"pn1e9TO\",\n",
" \"MQCIwzT\",\n",
" \"udLK6FS\",\n",
" \"ZNem5o3\",\n",
" \"DS2IZ6K\",\n",
" \"aydRUz8\",\n",
" \"MVUdQYK\",\n",
" \"kLvno0p\",\n",
" \"wScLiVz\",\n",
" \"Z0TII8i\",\n",
" \"F1SChho\",\n",
" \"9hRi2jN\",\n",
" \"lvzRF3W\",\n",
" \"fqHxOGI\",\n",
" \"1xeUYme\",\n",
" \"6tVqKyM\",\n",
" \"CCxZ6Wr\",\n",
" \"lMW0OPQ\",\n",
" \"wHVpHVG\",\n",
" \"Wj2PGRl\",\n",
" \"HlaTE8H\",\n",
" \"k5jALH0\",\n",
" \"3V37Hqr\",\n",
" \"Eq2uMTA\",\n",
" \"Vy9JShx\",\n",
" \"g9I2ZmK\",\n",
" \"Nu4RH7f\",\n",
" \"sWp0Dqd\",\n",
" \"bRKfspn\",\n",
" \"qawCMl5\",\n",
" \"2F6j2B4\",\n",
" \"fiJxCVA\",\n",
" \"pCAIlxD\",\n",
" \"zJx2skh\",\n",
" \"2Gdl1u7\",\n",
" \"aJJAY4c\",\n",
" \"ros6RLC\",\n",
" \"DKLBJh7\",\n",
" \"eyxH0Wc\",\n",
" \"rJEkEw4\"]\n",
" print(\"Success!\")\n",
" return HTML(\"\"\"\n",
" \n",
" \"\"\"%(random.sample(pups, 1)[0]))\n",
"SEQ = [2,1,3,2,4]\n",
"SEQ2 = [3, 4 ,3, -1, 2]"
]
},
{
"cell_type": "markdown",
"id": "57d753ac",
"metadata": {
"id": "57d753ac"
},
"source": [
"For each problem we will provide a Python specification. Your goal is to implement that specification with Transformers."
]
},
{
"cell_type": "markdown",
"id": "77441886",
"metadata": {
"id": "77441886"
},
"source": [
"### Challenge 0: Select the initial position\n",
"\n",
"Given a initial sequence compute a new sequence where all positions have the initial value. (1 line)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1da74d03",
"metadata": {
"id": "1da74d03"
},
"outputs": [],
"source": [
"def head_spec(seq):\n",
" return [seq[0] for _ in seq]\n",
"\n",
"def head(seq=tokens):\n",
" return (key(indices) == query(0)).value(seq)\n",
" \n",
"test_output(head, head_spec, [(SEQ,),(SEQ2,)])"
]
},
{
"cell_type": "markdown",
"id": "f565d61c",
"metadata": {
"id": "f565d61c"
},
"source": [
"### Challenge 1: Select a given index\n",
"\n",
"Produce a sequence where all the elements have the value at index `i`."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "10d1b909",
"metadata": {
"lines_to_next_cell": 1,
"id": "10d1b909"
},
"outputs": [],
"source": [
"def index_spec(i, seq):\n",
" return [seq[i] for _ in seq]\n",
"\n",
"def index(i, seq=tokens):\n",
" raise NotImplementedError\n",
"\n",
"test_output(index, index_spec, [(2, SEQ), (3, SEQ2), (1, SEQ)])"
]
},
{
"cell_type": "markdown",
"id": "a6c2b47d",
"metadata": {
"id": "a6c2b47d"
},
"source": [
"### Challenge 2: Shift\n",
"\n",
"Shift all of the tokens in a sequence to the right by `i` positions filling in the values with `default`. (1 line)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d9b5db79",
"metadata": {
"lines_to_next_cell": 1,
"id": "d9b5db79"
},
"outputs": [],
"source": [
"def shift_spec(i, default=\"0\", seq=None):\n",
" return [default]*i + [s for j, s in enumerate(seq) if j < len(seq) - i]\n",
"\n",
"def shift(i, default=\"0\", seq=tokens):\n",
" raise NotImplementedError\n",
"\n",
"test_output(shift, shift_spec, [(2, 0, SEQ), (3, 0, SEQ2), (1, 0, SEQ)])"
]
},
{
"cell_type": "markdown",
"id": "3f87e538",
"metadata": {
"lines_to_next_cell": 2,
"id": "3f87e538"
},
"source": [
"### Challenge 3: Right Align\n",
"\n",
"Right align a padded sequence e.g. ralign().inputs('xyz___') = '000xyz'\" (3 layers) (2 lines)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a841d6f4",
"metadata": {
"lines_to_next_cell": 1,
"id": "a841d6f4"
},
"outputs": [],
"source": [
"def ralign_spec(ldefault=\"0\", seq=tokens):\n",
" last = None\n",
" for i in range(len(seq)-1, -1, -1):\n",
" if seq[i] == \"_\":\n",
" last = i\n",
" else:\n",
" break\n",
" if last == None:\n",
" return seq\n",
" return [ldefault] * (len(seq) - last) + seq[:last]\n",
"\n",
"def ralign(ldefault=\"0\", seq=tokens):\n",
" raise NotImplementedError\n",
"\n",
"test_output(ralign, ralign_spec, [(\"-\", list(\"xyzabc__\"),), (\"0\", list(\"xyz___\"),)])"
]
},
{
"cell_type": "markdown",
"id": "a178203a",
"metadata": {
"id": "a178203a"
},
"source": [
"### Challenge 4: Split\n",
"\n",
"Split a sequence on a value. Get the first or second part. Right align. (5 lines)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e69995ed",
"metadata": {
"lines_to_next_cell": 1,
"id": "e69995ed"
},
"outputs": [],
"source": [
"def split_spec(v, get_first_part, seq):\n",
" out = []\n",
" mid = False\n",
" blank = \"0\" if not get_first_part else \"_\"\n",
" for j, s in enumerate(seq):\n",
" if s == v:\n",
" out.append(blank)\n",
" mid = True\n",
" elif (get_first_part and not mid) or (not get_first_part and mid):\n",
" out.append(s)\n",
" else:\n",
" out.append(blank)\n",
" return ralign_spec(\"0\", seq=out)\n",
"\n",
"def split(v, get_first_part, seq=tokens):\n",
" raise NotImplementedError\n",
"\n",
"test_output(split, split_spec,\n",
" [(\"-\", 1, list(\"xyz-ax\"),),\n",
" (\"-\", 0, list(\"xyz-ax\"),),\n",
" (\"+\", 0, list(\"xy+z-ax\"),)]\n",
" )"
]
},
{
"cell_type": "markdown",
"id": "f9d19ecd",
"metadata": {
"id": "f9d19ecd"
},
"source": [
"### Challenge 5: Minimum \n",
"\n",
"Compute the minimum value of the sequence. This one starts to get harder! (5 lines of code)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "53b19ac8",
"metadata": {
"lines_to_next_cell": 1,
"id": "53b19ac8"
},
"outputs": [],
"source": [
"def minimum_spec(seq):\n",
" m = min(seq)\n",
" return [m for _ in seq]\n",
"\n",
"def minimum(seq=tokens):\n",
" raise NotImplementedError\n",
"\n",
"test_output(minimum, minimum_spec, [(SEQ,), (SEQ2,), ([2, 1, 1],)])"
]
},
{
"cell_type": "markdown",
"id": "6d2fe8ec",
"metadata": {
"id": "6d2fe8ec"
},
"source": [
"### Challenge 6: First Index\n",
"\n",
"Compute the first index that has token `token`. (1 line)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a4a6a030",
"metadata": {
"lines_to_next_cell": 1,
"id": "a4a6a030"
},
"outputs": [],
"source": [
"def first_spec(token, seq):\n",
" first = None\n",
" for i, s in enumerate(seq):\n",
" if s == token and first is None:\n",
" first = i\n",
" return [first for _ in seq]\n",
"\n",
"def first(token, seq=tokens):\n",
" raise NotImplementedError\n",
"\n",
"test_output(first, first_spec, [(3, SEQ), (-1, SEQ2), ('l', list('hello'))])"
]
},
{
"cell_type": "markdown",
"id": "38236f00",
"metadata": {
"id": "38236f00"
},
"source": [
"### Challenge 7: Slide\n",
"\n",
"Replace special tokens \"<\" with the closest non \"<\" value to their right. (4 lines of code)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "7781c97c",
"metadata": {
"id": "7781c97c"
},
"outputs": [],
"source": [
"def slide_spec(match, seq):\n",
" out = []\n",
" for i, s in enumerate(seq):\n",
" if s == \"<\":\n",
" for v in seq[i+1:]:\n",
" if v != \"<\":\n",
" out.append(v)\n",
" break\n",
" else:\n",
" out.append(s)\n",
" return out\n",
"\n",
"def slide(match=\"<\", seq=tokens):\n",
" raise NotImplementedError\n",
"\n",
"test_output(slide, slide_spec,\n",
" [(\"<\", list(\"1<<2\"),),\n",
" (\"<\", list(\"2<<<3\"),),\n",
" (\"<\", list(\"3<<<1<<3\"),)]\n",
" )"
]
},
{
"cell_type": "markdown",
"id": "6be44511",
"metadata": {
"id": "6be44511"
},
"source": [
"### Final Challenge: Adder\n",
"\n",
"Now we will put everything together. Here are the steps. \n",
"\n",
"add().input(\"683+345\")\n",
"\n",
"0) Split into parts. Convert to ints. Add\n",
"\n",
"> \"683+345\" => [0, 0, 0, 9, 12, 8]\n",
"\n",
"1) Compute the carry terms. Three possibilities: 1 has carry, 0 no carry, < maybe has carry. \n",
"\n",
"> [0, 0, 0, 9, 12, 8] => \"00<100\"\n",
"\n",
"2) Slide the carry coefficients\n",
"\n",
"> \"00<100\" => 001100\"\n",
"\n",
"3) Complete the addition.\n",
"\n",
"Each of these is 1 line of code. "
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "11b27d5b",
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 96
},
"id": "11b27d5b",
"outputId": "b1224ef6-c889-40ab-f051-fd00e37ceb2a"
},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
"[1, 3, 2, 1]"
],
"image/svg+xml": "\n"
},
"metadata": {},
"execution_count": 32
}
],
"source": [
"# The function atoi lets us convert from string to sequences of integers\n",
"atoi(tokens).input(\"1321\")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "3370d2c8",
"metadata": {
"id": "3370d2c8"
},
"outputs": [],
"source": [
"def add_spec(seq):\n",
" a, b = \"\".join(seq).split(\"+\")\n",
" c = int(a) + int(b)\n",
" out = f\"{c}\"\n",
" return list(map(int, list((\"0\" * (len(seq) - len(out))) + out)))"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "98a331c6",
"metadata": {
"id": "98a331c6"
},
"outputs": [],
"source": [
"def add(seq=tokens):\n",
" x = atoi(split(\"+\", True, seq)) \\\n",
" + atoi(split(\"+\", False, seq))\n",
" # 1) Check for carries \n",
" gets_carry = shift(-1, \"0\", where(x > 9, \"1\", where(x == 9, \"<\", \"0\")))\n",
" # 2) Slide carries to their columns - all in one parallel go! \n",
" gets_carry = atoi(slide(\"<\", gets_carry))\n",
" # 3) Add in carries, and remove overflow from original addition. \n",
" return (x + gets_carry) % 10"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "7f1f708c",
"metadata": {
"id": "7f1f708c"
},
"outputs": [],
"source": [
"test_output(add, add_spec,\n",
" [(list(\"1+2\"),),\n",
" (list(\"22+384\"),),\n",
" (list(\"3+10\"),)]\n",
" )"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "raspy",
"language": "python",
"name": "raspy"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.10.6"
},
"colab": {
"provenance": [],
"include_colab_link": true
}
},
"nbformat": 4,
"nbformat_minor": 5
}
![\"Open](\"https://colab.research.google.com/assets/colab-badge.svg\"){kind=icon}
"
]
},
{
"cell_type": "markdown",
"id": "f8117f57",
"metadata": {
"id": "f8117f57"
},
"source": [
"# Transformer Puzzles"
]
},
{
"cell_type": "markdown",
"id": "e9e822cb",
"metadata": {
"id": "e9e822cb"
},
"source": [
"This notebook is a collection of short coding puzzles based on the internals of the Transformer. The puzzles are written in Python and can be done in this notebook. After completing these you will have a much better intutive sense of how a Transformer can compute certain logical operations. \n",
"\n",
"These puzzles are based on [Thinking Like Transformers](https://arxiv.org/pdf/2106.06981.pdf) by Gail Weiss, Yoav Goldberg, Eran Yahav and derived from this [blog post](https://srush.github.io/raspy/)."
]
},
{
"cell_type": "markdown",
"id": "8e962052",
"metadata": {
"id": "8e962052"
},
"source": [
"## Goal\n",
"\n",
"**Can we produce a Transformer that does basic addition?**\n",
"\n",
"i.e. given a string \"19492+23919\" can we produce the correct output? "
]
},
{
"cell_type": "markdown",
"id": "d332140b",
"metadata": {
"id": "d332140b"
},
"source": [
"## Rules\n",
"\n",
"Each exercise consists of a function with a argument `seq` and output `seq`. Like a transformer we cannot change length. Operations need to act on the entire sequence in parallel. There is a global `indices` which tells use the position in the sequence. If we want to do something different on certain positions we can use `where` like in Numpy or PyTorch. To run the seq we need to give it an initial input. "
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "6c5c885a",
"metadata": {
"id": "6c5c885a"
},
"outputs": [],
"source": [
"%%capture\n",
"!pip install -qqq git+https://github.com/chalk-diagrams/chalk git+https://github.com/srush/RASPy "
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "51724e11",
"metadata": {
"id": "51724e11"
},
"outputs": [],
"source": [
"from IPython.display import display, HTML\n",
"from raspy import key, query, tokens, indices, where, draw\n",
"import random"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1b28dc98",
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 96
},
"id": "1b28dc98",
"outputId": "f1ac1157-3db8-40c0-dbb2-7d9bad8943a0"
},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
"[0, -1, 2, -1, 4]"
],
"image/svg+xml": "\n"
},
"metadata": {},
"execution_count": 3
}
],
"source": [
"def even_vals(seq=tokens):\n",
" \"Keep even positions, set odd positions to -1\"\n",
" x = indices % 2\n",
" # Note that all operations broadcast so you can use scalars.\n",
" return where(x == 0, seq, -1)\n",
"seq = even_vals()\n",
"\n",
"# Give the initial input tokens\n",
"seq.input([0,1,2,3,4])"
]
},
{
"cell_type": "markdown",
"id": "9dc23f88",
"metadata": {
"id": "9dc23f88"
},
"source": [
"The main operation you can use is \"attention\". You do this by defining a selector which forms a matrix based on `key` and `query`."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e2ee0ff8",
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 176
},
"id": "e2ee0ff8",
"outputId": "a61ac19c-2550-4f3c-d653-50c323cdfd59"
},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
"Selector Layer 0\n",
" 0 1 2 3 4\n",
" |h e l l o|\n",
" 0 1 2 3 4\n",
" |---------|\n",
"0 h 0 | 1 1 1 1\n",
"1 e 1 | 1 1 1\n",
"2 l 2 | 1 1\n",
"3 l 3 | 1\n",
"4 o 4 | "
],
"image/svg+xml": "\n"
},
"metadata": {},
"execution_count": 4
}
],
"source": [
"before = key(indices) < query(indices)\n",
"before"
]
},
{
"cell_type": "markdown",
"id": "a4de0a14",
"metadata": {
"id": "a4de0a14"
},
"source": [
"We can combine selectors with logical operations."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c315ba6d",
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 201
},
"id": "c315ba6d",
"outputId": "270d50fa-649c-438b-8606-d3d078478162"
},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
"Selector Layer 0\n",
" 0 1 2 3 4\n",
" |h e l l o|\n",
" 0 1 2 3 4\n",
" 0 1 2 3 4\n",
" |---------|\n",
"0 h 0 | 1 1 1 1 1\n",
"1 e 1 | 1 1 1 1\n",
"2 l 2 | 1 1 1\n",
"3 l 3 | 1 1\n",
"4 o 4 | 1"
],
"image/svg+xml": "\n"
},
"metadata": {},
"execution_count": 5
}
],
"source": [
"before_or_same = before | (key(indices) == query(indices))\n",
"before_or_same"
]
},
{
"cell_type": "markdown",
"id": "00bc66a3",
"metadata": {
"id": "00bc66a3"
},
"source": [
"Once you have a selector, you can apply \"attention\" to sum over the grey positions. For example to compute cumulative such we run the following function. "
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e79c8c8b",
"metadata": {
"lines_to_next_cell": 1,
"colab": {
"base_uri": "https://localhost:8080/",
"height": 326
},
"id": "e79c8c8b",
"outputId": "44db7f90-502d-497c-c5ba-4062c09f0a9a"
},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
"[0, 1, 3, 6, 10]"
],
"image/svg+xml": "\n"
},
"metadata": {},
"execution_count": 6
}
],
"source": [
"def cumsum(seq=tokens):\n",
" return before_or_same.value(seq)\n",
"seq = cumsum()\n",
"seq.input([0, 1, 2, 3, 4])"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "9ae57559",
"metadata": {
"cellView": "form",
"id": "9ae57559"
},
"outputs": [],
"source": [
"#@title Test Code (Collapse)\n",
"def atoi(seq=tokens):\n",
" return seq.map(lambda x: ord(x) - ord('0'))\n",
"\n",
"def test_output(user, spec, token_sets):\n",
" for ex_num, token_set in enumerate(token_sets): \n",
" out1 = user(*token_set[:-1])((token_set[-1]))\n",
" out2 = spec(*token_set)\n",
" print(f\"Example {ex_num}. Args:\", token_set, \"Expected:\", out2)\n",
" display(out1)\n",
" out1 = out1.toseq()\n",
" for i, o in enumerate(out2):\n",
" assert out1[i] == o, f\"Output: {out1} Expected: {out2}\"\n",
"\n",
" pups = [\n",
" \"2m78jPG\",\n",
" \"pn1e9TO\",\n",
" \"MQCIwzT\",\n",
" \"udLK6FS\",\n",
" \"ZNem5o3\",\n",
" \"DS2IZ6K\",\n",
" \"aydRUz8\",\n",
" \"MVUdQYK\",\n",
" \"kLvno0p\",\n",
" \"wScLiVz\",\n",
" \"Z0TII8i\",\n",
" \"F1SChho\",\n",
" \"9hRi2jN\",\n",
" \"lvzRF3W\",\n",
" \"fqHxOGI\",\n",
" \"1xeUYme\",\n",
" \"6tVqKyM\",\n",
" \"CCxZ6Wr\",\n",
" \"lMW0OPQ\",\n",
" \"wHVpHVG\",\n",
" \"Wj2PGRl\",\n",
" \"HlaTE8H\",\n",
" \"k5jALH0\",\n",
" \"3V37Hqr\",\n",
" \"Eq2uMTA\",\n",
" \"Vy9JShx\",\n",
" \"g9I2ZmK\",\n",
" \"Nu4RH7f\",\n",
" \"sWp0Dqd\",\n",
" \"bRKfspn\",\n",
" \"qawCMl5\",\n",
" \"2F6j2B4\",\n",
" \"fiJxCVA\",\n",
" \"pCAIlxD\",\n",
" \"zJx2skh\",\n",
" \"2Gdl1u7\",\n",
" \"aJJAY4c\",\n",
" \"ros6RLC\",\n",
" \"DKLBJh7\",\n",
" \"eyxH0Wc\",\n",
" \"rJEkEw4\"]\n",
" print(\"Success!\")\n",
" return HTML(\"\"\"\n",
" \n",
" \"\"\"%(random.sample(pups, 1)[0]))\n",
"SEQ = [2,1,3,2,4]\n",
"SEQ2 = [3, 4 ,3, -1, 2]"
]
},
{
"cell_type": "markdown",
"id": "57d753ac",
"metadata": {
"id": "57d753ac"
},
"source": [
"For each problem we will provide a Python specification. Your goal is to implement that specification with Transformers."
]
},
{
"cell_type": "markdown",
"id": "77441886",
"metadata": {
"id": "77441886"
},
"source": [
"### Challenge 0: Select the initial position\n",
"\n",
"Given a initial sequence compute a new sequence where all positions have the initial value. (1 line)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1da74d03",
"metadata": {
"id": "1da74d03"
},
"outputs": [],
"source": [
"def head_spec(seq):\n",
" return [seq[0] for _ in seq]\n",
"\n",
"def head(seq=tokens):\n",
" return (key(indices) == query(0)).value(seq)\n",
" \n",
"test_output(head, head_spec, [(SEQ,),(SEQ2,)])"
]
},
{
"cell_type": "markdown",
"id": "f565d61c",
"metadata": {
"id": "f565d61c"
},
"source": [
"### Challenge 1: Select a given index\n",
"\n",
"Produce a sequence where all the elements have the value at index `i`."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "10d1b909",
"metadata": {
"lines_to_next_cell": 1,
"id": "10d1b909"
},
"outputs": [],
"source": [
"def index_spec(i, seq):\n",
" return [seq[i] for _ in seq]\n",
"\n",
"def index(i, seq=tokens):\n",
" raise NotImplementedError\n",
"\n",
"test_output(index, index_spec, [(2, SEQ), (3, SEQ2), (1, SEQ)])"
]
},
{
"cell_type": "markdown",
"id": "a6c2b47d",
"metadata": {
"id": "a6c2b47d"
},
"source": [
"### Challenge 2: Shift\n",
"\n",
"Shift all of the tokens in a sequence to the right by `i` positions filling in the values with `default`. (1 line)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d9b5db79",
"metadata": {
"lines_to_next_cell": 1,
"id": "d9b5db79"
},
"outputs": [],
"source": [
"def shift_spec(i, default=\"0\", seq=None):\n",
" return [default]*i + [s for j, s in enumerate(seq) if j < len(seq) - i]\n",
"\n",
"def shift(i, default=\"0\", seq=tokens):\n",
" raise NotImplementedError\n",
"\n",
"test_output(shift, shift_spec, [(2, 0, SEQ), (3, 0, SEQ2), (1, 0, SEQ)])"
]
},
{
"cell_type": "markdown",
"id": "3f87e538",
"metadata": {
"lines_to_next_cell": 2,
"id": "3f87e538"
},
"source": [
"### Challenge 3: Right Align\n",
"\n",
"Right align a padded sequence e.g. ralign().inputs('xyz___') = '000xyz'\" (3 layers) (2 lines)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a841d6f4",
"metadata": {
"lines_to_next_cell": 1,
"id": "a841d6f4"
},
"outputs": [],
"source": [
"def ralign_spec(ldefault=\"0\", seq=tokens):\n",
" last = None\n",
" for i in range(len(seq)-1, -1, -1):\n",
" if seq[i] == \"_\":\n",
" last = i\n",
" else:\n",
" break\n",
" if last == None:\n",
" return seq\n",
" return [ldefault] * (len(seq) - last) + seq[:last]\n",
"\n",
"def ralign(ldefault=\"0\", seq=tokens):\n",
" raise NotImplementedError\n",
"\n",
"test_output(ralign, ralign_spec, [(\"-\", list(\"xyzabc__\"),), (\"0\", list(\"xyz___\"),)])"
]
},
{
"cell_type": "markdown",
"id": "a178203a",
"metadata": {
"id": "a178203a"
},
"source": [
"### Challenge 4: Split\n",
"\n",
"Split a sequence on a value. Get the first or second part. Right align. (5 lines)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e69995ed",
"metadata": {
"lines_to_next_cell": 1,
"id": "e69995ed"
},
"outputs": [],
"source": [
"def split_spec(v, get_first_part, seq):\n",
" out = []\n",
" mid = False\n",
" blank = \"0\" if not get_first_part else \"_\"\n",
" for j, s in enumerate(seq):\n",
" if s == v:\n",
" out.append(blank)\n",
" mid = True\n",
" elif (get_first_part and not mid) or (not get_first_part and mid):\n",
" out.append(s)\n",
" else:\n",
" out.append(blank)\n",
" return ralign_spec(\"0\", seq=out)\n",
"\n",
"def split(v, get_first_part, seq=tokens):\n",
" raise NotImplementedError\n",
"\n",
"test_output(split, split_spec,\n",
" [(\"-\", 1, list(\"xyz-ax\"),),\n",
" (\"-\", 0, list(\"xyz-ax\"),),\n",
" (\"+\", 0, list(\"xy+z-ax\"),)]\n",
" )"
]
},
{
"cell_type": "markdown",
"id": "f9d19ecd",
"metadata": {
"id": "f9d19ecd"
},
"source": [
"### Challenge 5: Minimum \n",
"\n",
"Compute the minimum value of the sequence. This one starts to get harder! (5 lines of code)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "53b19ac8",
"metadata": {
"lines_to_next_cell": 1,
"id": "53b19ac8"
},
"outputs": [],
"source": [
"def minimum_spec(seq):\n",
" m = min(seq)\n",
" return [m for _ in seq]\n",
"\n",
"def minimum(seq=tokens):\n",
" raise NotImplementedError\n",
"\n",
"test_output(minimum, minimum_spec, [(SEQ,), (SEQ2,), ([2, 1, 1],)])"
]
},
{
"cell_type": "markdown",
"id": "6d2fe8ec",
"metadata": {
"id": "6d2fe8ec"
},
"source": [
"### Challenge 6: First Index\n",
"\n",
"Compute the first index that has token `token`. (1 line)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a4a6a030",
"metadata": {
"lines_to_next_cell": 1,
"id": "a4a6a030"
},
"outputs": [],
"source": [
"def first_spec(token, seq):\n",
" first = None\n",
" for i, s in enumerate(seq):\n",
" if s == token and first is None:\n",
" first = i\n",
" return [first for _ in seq]\n",
"\n",
"def first(token, seq=tokens):\n",
" raise NotImplementedError\n",
"\n",
"test_output(first, first_spec, [(3, SEQ), (-1, SEQ2), ('l', list('hello'))])"
]
},
{
"cell_type": "markdown",
"id": "38236f00",
"metadata": {
"id": "38236f00"
},
"source": [
"### Challenge 7: Slide\n",
"\n",
"Replace special tokens \"<\" with the closest non \"<\" value to their right. (4 lines of code)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "7781c97c",
"metadata": {
"id": "7781c97c"
},
"outputs": [],
"source": [
"def slide_spec(match, seq):\n",
" out = []\n",
" for i, s in enumerate(seq):\n",
" if s == \"<\":\n",
" for v in seq[i+1:]:\n",
" if v != \"<\":\n",
" out.append(v)\n",
" break\n",
" else:\n",
" out.append(s)\n",
" return out\n",
"\n",
"def slide(match=\"<\", seq=tokens):\n",
" raise NotImplementedError\n",
"\n",
"test_output(slide, slide_spec,\n",
" [(\"<\", list(\"1<<2\"),),\n",
" (\"<\", list(\"2<<<3\"),),\n",
" (\"<\", list(\"3<<<1<<3\"),)]\n",
" )"
]
},
{
"cell_type": "markdown",
"id": "6be44511",
"metadata": {
"id": "6be44511"
},
"source": [
"### Final Challenge: Adder\n",
"\n",
"Now we will put everything together. Here are the steps. \n",
"\n",
"add().input(\"683+345\")\n",
"\n",
"0) Split into parts. Convert to ints. Add\n",
"\n",
"> \"683+345\" => [0, 0, 0, 9, 12, 8]\n",
"\n",
"1) Compute the carry terms. Three possibilities: 1 has carry, 0 no carry, < maybe has carry. \n",
"\n",
"> [0, 0, 0, 9, 12, 8] => \"00<100\"\n",
"\n",
"2) Slide the carry coefficients\n",
"\n",
"> \"00<100\" => 001100\"\n",
"\n",
"3) Complete the addition.\n",
"\n",
"Each of these is 1 line of code. "
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "11b27d5b",
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 96
},
"id": "11b27d5b",
"outputId": "b1224ef6-c889-40ab-f051-fd00e37ceb2a"
},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
"[1, 3, 2, 1]"
],
"image/svg+xml": "\n"
},
"metadata": {},
"execution_count": 32
}
],
"source": [
"# The function atoi lets us convert from string to sequences of integers\n",
"atoi(tokens).input(\"1321\")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "3370d2c8",
"metadata": {
"id": "3370d2c8"
},
"outputs": [],
"source": [
"def add_spec(seq):\n",
" a, b = \"\".join(seq).split(\"+\")\n",
" c = int(a) + int(b)\n",
" out = f\"{c}\"\n",
" return list(map(int, list((\"0\" * (len(seq) - len(out))) + out)))"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "98a331c6",
"metadata": {
"id": "98a331c6"
},
"outputs": [],
"source": [
"def add(seq=tokens):\n",
" x = atoi(split(\"+\", True, seq)) \\\n",
" + atoi(split(\"+\", False, seq))\n",
" # 1) Check for carries \n",
" gets_carry = shift(-1, \"0\", where(x > 9, \"1\", where(x == 9, \"<\", \"0\")))\n",
" # 2) Slide carries to their columns - all in one parallel go! \n",
" gets_carry = atoi(slide(\"<\", gets_carry))\n",
" # 3) Add in carries, and remove overflow from original addition. \n",
" return (x + gets_carry) % 10"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "7f1f708c",
"metadata": {
"id": "7f1f708c"
},
"outputs": [],
"source": [
"test_output(add, add_spec,\n",
" [(list(\"1+2\"),),\n",
" (list(\"22+384\"),),\n",
" (list(\"3+10\"),)]\n",
" )"
]
}
],
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"kernelspec": {
"display_name": "raspy",
"language": "python",
"name": "raspy"
},
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"codemirror_mode": {
"name": "ipython",
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"name": "python",
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