transformer_puzz.ipynb

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   "source": [
    "<a href=\"https://colab.research.google.com/github/srush/Transformer-Puzzles/blob/main/TransformerPuzzlers.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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   "source": [
    "# Transformer Puzzles"
   ]
  },
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   "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/)."
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    "## 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? "
   ]
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    "## 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. "
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   "cell_type": "code",
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   "source": [
    "%%capture\n",
    "!pip install -qqq git+https://github.com/chalk-diagrams/chalk git+https://github.com/srush/RASPy "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "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"
   ]
  },
  {
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   "execution_count": 4,
   "id": "1b28dc98",
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    "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])"
   ]
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   "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`."
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      ],
      "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 |          "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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": 6,
   "id": "c315ba6d",
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 201
    },
    "id": "c315ba6d",
    "outputId": "270d50fa-649c-438b-8606-d3d078478162"
   },
   "outputs": [
    {
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      ],
      "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"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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": 7,
   "id": "e79c8c8b",
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 326
    },
    "id": "e79c8c8b",
    "lines_to_next_cell": 1,
    "outputId": "44db7f90-502d-497c-c5ba-4062c09f0a9a"
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "[0, 1, 3, 6, 10]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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": 8,
   "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",
    "    <video alt=\"test\" controls autoplay=1>\n",
    "        <source src=\"https://openpuppies.com/mp4/%s.mp4\"  type=\"video/mp4\">\n",
    "    </video>\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": 9,
   "id": "1da74d03",
   "metadata": {
    "id": "1da74d03"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Example 0. Args: ([2, 1, 3, 2, 4],) Expected: [2, 2, 2, 2, 2]\n"
     ]
    },
    {
     "data": {
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transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 4.0, 0.0)\"><g><g style=\"fill: #ee2e31;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">4</text></g><g /></g></g></g></g></g></g></g></g></g></g><g style=\"stroke: black;stroke-width: 0.9;\" transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 9.2)\"><g /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 10.2)\"><g><g><g><g style=\"fill: #1d7874;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">Final</text></g><g /></g><g /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.0, 0.0)\"><g><g><g><g><g><g><g><g><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">2</text></g><g /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 1.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">2</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 1.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">2</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">2</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 4.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">2</text></g><g /></g></g></g></g></g></g></g></g></g><g /></g></g></g></g></svg>"
      ],
      "text/plain": [
       "[2, 2, 2, 2, 2]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Example 1. Args: ([3, 4, 3, -1, 2],) Expected: [3, 3, 3, 3, 3]\n"
     ]
    },
    {
     "data": {
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dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">Final</text></g><g /></g><g /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.0, 0.0)\"><g><g><g><g><g><g><g><g><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">3</text></g><g /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 1.0, 0.0)\"><g><g style=\"fill: #000000;stroke: 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/></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">3</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 4.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">3</text></g><g /></g></g></g></g></g></g></g></g></g><g /></g></g></g></g></svg>"
      ],
      "text/plain": [
       "[3, 3, 3, 3, 3]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Success!\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "\n",
       "    <video alt=\"test\" controls autoplay=1>\n",
       "        <source src=\"https://openpuppies.com/mp4/lvzRF3W.mp4\"  type=\"video/mp4\">\n",
       "    </video>\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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": 11,
   "id": "10d1b909",
   "metadata": {
    "id": "10d1b909",
    "lines_to_next_cell": 1
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Example 0. Args: (2, [2, 1, 3, 2, 4]) Expected: [3, 3, 3, 3, 3]\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
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transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 4.0, 0.0)\"><g><g style=\"fill: #ee2e31;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">4</text></g><g /></g></g></g></g></g></g></g></g></g></g><g style=\"stroke: black;stroke-width: 0.9;\" transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 9.2)\"><g /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 10.2)\"><g><g><g><g style=\"fill: #1d7874;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">Final</text></g><g /></g><g /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.0, 0.0)\"><g><g><g><g><g><g><g><g><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">3</text></g><g /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 1.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">3</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 1.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">3</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">3</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 4.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">3</text></g><g /></g></g></g></g></g></g></g></g></g><g /></g></g></g></g></svg>"
      ],
      "text/plain": [
       "[3, 3, 3, 3, 3]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Example 1. Args: (3, [3, 4, 3, -1, 2]) Expected: [-1, -1, -1, -1, -1]\n"
     ]
    },
    {
     "data": {
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dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">-1</text></g><g /></g></g></g></g></g></g></g></g></g><g /></g></g></g></g></svg>"
      ],
      "text/plain": [
       "[-1, -1, -1, -1, -1]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Example 2. Args: (1, [2, 1, 3, 2, 4]) Expected: [1, 1, 1, 1, 1]\n"
     ]
    },
    {
     "data": {
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     "text": [
      "Success!\n"
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      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def index_spec(i, seq):\n",
    "    return [seq[i] for _ in seq]\n",
    "\n",
    "def index(i, seq=tokens):\n",
    "    return (key(indices) == query(i)).value(seq)\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": "a356979c",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "d9b5db79",
   "metadata": {
    "id": "d9b5db79",
    "lines_to_next_cell": 1
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Example 0. Args: (2, 0, [2, 1, 3, 2, 4]) Expected: [0, 0, 2, 1, 3]\n"
     ]
    },
    {
     "data": {
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dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">Final</text></g><g /></g><g /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.0, 0.0)\"><g><g><g><g><g><g><g><g><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">0</text></g><g /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 1.0, 0.0)\"><g><g style=\"fill: #000000;stroke: 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/></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">1</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 4.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">3</text></g><g /></g></g></g></g></g></g></g></g></g><g /></g></g></g></g></svg>"
      ],
      "text/plain": [
       "[0, 0, 2, 1, 3]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Example 1. Args: (3, 0, [3, 4, 3, -1, 2]) Expected: [0, 0, 0, 3, 4]\n"
     ]
    },
    {
     "data": {
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1.0, 3.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">-1</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 4.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">2</text></g><g /></g></g></g></g></g><g style=\"stroke: black;stroke-width: 0.9;\" transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 1.0)\"><g /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 5.1)\"><g><g><g><g><g style=\"fill: #1d7874;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">Layer 1</text></g><g /></g><g /></g><g style=\"stroke: black;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.0, 0.0)\"><g /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 6.1, 0.0)\"><g><g><g><g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, -4.0, -1.9999999999999998)\"><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.0, 0.0)\"><g><g><g><g><g><g><g><g><g><g><g><g><g><g><g><g><g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, -0.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 0.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 1.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 1.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" 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transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, -0.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 0.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 1.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 1.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 2.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g 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style=\"vector-effect: non-scaling-stroke;\" /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 0.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 1.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 1.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 2.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 2.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 3.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 3.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.0, 0.0)\"><g><g><g><g><g><g><g><g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, -0.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 0.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 1.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 1.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 2.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 2.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 3.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 3.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 4.0, 0.0)\"><g><g><g><g><g><g><g><g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, -0.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 0.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 1.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 1.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 2.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 2.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 3.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g style=\"fill: #ffffff;stroke: #071e22;stroke-width: 0.9;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.5, 3.5)\"><path d=\"M 0.0 0.0 L 1.0 0.0 L 1.0 1.0 L 0.0 1.0 L 0.0 0.0 Z\" style=\"vector-effect: non-scaling-stroke;\" /></g></g></g></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.0, -3.0999999999999996)\"><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, -2.0, -0.5)\"><g><g /><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.5)\"><g><g><g><g><g><g><g><g><g><g style=\"fill: #f4c095;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">0</text></g><g /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 1.0, 0.0)\"><g><g style=\"fill: #f4c095;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">1</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 1.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.0, 0.0)\"><g><g style=\"fill: #f4c095;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">2</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.0, 0.0)\"><g><g style=\"fill: #f4c095;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">3</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 4.0, 0.0)\"><g><g style=\"fill: 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transform=\"translate(-5e-05, 0)\">-3</text></g><g /></g><g style=\"stroke: black;stroke-width: 0.9;\" transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.5)\"><g /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 1.0)\"><g><g style=\"fill: #679289;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">-2</text></g><g /></g></g></g><g style=\"stroke: black;stroke-width: 0.9;\" transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 1.5)\"><g /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 2.0)\"><g><g style=\"fill: #679289;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">-1</text></g><g /></g></g></g><g style=\"stroke: black;stroke-width: 0.9;\" transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 2.5)\"><g /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 3.0)\"><g><g style=\"fill: #679289;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">0</text></g><g /></g></g></g><g style=\"stroke: black;stroke-width: 0.9;\" transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 3.5)\"><g /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 4.0)\"><g><g style=\"fill: #679289;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">1</text></g><g /></g></g></g></g></g></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.0999999999999996, 0.0)\"><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, -0.0, -2.0)\"><g><g><g><g><g><g><g><g><g><g style=\"fill: #071e22;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: 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style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">0</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 2.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 3.0)\"><g><g style=\"fill: #071e22;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">3</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 3.5)\"><g transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 4.0)\"><g><g style=\"fill: #071e22;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">4</text></g><g /></g></g></g></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 3.0999999999999996)\"><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, -2.0, -0.0)\"><g><g><g><g><g><g><g><g><g><g style=\"fill: #ee2e31;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                  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transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 4.0, 0.0)\"><g><g style=\"fill: #ee2e31;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">2</text></g><g /></g></g></g></g></g></g></g></g></g></g><g style=\"stroke: black;stroke-width: 0.9;\" transform=\"matrix(0.0, -1.0, 1.0, 0.0, 0.0, 9.2)\"><g /></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 10.2)\"><g><g><g><g style=\"fill: #1d7874;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">Final</text></g><g /></g><g /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.0, 0.0)\"><g><g><g><g><g><g><g><g><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">0</text></g><g /></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 1.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">0</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 1.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">0</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 2.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">3</text></g><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 3.5, 0.0)\"><g transform=\"matrix(1.0, -0.0, 0.0, 1.0, 0.0, 0.0)\"><g /></g></g></g><g transform=\"matrix(1.0, 0.0, 0.0, 1.0, 4.0, 0.0)\"><g><g style=\"fill: #000000;stroke: black;stroke-width: 0.0;\" transform=\"matrix(1.0, 0.0, 0.0, 1.0, 0.0, 0.0)\"><text style=\"text-align:center; text-anchor:middle; dominant-baseline:middle;&#10;                      font-family:sans-serif; font-weight: bold;&#10;                      font-size:0.85px;&#10;                      vector-effect: non-scaling-stroke;\" transform=\"translate(-5e-05, 0)\">4</text></g><g /></g></g></g></g></g></g></g></g></g><g /></g></g></g></g></svg>"
      ],
      "text/plain": [
       "[0, 0, 0, 3, 4]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Example 2. Args: (1, 0, [2, 1, 3, 2, 4]) Expected: [0, 2, 1, 3, 2]\n"
     ]
    },
    {
     "data": {
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      "text/plain": [
       "[0, 2, 1, 3, 2]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Success!\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "\n",
       "    <video alt=\"test\" controls autoplay=1>\n",
       "        <source src=\"https://openpuppies.com/mp4/aydRUz8.mp4\"  type=\"video/mp4\">\n",
       "    </video>\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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",
    "    return (key(indices) == query(indices-i)).value(seq, default)\n",
    "\n",
    "test_output(shift, shift_spec, [(2, 0, SEQ), (3, 0, SEQ2), (1, 0, SEQ)])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f87e538",
   "metadata": {
    "id": "3f87e538",
    "lines_to_next_cell": 2
   },
   "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": {
    "id": "a841d6f4",
    "lines_to_next_cell": 1
   },
   "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",
    "    selector =  (key(seq) == '_').value(1)\n",
    "    selec2 = \n",
    "    return \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": {
    "id": "e69995ed",
    "lines_to_next_cell": 1
   },
   "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": {
    "id": "53b19ac8",
    "lines_to_next_cell": 1
   },
   "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": {
    "id": "a4a6a030",
    "lines_to_next_cell": 1
   },
   "outputs": [],
   "source": [
    "def first_spec(token, seq):\n",
    "    first = None\n",
    "    for i, s in enumerate(seq):\n",
    "        if s == token:\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)])\n"
   ]
  },
  {
   "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": [
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       "[1, 3, 2, 1]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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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