{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": true,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Efficient Grammar Fuzzing\n",
    "\n",
    "In the [chapter on grammars](Grammars.ipynb), we have seen how to use _grammars_ for very effective and efficient testing.  In this chapter, we refine the previous _string-based_ algorithm into a _tree-based_ algorithm, which is much faster and allows for much more control over the production of fuzz inputs."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": true,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "The algorithm in this chapter serves as a foundation for several more techniques; this chapter thus is a \"hub\" in the book."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:18.874021Z",
     "iopub.status.busy": "2025-01-16T09:40:18.873922Z",
     "iopub.status.idle": "2025-01-16T09:40:18.958811Z",
     "shell.execute_reply": "2025-01-16T09:40:18.958540Z"
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"640\"\n",
       "            height=\"360\"\n",
       "            src=\"https://www.youtube-nocookie.com/embed/Ohl8TLcLl3A\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "            \n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.IFrame at 0x105f24410>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from bookutils import YouTubeVideo\n",
    "YouTubeVideo('Ohl8TLcLl3A')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "**Prerequisites**\n",
    "\n",
    "* You should know how grammar-based fuzzing works, e.g. from the [chapter on grammars](Grammars.ipynb)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "## Synopsis\n",
    "<!-- Automatically generated. Do not edit. -->\n",
    "\n",
    "To [use the code provided in this chapter](Importing.ipynb), write\n",
    "\n",
    "```python\n",
    ">>> from fuzzingbook.GrammarFuzzer import <identifier>\n",
    "```\n",
    "\n",
    "and then make use of the following features.\n",
    "\n",
    "\n",
    "### Efficient Grammar Fuzzing\n",
    "\n",
    "This chapter introduces `GrammarFuzzer`, an efficient grammar fuzzer that takes a grammar to produce syntactically valid input strings.  Here's a typical usage:\n",
    "\n",
    "```python\n",
    ">>> from Grammars import US_PHONE_GRAMMAR\n",
    ">>> phone_fuzzer = GrammarFuzzer(US_PHONE_GRAMMAR)\n",
    ">>> phone_fuzzer.fuzz()\n",
    "'(771)306-0659'\n",
    "```\n",
    "The `GrammarFuzzer` constructor takes a number of keyword arguments to control its behavior.  `start_symbol`, for instance, allows setting the symbol that expansion starts with (instead of `<start>`):\n",
    "\n",
    "```python\n",
    ">>> area_fuzzer = GrammarFuzzer(US_PHONE_GRAMMAR, start_symbol='<area>')\n",
    ">>> area_fuzzer.fuzz()\n",
    "'409'\n",
    "```\n",
    "Here's how to parameterize the `GrammarFuzzer` constructor:\n",
    "\n",
    "```python\n",
    "Produce strings from `grammar`, starting with `start_symbol`.\n",
    "If `min_nonterminals` or `max_nonterminals` is given, use them as limits \n",
    "for the number of nonterminals produced.  \n",
    "If `disp` is set, display the intermediate derivation trees.\n",
    "If `log` is set, show intermediate steps as text on standard output.\n",
    "\n",
    "```\n",
    "![](PICS/GrammarFuzzer-synopsis-1.svg)\n",
    "\n",
    "### Derivation Trees\n",
    "\n",
    "Internally, `GrammarFuzzer` makes use of [derivation trees](#Derivation-Trees), which it expands step by step.  After producing a string, the tree produced can be accessed in the `derivation_tree` attribute.\n",
    "\n",
    "```python\n",
    ">>> display_tree(phone_fuzzer.derivation_tree)\n",
    "```\n",
    "![](PICS/GrammarFuzzer-synopsis-2.svg)\n",
    "\n",
    "In the internal representation of a derivation tree, a _node_ is a pair (`symbol`, `children`).  For nonterminals, `symbol` is the symbol that is being expanded, and `children` is a list of further nodes.  For terminals, `symbol` is the terminal string, and `children` is empty.\n",
    "\n",
    "```python\n",
    ">>> phone_fuzzer.derivation_tree\n",
    "('<start>',\n",
    " [('<phone-number>',\n",
    "   [('(', []),\n",
    "    ('<area>',\n",
    "     [('<lead-digit>', [('7', [])]),\n",
    "      ('<digit>', [('7', [])]),\n",
    "      ('<digit>', [('1', [])])]),\n",
    "    (')', []),\n",
    "    ('<exchange>',\n",
    "     [('<lead-digit>', [('3', [])]),\n",
    "      ('<digit>', [('0', [])]),\n",
    "      ('<digit>', [('6', [])])]),\n",
    "    ('-', []),\n",
    "    ('<line>',\n",
    "     [('<digit>', [('0', [])]),\n",
    "      ('<digit>', [('6', [])]),\n",
    "      ('<digit>', [('5', [])]),\n",
    "      ('<digit>', [('9', [])])])])])\n",
    "```\n",
    "The chapter contains various helpers to work with derivation trees, including visualization tools – notably, `display_tree()`, above.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## An Insufficient Algorithm\n",
    "\n",
    "In the [previous chapter](Grammars.ipynb), we have introduced the `simple_grammar_fuzzer()` function which takes a grammar and automatically produces a syntactically valid string from it.  However, `simple_grammar_fuzzer()` is just what its name suggests – simple.  To illustrate the problem, let us get back to the `expr_grammar` we created from `EXPR_GRAMMAR_BNF` in the [chapter on grammars](Grammars.ipynb):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:18.979645Z",
     "iopub.status.busy": "2025-01-16T09:40:18.979498Z",
     "iopub.status.idle": "2025-01-16T09:40:18.981866Z",
     "shell.execute_reply": "2025-01-16T09:40:18.981616Z"
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "import bookutils.setup"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:18.983519Z",
     "iopub.status.busy": "2025-01-16T09:40:18.983414Z",
     "iopub.status.idle": "2025-01-16T09:40:18.985043Z",
     "shell.execute_reply": "2025-01-16T09:40:18.984796Z"
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from bookutils import quiz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:18.986692Z",
     "iopub.status.busy": "2025-01-16T09:40:18.986578Z",
     "iopub.status.idle": "2025-01-16T09:40:18.988267Z",
     "shell.execute_reply": "2025-01-16T09:40:18.988009Z"
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from typing import Tuple, List, Optional, Any, Union, Set, Callable, Dict"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:18.989658Z",
     "iopub.status.busy": "2025-01-16T09:40:18.989572Z",
     "iopub.status.idle": "2025-01-16T09:40:18.991173Z",
     "shell.execute_reply": "2025-01-16T09:40:18.990932Z"
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from bookutils import unicode_escape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:18.992812Z",
     "iopub.status.busy": "2025-01-16T09:40:18.992695Z",
     "iopub.status.idle": "2025-01-16T09:40:19.348741Z",
     "shell.execute_reply": "2025-01-16T09:40:19.348411Z"
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from Grammars import EXPR_EBNF_GRAMMAR, convert_ebnf_grammar, Grammar, Expansion\n",
    "from Grammars import simple_grammar_fuzzer, is_valid_grammar, exp_string"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:19.350867Z",
     "iopub.status.busy": "2025-01-16T09:40:19.350704Z",
     "iopub.status.idle": "2025-01-16T09:40:19.353307Z",
     "shell.execute_reply": "2025-01-16T09:40:19.353045Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'<start>': ['<expr>'],\n",
       " '<expr>': ['<term> + <expr>', '<term> - <expr>', '<term>'],\n",
       " '<term>': ['<factor> * <term>', '<factor> / <term>', '<factor>'],\n",
       " '<factor>': ['<sign-1><factor>', '(<expr>)', '<integer><symbol-1>'],\n",
       " '<sign>': ['+', '-'],\n",
       " '<integer>': ['<digit-1>'],\n",
       " '<digit>': ['0', '1', '2', '3', '4', '5', '6', '7', '8', '9'],\n",
       " '<symbol>': ['.<integer>'],\n",
       " '<sign-1>': ['', '<sign>'],\n",
       " '<symbol-1>': ['', '<symbol>'],\n",
       " '<digit-1>': ['<digit>', '<digit><digit-1>']}"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_grammar = convert_ebnf_grammar(EXPR_EBNF_GRAMMAR)\n",
    "expr_grammar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "`expr_grammar` has an interesting property.  If we feed it into `simple_grammar_fuzzer()`, the function gets stuck:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:19.354960Z",
     "iopub.status.busy": "2025-01-16T09:40:19.354838Z",
     "iopub.status.idle": "2025-01-16T09:40:19.356361Z",
     "shell.execute_reply": "2025-01-16T09:40:19.356091Z"
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from ExpectError import ExpectTimeout"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:19.357860Z",
     "iopub.status.busy": "2025-01-16T09:40:19.357772Z",
     "iopub.status.idle": "2025-01-16T09:40:20.364506Z",
     "shell.execute_reply": "2025-01-16T09:40:20.364186Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Traceback (most recent call last):\n",
      "  File \"/var/folders/n2/xd9445p97rb3xh7m1dfx8_4h0006ts/T/ipykernel_9136/3259437052.py\", line 2, in <module>\n",
      "    simple_grammar_fuzzer(grammar=expr_grammar, max_nonterminals=3)\n",
      "  File \"Grammars.ipynb\", line 87, in simple_grammar_fuzzer\n",
      "    symbol_to_expand = random.choice(nonterminals(term))\n",
      "                                     ^^^^^^^^^^^^^^^^^^\n",
      "  File \"Grammars.ipynb\", line 61, in nonterminals\n",
      "    return RE_NONTERMINAL.findall(expansion)\n",
      "           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n",
      "  File \"Timeout.ipynb\", line 43, in timeout_handler\n",
      "    raise TimeoutError()\n",
      "TimeoutError (expected)\n"
     ]
    }
   ],
   "source": [
    "with ExpectTimeout(1):\n",
    "    simple_grammar_fuzzer(grammar=expr_grammar, max_nonterminals=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Why is that so?  Have a look at the grammar; remember what you know about `simple_grammar_fuzzer()`; and run `simple_grammar_fuzzer()` with `log=true` argument to see the expansions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:20.366294Z",
     "iopub.status.busy": "2025-01-16T09:40:20.366181Z",
     "iopub.status.idle": "2025-01-16T09:40:20.370684Z",
     "shell.execute_reply": "2025-01-16T09:40:20.370395Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "    \n",
       "    <script>\n",
       "    var bad_answers = new Map();\n",
       "\n",
       "    function answer(quiz_id) {\n",
       "        ans = 0;\n",
       "        for (i = 1;; i++) {\n",
       "            checkbox = document.getElementById(quiz_id + \"-\" + i.toString());\n",
       "            if (!checkbox)\n",
       "                break;\n",
       "            if (checkbox.checked)\n",
       "                ans |= (1 << i);\n",
       "        }\n",
       "        return ans;\n",
       "    }\n",
       "    function check_selection(quiz_id, correct_answer, multiple_choice, hint) {\n",
       "        given_answer = answer(quiz_id);\n",
       "        if (given_answer == correct_answer)\n",
       "        {\n",
       "            document.getElementById(quiz_id + \"-submit\").value = \"Correct!\";\n",
       "            document.getElementById(quiz_id + \"-hint\").innerHTML = \"\";\n",
       "\n",
       "            for (i = 1;; i++) {\n",
       "                checkbox = document.getElementById(quiz_id + \"-\" + i.toString());\n",
       "                label = document.getElementById(quiz_id + \"-\" + i.toString() + \"-label\")\n",
       "                if (!checkbox)\n",
       "                    break;\n",
       "\n",
       "                if (checkbox.checked) {\n",
       "                    label.style.fontWeight = \"bold\";\n",
       "                }\n",
       "                else {\n",
       "                    label.style.textDecoration = \"line-through\";\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "        else \n",
       "        {\n",
       "            document.getElementById(quiz_id + \"-submit\").value = \"Try again\";\n",
       "\n",
       "            if (!bad_answers.has(quiz_id)) {\n",
       "                bad_answers.set(quiz_id, 1);\n",
       "            }\n",
       "            else {\n",
       "                bad_answers.set(quiz_id, bad_answers.get(quiz_id) + 1);\n",
       "            }\n",
       "\n",
       "            if (bad_answers.get(quiz_id) >= 2 && hint.length > 0) {\n",
       "                document.getElementById(quiz_id + \"-hint\").innerHTML = \n",
       "                    \"&nbsp;&nbsp;(Hint: <code>\" + hint + \"</code>)\";\n",
       "            }\n",
       "\n",
       "            if (!multiple_choice) {\n",
       "                for (i = 1;; i++) {\n",
       "                    checkbox = document.getElementById(quiz_id + \"-\" + i.toString());\n",
       "                    label = document.getElementById(quiz_id + \"-\" + i.toString() + \"-label\")\n",
       "\n",
       "                    if (!checkbox)\n",
       "                        break;\n",
       "                    if (checkbox.checked) {\n",
       "                        label.style.textDecoration = \"line-through\";\n",
       "                    }\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "    function clear_selection(quiz_id) {\n",
       "        document.getElementById(quiz_id + \"-submit\").value = \"Submit\";\n",
       "        document.getElementById(quiz_id + \"-hint\").innerHTML = \"\";\n",
       "    }\n",
       "    </script>\n",
       "    \n",
       "    <div class=\"quiz\">\n",
       "    <h3 class=\"quiz_title\">Quiz</h3>\n",
       "    <p>\n",
       "    <div class=\"quiz_question\">Why does <code>simple_grammar_fuzzer()</code> hang?</div>\n",
       "    </p>\n",
       "    <p>\n",
       "    <div class=\"quiz_options\" title=\"Pick a choice.\">\n",
       "    \n",
       "        <input type=\"radio\" name=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790\" id=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790-1\" onclick=\"clear_selection('e6bd93ec-d3ed-11ef-8343-6298cf1a5790')\">\n",
       "        <label id=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790-1-label\" for=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790-1\">It produces an infinite number of additions</label><br>\n",
       "    \n",
       "        <input type=\"radio\" name=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790\" id=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790-2\" onclick=\"clear_selection('e6bd93ec-d3ed-11ef-8343-6298cf1a5790')\">\n",
       "        <label id=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790-2-label\" for=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790-2\">It produces an infinite number of digits</label><br>\n",
       "    \n",
       "        <input type=\"radio\" name=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790\" id=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790-3\" onclick=\"clear_selection('e6bd93ec-d3ed-11ef-8343-6298cf1a5790')\">\n",
       "        <label id=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790-3-label\" for=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790-3\">It produces an infinite number of parentheses</label><br>\n",
       "    \n",
       "        <input type=\"radio\" name=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790\" id=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790-4\" onclick=\"clear_selection('e6bd93ec-d3ed-11ef-8343-6298cf1a5790')\">\n",
       "        <label id=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790-4-label\" for=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790-4\">It produces an infinite number of signs</label><br>\n",
       "    \n",
       "    </div>\n",
       "    </p>\n",
       "    <input id=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790-submit\" type=\"submit\" value=\"Submit\" onclick=\"check_selection('e6bd93ec-d3ed-11ef-8343-6298cf1a5790', 8, 0, '(3 * 3 * 3) ** (3 / (3 * 3))')\">\n",
       "    <span class=\"quiz_hint\" id=\"e6bd93ec-d3ed-11ef-8343-6298cf1a5790-hint\"></span>\n",
       "    </div>\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "quiz(\"Why does `simple_grammar_fuzzer()` hang?\",\n",
    "     [\n",
    "         \"It produces an infinite number of additions\",\n",
    "         \"It produces an infinite number of digits\",\n",
    "         \"It produces an infinite number of parentheses\",\n",
    "         \"It produces an infinite number of signs\",\n",
    "     ], '(3 * 3 * 3) ** (3 / (3 * 3))')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Indeed! The problem is in this rule:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:20.372829Z",
     "iopub.status.busy": "2025-01-16T09:40:20.372710Z",
     "iopub.status.idle": "2025-01-16T09:40:20.374845Z",
     "shell.execute_reply": "2025-01-16T09:40:20.374590Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['<sign-1><factor>', '(<expr>)', '<integer><symbol-1>']"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_grammar['<factor>']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Here, any choice except for `(expr)` increases the number of symbols, even if only temporary.  Since we place a hard limit on the number of symbols to expand, the only choice left for expanding `<factor>` is `(<expr>)`, which leads to an _infinite addition of parentheses._"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The problem of potentially infinite expansion is only one of the problems with `simple_grammar_fuzzer()`.  More problems include:\n",
    "\n",
    "1. *It is inefficient*.  With each iteration, this fuzzer would go search the string produced so far for symbols to expand.  This becomes inefficient as the production string grows.\n",
    "\n",
    "2. *It is hard to control.*  Even while limiting the number of symbols, it is still possible to obtain very long strings – and even infinitely long ones, as discussed above.\n",
    "\n",
    "Let us illustrate both problems by plotting the time required for strings of different lengths."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:20.376360Z",
     "iopub.status.busy": "2025-01-16T09:40:20.376253Z",
     "iopub.status.idle": "2025-01-16T09:40:20.377813Z",
     "shell.execute_reply": "2025-01-16T09:40:20.377567Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from Grammars import simple_grammar_fuzzer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:20.379154Z",
     "iopub.status.busy": "2025-01-16T09:40:20.379063Z",
     "iopub.status.idle": "2025-01-16T09:40:20.380653Z",
     "shell.execute_reply": "2025-01-16T09:40:20.380417Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from Grammars import START_SYMBOL, EXPR_GRAMMAR, URL_GRAMMAR, CGI_GRAMMAR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:20.382163Z",
     "iopub.status.busy": "2025-01-16T09:40:20.382066Z",
     "iopub.status.idle": "2025-01-16T09:40:20.383490Z",
     "shell.execute_reply": "2025-01-16T09:40:20.383265Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from Grammars import RE_NONTERMINAL, nonterminals, is_nonterminal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:20.384868Z",
     "iopub.status.busy": "2025-01-16T09:40:20.384771Z",
     "iopub.status.idle": "2025-01-16T09:40:20.386281Z",
     "shell.execute_reply": "2025-01-16T09:40:20.386051Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from Timer import Timer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:20.387571Z",
     "iopub.status.busy": "2025-01-16T09:40:20.387480Z",
     "iopub.status.idle": "2025-01-16T09:40:25.392152Z",
     "shell.execute_reply": "2025-01-16T09:40:25.391836Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 \n"
     ]
    }
   ],
   "source": [
    "trials = 50\n",
    "xs = []\n",
    "ys = []\n",
    "for i in range(trials):\n",
    "    with Timer() as t:\n",
    "        s = simple_grammar_fuzzer(EXPR_GRAMMAR, max_nonterminals=15)\n",
    "    xs.append(len(s))\n",
    "    ys.append(t.elapsed_time())\n",
    "    print(i, end=\" \")\n",
    "print()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:25.394006Z",
     "iopub.status.busy": "2025-01-16T09:40:25.393882Z",
     "iopub.status.idle": "2025-01-16T09:40:25.395718Z",
     "shell.execute_reply": "2025-01-16T09:40:25.395468Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Average time: 0.10001757324207575\n"
     ]
    }
   ],
   "source": [
    "average_time = sum(ys) / trials\n",
    "print(\"Average time:\", average_time)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:25.397397Z",
     "iopub.status.busy": "2025-01-16T09:40:25.397270Z",
     "iopub.status.idle": "2025-01-16T09:40:25.489703Z",
     "shell.execute_reply": "2025-01-16T09:40:25.489340Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAiwAAAGzCAYAAAAMr0ziAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjkuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8hTgPZAAAACXBIWXMAAA9hAAAPYQGoP6dpAABDS0lEQVR4nO3deVxVdeL/8feF5IILVxTZFHHNwrUoibLMxNAc0xa3qREdtbKyhVZnJrGpR7RN2aijbe5NtltNRimJfivMXKjIdNRQU7m4AoKCCp/fH/244xVQLgoc4PV8PO6j7jmf87mf8+Hce9+ecz6fazPGGAEAAFiYV203AAAA4GwILAAAwPIILAAAwPIILAAAwPIILAAAwPIILAAAwPIILAAAwPIILAAAwPIILAAAwPIILDhvxo4dq3bt2tV2Myxv2rRpstlsNfJaqampstlsSk1NPWvZ77//XldeeaWaNGkim82m9PT0am9fQ+bJ3wYAgQVnYbPZKvXgQ7duO3HihIYPH65Dhw7p5Zdf1qJFixQREVHbzaoX/vWvf2n+/Pm13Yx645lnntHSpUtr5LW+/fZbTZs2TTk5OTXyejgzG78lhDNZvHix2/OFCxdq+fLlWrRokdvyAQMGqEWLFiopKZHdbq/JJtY5J0+e1MmTJ+Xr61vtr5Wamqp+/fpp5cqVuvbaaysst3nzZl188cV6/fXXNWHChGpvV0PSrVs3BQYGlgn1JSUlOn78uHx8fOTlxb8dK6tp06a69dZbayQEvvjii3rkkUeUmZnJ2WMLuKC2GwBru/32292er1mzRsuXLy+zvK4oKChQkyZNarUNF1xwgS644MxvvdIvs5oINZK0b98+SVLz5s3PW51W6OvzzRijwsJC+fn5nXNdXl5eNfb3BeoDYj3Om9PvYdmxY4dsNptefPFFzZo1Sx06dFDjxo11/fXX67fffpMxRk899ZTatGkjPz8/DR06VIcOHSpT7+eff66rr75aTZo0UbNmzTR48GD9/PPPZ23P/PnzZbPZtGrVKt19990KCgpSmzZtPK536dKl6tatm3x9fdWtWzd99NFHZfa1ovsRSvvg1H8NlncPi81m07333qu33npLXbt2ld1uV3JysiRpz549+vOf/6zg4GDZ7XZ17dpVc+fOLdPO3bt3a9iwYWrSpImCgoL04IMPqqio6Kz9NHbsWPXt21eSNHz4cNlsNrezMV999ZWrn5o3b66hQ4fql19+caujdJ82bdqkP/7xjwoICFCfPn3O+Lo//vij+vbtKz8/P7Vp00ZPP/205s2bJ5vNph07driVrczfauzYsWratKn27NmjYcOGqWnTpmrVqpUefvhhFRcXu5UtKSnR9OnT1bVrV/n6+io4OFh33nmnDh8+7FauXbt2+sMf/qAvvvhCl112mfz8/PTqq69KkubNm6frrrtOQUFBstvtioyM1OzZs8ts//PPP2vVqlWuy6elfVveMXPttdeqW7du2rRpk/r166fGjRurdevWev7558v0386dO3XjjTe6/b2/+OKLSl2i3blzp+6++2516dJFfn5+atmypYYPH16m30vfQ998840SEhLUqlUrNWnSRDfddJP2799/xtcoVZnjp6L7305/r9hsNhUUFGjBggWu/hw7dqxb2c2bN2vEiBHy9/dXy5Ytdf/996uwsNBVR3nvyVPrnzZtmqu+Rx55RJLUvn171+ud3keoOZxhQbV76623dPz4cU2ePFmHDh3S888/rxEjRui6665TamqqHnvsMW3btk0zZszQww8/7PZlvGjRIsXHxysuLk7PPfecjh49qtmzZ6tPnz7auHFjpU7T3n333WrVqpWmTp2qgoICj+r98ssvdcsttygyMlJJSUk6ePCgxo0b5xZ8zpevvvpK7777ru69914FBgaqXbt2ys7O1hVXXOEKNK1atdLnn3+u8ePHKy8vTw888IAk6dixY+rfv7927dql++67T2FhYVq0aJG++uqrs77unXfeqdatW+uZZ57Rfffdp8svv1zBwcGSpBUrVmjQoEHq0KGDpk2bpmPHjmnGjBm66qqrtGHDhjL9P3z4cHXu3FnPPPOMznS1ec+ePerXr59sNpumTJmiJk2a6I033ij3cqInx0BxcbHi4uIUHR2tF198UStWrNA//vEPdezYUZMmTXLb5/nz52vcuHG67777lJmZqZkzZ2rjxo365ptv1KhRI1fZLVu2aPTo0brzzjs1ceJEdenSRZI0e/Zsde3aVTfeeKMuuOACffrpp7r77rtVUlKie+65R5I0ffp0TZ48WU2bNtVf//pXSXL1bUUOHz6sgQMH6uabb9aIESP0/vvv67HHHlP37t01aNAgSb+fvbruuuuUlZWl+++/XyEhIfr3v/+tlStXnrHuUt9//72+/fZbjRo1Sm3atNGOHTs0e/ZsXXvttdq0aZMaN27sVn7y5MkKCAhQYmKiduzYoenTp+vee+/VO++8c8bX8fT4OZtFixZpwoQJ6t27t+644w5JUseOHd3KjBgxQu3atVNSUpLWrFmjf/7znzp8+LAWLlzo0WvdfPPN+u9//6u3335bL7/8sgIDAyVJrVq18qgenEcG8MA999xjKjps4uPjTUREhOt5ZmamkWRatWplcnJyXMunTJliJJmePXuaEydOuJaPHj3a+Pj4mMLCQmOMMUeOHDHNmzc3EydOdHsdp9NpHA5HmeWnmzdvnpFk+vTpY06ePOla7km9vXr1MqGhoW7t//LLL40kt31duXKlkWRWrlzpVmdpH8ybN8+1LDExsUwfSjJeXl7m559/dls+fvx4Exoaag4cOOC2fNSoUcbhcJijR48aY4yZPn26kWTeffddV5mCggLTqVOnctt1utL2v/fee27Le/XqZYKCgszBgwddy3744Qfj5eVlxowZU2afRo8efcbXKTV58mRjs9nMxo0bXcsOHjxoWrRoYSSZzMxMY4xnf6v4+Hgjyfz97393K3vJJZeYqKgo1/P/+7//M5LMW2+95VYuOTm5zPKIiAgjySQnJ5fZh9K+P1VcXJzp0KGD27KuXbuavn37lilb3jHTt29fI8ksXLjQtayoqMiEhISYW265xbXsH//4h5Fkli5d6lp27Ngxc9FFF1Xq711e29PS0sq8dul7KDY21pSUlLiWP/jgg8bb29vtfVGeyh4/p392lCrvvdKkSRMTHx9fYdkbb7zRbfndd99tJJkffvjBGFP+e7KUJJOYmOh6/sILL7gdj6hdXBJCtRs+fLgcDofreXR0tKTf74859V6O6OhoHT9+XHv27JEkLV++XDk5ORo9erQOHDjgenh7eys6OrrS/5qcOHGivL29Xc8rW29WVpbS09MVHx/v1v4BAwYoMjKy6h1Sgb59+7rVa4zRBx98oCFDhsgY49bWuLg45ebmasOGDZKkZcuWKTQ0VLfeeqtr+8aNG7v+FVoVpfs/duxYtWjRwrW8R48eGjBggJYtW1Zmm7vuuqtSdScnJysmJka9evVyLWvRooVuu+02t3JVOQZOb8PVV1+tX3/91fX8vffek8Ph0IABA9zqjIqKUtOmTcvU2b59e8XFxZV5nVPvY8nNzdWBAwfUt29f/frrr8rNza1UP5SnadOmbveI+fj4qHfv3m77kJycrNatW+vGG290LfP19dXEiRMr9Rqntv3EiRM6ePCgOnXqpObNm7uOqVPdcccdbpdmrr76ahUXF2vnzp0VvkZVjp/zofTsVqnJkydLUrW9HmoOl4RQ7dq2bev2vPTLPzw8vNzlpfcRbN26VZJ03XXXlVuvv79/pV6/ffv2bs8rW2/ph3Hnzp3LlOnSpUu5H+zn4vR27t+/Xzk5OXrttdf02muvlbtN6c2yO3fuVKdOncrcG1N6+aIqSve/vDouvvhiffHFF2VurD19H85Ud0xMTJnlnTp1cnvu6THg6+tb5pR9QECA270pW7duVW5uroKCgsqts7RPS1W0T998840SExOVlpamo0ePuq3Lzc11C7meaNOmTZm/Y0BAgH788UfX8507d6pjx45lyp3efxU5duyYkpKSNG/ePO3Zs8ft8l15Yev093BAQIAklbnn51RVOX7Oh9Pfrx07dpSXlxf3ntQDBBZUu1PPblRmeemHZ0lJiaTfr1uHhISUKXe2kTalTh/Rcb7qPVVFE8GdfrPnmVTUzttvv13x8fHlbtOjR49K118TzsfomVN5+req6Jg6vc6goCC99dZb5a4/PfCUt0/bt29X//79ddFFF+mll15SeHi4fHx8tGzZMr388suudlfF2d4X58PkyZM1b948PfDAA4qJiZHD4ZDNZtOoUaPKbXt1t+l8vH8qW3d1vhaqF4EFllV6M11QUJBiY2NrvN7SidNK/5V/qi1btrg9L/0X5+kTTJ3plPnZtGrVSs2aNVNxcfFZ9z8iIkIZGRkyxrh9IJ/eTk+U7n95dWzevFmBgYFV/tdxRESEtm3bVmb56cuq4xjo2LGjVqxYoauuuqrKAevTTz9VUVGRPvnkE7ezD+VdoqqOWY0jIiK0adOmMn/v8vq0PO+//77i4+P1j3/8w7WssLDwvE6Q5snxExAQUO5rl/f+OVt/bt261e2s2LZt21RSUuK6wdeT92pNzUiNyuEeFlhWXFyc/P399cwzz+jEiRNl1ld2WGVV6w0NDVWvXr20YMECt9Pky5cv16ZNm9y2iYiIkLe3t1avXu22/F//+leV2ij9/q/aW265RR988IEyMjIqbKck3XDDDdq7d6/ef/9917KjR49WeCmpMk7d/1M/3DMyMvTll1/qhhtuqHLdcXFxSktLc5v+/9ChQ2XOelTHMTBixAgVFxfrqaeeKrPu5MmTlfrSLj3jcPqllHnz5pUp26RJk/M+U2pcXJz27NmjTz75xLWssLBQr7/+eqW29/b2LnN2ZMaMGef1LIMnx0/Hjh2Vm5vrdtkrKytLH330UZl6z9afs2bNcns+Y8YMSXKNsPL391dgYGCl3qulgYqZbq2BMyywLH9/f82ePVt/+tOfdOmll2rUqFFq1aqVdu3apc8++0xXXXWVZs6cWa31JiUlafDgwerTp4/+/Oc/69ChQ5oxY4a6du2q/Px8V50Oh0PDhw/XjBkzZLPZ1LFjR/3nP/8pcz+Ep5599lmtXLlS0dHRmjhxoiIjI3Xo0CFt2LBBK1ascM1bM3HiRM2cOVNjxozR+vXrFRoaqkWLFpUZnuqpF154QYMGDVJMTIzGjx/vGpbqcDhc81VUxaOPPqrFixdrwIABmjx5smtYc9u2bXXo0CHXv2yr4xjo27ev7rzzTiUlJSk9PV3XX3+9GjVqpK1bt+q9997TK6+84nbzcnmuv/56+fj4aMiQIbrzzjuVn5+v119/XUFBQcrKynIrGxUVpdmzZ+vpp59Wp06dFBQUVOE9OZV15513aubMmRo9erTuv/9+hYaG6q233nJNRHe2MwN/+MMftGjRIjkcDkVGRiotLU0rVqxQy5Ytz6ldp6vs8TNq1Cg99thjuummm3Tfffe5hq5feOGFZe4Vi4qK0ooVK/TSSy8pLCxM7du3d93IL0mZmZm68cYbNXDgQKWlpWnx4sX64x//qJ49e7rKTJgwQc8++6wmTJigyy67TKtXr9Z///vfMu2PioqSJP31r3/VqFGj1KhRIw0ZMqTeTYhYZ9TW8CTUTVUZ1vzCCy+4latoCG3pEMrvv/++TPm4uDjjcDiMr6+v6dixoxk7dqxZt27dGdtaUX2e1vvBBx+Yiy++2NjtdhMZGWk+/PDDcodh7t+/39xyyy2mcePGJiAgwNx5550mIyOj0sOa77nnnnLbmZ2dbe655x4THh5uGjVqZEJCQkz//v3Na6+95lZu586d5sYbbzSNGzc2gYGB5v7773cN1a3qsGZjjFmxYoW56qqrjJ+fn/H39zdDhgwxmzZtcitTuk/79+8/4+ucauPGjebqq682drvdtGnTxiQlJZl//vOfRpJxOp1l2ne2v1V8fLxp0qRJmdcpr7+NMea1114zUVFRxs/PzzRr1sx0797dPProo2bv3r2uMhEREWbw4MHltv+TTz4xPXr0ML6+vqZdu3bmueeeM3Pnzi0zDNbpdJrBgwebZs2aGUmuIc4VDWvu2rVrmdcq73j79ddfzeDBg42fn59p1aqVeeihh8wHH3xgJJk1a9aU2+ZShw8fNuPGjTOBgYGmadOmJi4uzmzevNlERES4DRk+03uyMseVMZU7foz5fbqAbt26GR8fH9OlSxezePHicv92mzdvNtdcc43x8/MzklztLS27adMmc+utt5pmzZqZgIAAc++995pjx4651XH06FEzfvx443A4TLNmzcyIESPMvn37ygxrNsaYp556yrRu3dp4eXkxxLmW8VtCQBWMHTtWqampjDw4zx544AG9+uqrys/Pr9QNtHA3ffp0Pfjgg9q9e7dat25d282pUdOmTdOTTz6p/fv3uyZ5Q/3CPSwAasWxY8fcnh88eFCLFi1Snz59CCuVcHr/FRYW6tVXX1Xnzp0bXFhBw8A9LABqRUxMjK699lpdfPHFys7O1ptvvqm8vDw98cQTtd20OuHmm29W27Zt1atXL+Xm5mrx4sXavHlzhcO1gbqOwAKgVtxwww16//339dprr8lms+nSSy/Vm2++qWuuuaa2m1YnxMXF6Y033tBbb72l4uJiRUZGasmSJRo5cmRtNw2oFtzDAgAALI97WAAAgOURWAAAgOXVi3tYSkpKtHfvXjVr1oyplAEAqCOMMTpy5IjCwsLk5XXmcyj1IrDs3bu3zC//AgCAuuG3335TmzZtzlimXgSWZs2aSfp9h0//uXkAAGBNeXl5Cg8Pd32Pn0m9CCyn/u4IgQUAgLqlMrdzcNMtAACwPAILAACwPAILAACwPAILAACwPAILAACwPAILAACwPI8Dy+rVqzVkyBCFhYXJZrNp6dKlZyw/duxY2Wy2Mo+uXbu6ykybNq3M+osuusjjnQEAAPWTx4GloKBAPXv21KxZsypV/pVXXlFWVpbr8dtvv6lFixYaPny4W7muXbu6lfv66689bRoAAKinPJ44btCgQRo0aFClyzscDjkcDtfzpUuX6vDhwxo3bpx7Qy64QCEhIZ42BwAAVKPiEqO1mYe070ihgpr5qnf7FvL2qvnf7avxmW7ffPNNxcbGKiIiwm351q1bFRYWJl9fX8XExCgpKUlt27Ytt46ioiIVFRW5nufl5VVrmwEAaIiSM7L05KeblJVb6FoW6vBV4pBIDewWWqNtqdGbbvfu3avPP/9cEyZMcFseHR2t+fPnKzk5WbNnz1ZmZqauvvpqHTlypNx6kpKSXGduHA4HP3wIAMB5lpyRpUmLN7iFFUly5hZq0uINSs7IqtH21GhgWbBggZo3b65hw4a5LR80aJCGDx+uHj16KC4uTsuWLVNOTo7efffdcuuZMmWKcnNzXY/ffvutBloPAEDDUFxi9OSnm2TKWVe67MlPN6m4pLwS1aPGLgkZYzR37lz96U9/ko+PzxnLNm/eXBdeeKG2bdtW7nq73S673V4dzQQAoMFbm3mozJmVUxlJWbmFWpt5SDEdW9ZIm2rsDMuqVau0bds2jR8//qxl8/PztX37doWG1uz1MQAAIO07UnFYqUq588HjwJKfn6/09HSlp6dLkjIzM5Wenq5du3ZJ+v1yzZgxY8ps9+abbyo6OlrdunUrs+7hhx/WqlWrtGPHDn377be66aab5O3trdGjR3vaPAAAcI6Cmvme13Lng8eXhNatW6d+/fq5nickJEiS4uPjNX/+fGVlZbnCS6nc3Fx98MEHeuWVV8qtc/fu3Ro9erQOHjyoVq1aqU+fPlqzZo1atWrlafMAAMA56t2+hUIdvnLmFpZ7H4tNUojj9yHONcVmjKm5O2aqSV5enhwOh3Jzc+Xv71/bzQEAoM4rHSUkyS20lM7AMvv2S895aLMn39/8lhAAAChjYLdQzb79UoU43C/7hDh8z0tY8VSNTxwHAADqhoHdQjUgMqRhznQLAADqDm8vW40NXT4TLgkBAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADLI7AAAADL8ziwrF69WkOGDFFYWJhsNpuWLl16xvKpqamy2WxlHk6n063crFmz1K5dO/n6+io6Olpr1671tGkAAKCe8jiwFBQUqGfPnpo1a5ZH223ZskVZWVmuR1BQkGvdO++8o4SEBCUmJmrDhg3q2bOn4uLitG/fPk+bBwAA6qELPN1g0KBBGjRokMcvFBQUpObNm5e77qWXXtLEiRM1btw4SdKcOXP02Wefae7cuXr88cc9fi0AAFC/1Ng9LL169VJoaKgGDBigb775xrX8+PHjWr9+vWJjY//XKC8vxcbGKi0trdy6ioqKlJeX5/YAAAD1V7UHltDQUM2ZM0cffPCBPvjgA4WHh+vaa6/Vhg0bJEkHDhxQcXGxgoOD3bYLDg4uc59LqaSkJDkcDtcjPDy8uncDAADUIo8vCXmqS5cu6tKli+v5lVdeqe3bt+vll1/WokWLqlTnlClTlJCQ4Hqel5dHaAEAoB6r9sBSnt69e+vrr7+WJAUGBsrb21vZ2dluZbKzsxUSElLu9na7XXa7vdrbCQAArKFW5mFJT09XaGioJMnHx0dRUVFKSUlxrS8pKVFKSopiYmJqo3kAAMBiPD7Dkp+fr23btrmeZ2ZmKj09XS1atFDbtm01ZcoU7dmzRwsXLpQkTZ8+Xe3bt1fXrl1VWFioN954Q1999ZW+/PJLVx0JCQmKj4/XZZddpt69e2v69OkqKChwjRoCAAANm8eBZd26derXr5/reem9JPHx8Zo/f76ysrK0a9cu1/rjx4/roYce0p49e9S4cWP16NFDK1ascKtj5MiR2r9/v6ZOnSqn06levXopOTm5zI24AACgYbIZY0xtN+Jc5eXlyeFwKDc3V/7+/rXdHAAAUAmefH/zW0IAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyPA4sq1ev1pAhQxQWFiabzaalS5eesfyHH36oAQMGqFWrVvL391dMTIy++OILtzLTpk2TzWZze1x00UWeNg0AANRTHgeWgoIC9ezZU7NmzapU+dWrV2vAgAFatmyZ1q9fr379+mnIkCHauHGjW7muXbsqKyvL9fj66689bRoAAKinLvB0g0GDBmnQoEGVLj99+nS3588884w+/vhjffrpp7rkkkv+15ALLlBISIinzQEAAA1Ajd/DUlJSoiNHjqhFixZuy7du3aqwsDB16NBBt912m3bt2lVhHUVFRcrLy3N7AACA+qvGA8uLL76o/Px8jRgxwrUsOjpa8+fPV3JysmbPnq3MzExdffXVOnLkSLl1JCUlyeFwuB7h4eE11XwAAFALbMYYU+WNbTZ99NFHGjZsWKXK//vf/9bEiRP18ccfKzY2tsJyOTk5ioiI0EsvvaTx48eXWV9UVKSioiLX87y8PIWHhys3N1f+/v4e7wcAAKh5eXl5cjgclfr+9vgelqpasmSJJkyYoPfee++MYUWSmjdvrgsvvFDbtm0rd73dbpfdbq+OZgIAAAuqkUtCb7/9tsaNG6e3335bgwcPPmv5/Px8bd++XaGhoTXQOgAAYHUen2HJz893O/ORmZmp9PR0tWjRQm3bttWUKVO0Z88eLVy4UNLvl4Hi4+P1yiuvKDo6Wk6nU5Lk5+cnh8MhSXr44Yc1ZMgQRUREaO/evUpMTJS3t7dGjx59PvYRAADUcR6fYVm3bp0uueQS15DkhIQEXXLJJZo6daokKSsry22Ez2uvvaaTJ0/qnnvuUWhoqOtx//33u8rs3r1bo0ePVpcuXTRixAi1bNlSa9asUatWrc51/wAAQD1wTjfdWoUnN+0AAABr8OT7m98SAgAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAlkdgAQAAludxYFm9erWGDBmisLAw2Ww2LV269KzbpKam6tJLL5XdblenTp00f/78MmVmzZqldu3aydfXV9HR0Vq7dq2nTQMAAPWUx4GloKBAPXv21KxZsypVPjMzU4MHD1a/fv2Unp6uBx54QBMmTNAXX3zhKvPOO+8oISFBiYmJ2rBhg3r27Km4uDjt27fP0+YBAIB6yGaMMVXe2GbTRx99pGHDhlVY5rHHHtNnn32mjIwM17JRo0YpJydHycnJkqTo6GhdfvnlmjlzpiSppKRE4eHhmjx5sh5//PGztiMvL08Oh0O5ubny9/ev6u4AAIAa5Mn3d7Xfw5KWlqbY2Fi3ZXFxcUpLS5MkHT9+XOvXr3cr4+XlpdjYWFeZ0xUVFSkvL8/tAQAA6q9qDyxOp1PBwcFuy4KDg5WXl6djx47pwIEDKi4uLreM0+kst86kpCQ5HA7XIzw8vNraDwAAal+dHCU0ZcoU5ebmuh6//fZbbTcJAABUowuq+wVCQkKUnZ3ttiw7O1v+/v7y8/OTt7e3vL29yy0TEhJSbp12u112u73a2gwAAKyl2s+wxMTEKCUlxW3Z8uXLFRMTI0ny8fFRVFSUW5mSkhKlpKS4ygAAgIbN48CSn5+v9PR0paenS/p92HJ6erp27dol6ffLNWPGjHGVv+uuu/Trr7/q0Ucf1ebNm/Wvf/1L7777rh588EFXmYSEBL3++utasGCBfvnlF02aNEkFBQUaN27cOe4eAACoDzy+JLRu3Tr169fP9TwhIUGSFB8fr/nz5ysrK8sVXiSpffv2+uyzz/Tggw/qlVdeUZs2bfTGG28oLi7OVWbkyJHav3+/pk6dKqfTqV69eik5ObnMjbgAAKBhOqd5WKyCeVgAAKh7LDUPCwAAwLkisAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMurUmCZNWuW2rVrJ19fX0VHR2vt2rUVlr322mtls9nKPAYPHuwqM3bs2DLrBw4cWJWmAQCAeugCTzd45513lJCQoDlz5ig6OlrTp09XXFyctmzZoqCgoDLlP/zwQx0/ftz1/ODBg+rZs6eGDx/uVm7gwIGaN2+e67ndbve0aQAAoJ7y+AzLSy+9pIkTJ2rcuHGKjIzUnDlz1LhxY82dO7fc8i1atFBISIjrsXz5cjVu3LhMYLHb7W7lAgICqrZHAACg3vEosBw/flzr169XbGzs/yrw8lJsbKzS0tIqVcebb76pUaNGqUmTJm7LU1NTFRQUpC5dumjSpEk6ePBghXUUFRUpLy/P7QEAAOovjwLLgQMHVFxcrODgYLflwcHBcjqdZ91+7dq1ysjI0IQJE9yWDxw4UAsXLlRKSoqee+45rVq1SoMGDVJxcXG59SQlJcnhcLge4eHhnuwGAACoYzy+h+VcvPnmm+revbt69+7ttnzUqFGu/+/evbt69Oihjh07KjU1Vf379y9Tz5QpU5SQkOB6npeXR2gBAKAe8+gMS2BgoLy9vZWdne22PDs7WyEhIWfctqCgQEuWLNH48ePP+jodOnRQYGCgtm3bVu56u90uf39/twcAAKi/PAosPj4+ioqKUkpKimtZSUmJUlJSFBMTc8Zt33vvPRUVFen2228/6+vs3r1bBw8eVGhoqCfNAwAA9ZTHo4QSEhL0+uuva8GCBfrll180adIkFRQUaNy4cZKkMWPGaMqUKWW2e/PNNzVs2DC1bNnSbXl+fr4eeeQRrVmzRjt27FBKSoqGDh2qTp06KS4uroq7BQAA6hOP72EZOXKk9u/fr6lTp8rpdKpXr15KTk523Yi7a9cueXm556AtW7bo66+/1pdfflmmPm9vb/34449asGCBcnJyFBYWpuuvv15PPfUUc7EAAABJks0YY2q7EecqLy9PDodDubm53M8CAEAd4cn3N78lBAAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALI/AAgAALK9KgWXWrFlq166dfH19FR0drbVr11ZYdv78+bLZbG4PX19ftzLGGE2dOlWhoaHy8/NTbGystm7dWpWmAQCAesjjwPLOO+8oISFBiYmJ2rBhg3r27Km4uDjt27evwm38/f2VlZXleuzcudNt/fPPP69//vOfmjNnjr777js1adJEcXFxKiws9HyPAABAveNxYHnppZc0ceJEjRs3TpGRkZozZ44aN26suXPnVriNzWZTSEiI6xEcHOxaZ4zR9OnT9be//U1Dhw5Vjx49tHDhQu3du1dLly4tt76ioiLl5eW5PQAAQP3lUWA5fvy41q9fr9jY2P9V4OWl2NhYpaWlVbhdfn6+IiIiFB4erqFDh+rnn392rcvMzJTT6XSr0+FwKDo6usI6k5KS5HA4XI/w8HBPdgMAANQxHgWWAwcOqLi42O0MiSQFBwfL6XSWu02XLl00d+5cffzxx1q8eLFKSkp05ZVXavfu3ZLk2s6TOqdMmaLc3FzX47fffvNkNwAAQB1zQXW/QExMjGJiYlzPr7zySl188cV69dVX9dRTT1WpTrvdLrvdfr6aCAAALM6jMyyBgYHy9vZWdna22/Ls7GyFhIRUqo5GjRrpkksu0bZt2yTJtd251AkAAOo3jwKLj4+PoqKilJKS4lpWUlKilJQUt7MoZ1JcXKyffvpJoaGhkqT27dsrJCTErc68vDx99913la4TAADUbx5fEkpISFB8fLwuu+wy9e7dW9OnT1dBQYHGjRsnSRozZoxat26tpKQkSdLf//53XXHFFerUqZNycnL0wgsvaOfOnZowYYKk30cQPfDAA3r66afVuXNntW/fXk888YTCwsI0bNiw87enAACgzvI4sIwcOVL79+/X1KlT5XQ61atXLyUnJ7tumt21a5e8vP534ubw4cOaOHGinE6nAgICFBUVpW+//VaRkZGuMo8++qgKCgp0xx13KCcnR3369FFycnKZCeYAAEDDZDPGmNpuxLnKy8uTw+FQbm6u/P39a7s5AACgEjz5/ua3hAAAgOURWAAAgOURWAAAgOURWAAAgOVV+0y3AOqf4hKjtZmHtO9IoYKa+ap3+xby9rLVdrMA1GMEFgAeSc7I0pOfblJWbqFrWajDV4lDIjWwW2gttgxAfcYlIQCVlpyRpUmLN7iFFUly5hZq0uINSs7IqqWWAajvCCwAyiguMUrbflAfp+9R2vaDKi4xKi4xevLTTSpv4qbSZU9+uknFJXV+aicAFsQlIQBuKrrkM+ry8DJnVk5lJGXlFmpt5iHFdGxZAy0F0JAQWAC4lF7yOf0ciTO3UC+v2FqpOvYdqTjUAEBVEViABq50xI8z95ie+uyXM17yqYygZvwGGIDzj8ACNGDJGVma9snPcuYVnXNdNkkhjt+HOAPA+UZgARqo5Iws3bV4Q5W2tcn9rEvpDCyJQyKZjwVAtWCUENAAFZcYPf7hT1Xa9sHYCxXicL/sE+Lw1ezbL2UeFgDVhjMsQAO0ZvtB5Rw94dE2pZd87r2uk+69rhMz3QKoUQQWoAFK+/WAR+XLu+TD0GUANYnAAjRInp0NCWHqfQC1jMACNEAxHVtq5sptZy13T7+O6tOpFZd8ANQ6AgvQAF3RoaWaN250xvtYAho3UsKALgQVAJbAKCGgAfL2sunZm7ufsUzSzd0JKwAsg8ACNFADu4Vqzu2XKsTffYhyqMNXcxiiDMBiuCQE1FGlU+qfy9Digd1CNSAyhCHKACyPwALUQRX9onJVRvJ4e9kYogzA8rgkBNQxpb+ofGpYkX7/ReVJizcoOSOrlloGANWHwALUIcUlRk9+uumMv6j85KebVFziye8rA4D1EViAOmRt5qEyZ1ZOZSRl5RZqbeahmmsUANQAAgtQh+w7UnFYqUo5AKgruOkWsKCKRgAFNfM9+8ZSpcsBQF1BYAEs5kwjgAZEhijU4StnbmG597GU/qJy7/Ytaqy9AFATuCQEWMjZRgAt3+RU4pBISWV/vrC8X1QGgPqCwAJYRGVHAA2IDNHs2y9ViMP9sk+Iw1ezmaEWQD3FJSHAIjwZAcQMtQAaGgILYBGejgBihloADQmXhACLYAQQAFSMwAJYRO/2LRTq8C1zM20pm34fLcQIIAANEYEFsAhvLxsjgACgAgQW4DwpLjFK235QH6fvUdr2g1X6PZ+B3UIZAQQA5ajSTbezZs3SCy+8IKfTqZ49e2rGjBnq3bt3uWVff/11LVy4UBkZGZKkqKgoPfPMM27lx44dqwULFrhtFxcXp+Tk5Ko0D6hxZ5rszdOQwQggACjL4zMs77zzjhISEpSYmKgNGzaoZ8+eiouL0759+8otn5qaqtGjR2vlypVKS0tTeHi4rr/+eu3Zs8et3MCBA5WVleV6vP3221XbI6CGnW2yt+SMLI/rLB0BNLRXa8V0bElYAdDg2YwxHp23jo6O1uWXX66ZM2dKkkpKShQeHq7Jkyfr8ccfP+v2xcXFCggI0MyZMzVmzBhJv59hycnJ0dKlSz3fA0l5eXlyOBzKzc2Vv79/leoAqqK4xKjPc19VOH9K6VT5Xz92HaEDAE7jyfe3R2dYjh8/rvXr1ys2NvZ/FXh5KTY2VmlpaZWq4+jRozpx4oRatHAf6ZCamqqgoCB16dJFkyZN0sGDByuso6ioSHl5eW4PoDZ4MtkbAKDqPAosBw4cUHFxsYKDg92WBwcHy+l0VqqOxx57TGFhYW6hZ+DAgVq4cKFSUlL03HPPadWqVRo0aJCKi4vLrSMpKUkOh8P1CA8P92Q3gPPG08neAABVU6Mz3T777LNasmSJUlNT5ev7v1EQo0aNcv1/9+7d1aNHD3Xs2FGpqanq379/mXqmTJmihIQE1/O8vDxCC2oFk70BQM3w6AxLYGCgvL29lZ2d7bY8OztbISEhZ9z2xRdf1LPPPqsvv/xSPXr0OGPZDh06KDAwUNu2bSt3vd1ul7+/v9sDqA1M9gYANcOjwOLj46OoqCilpKS4lpWUlCglJUUxMTEVbvf888/rqaeeUnJysi677LKzvs7u3bt18OBBhYYy5wSsjcneAKBmeDysOSEhQa+//roWLFigX375RZMmTVJBQYHGjRsnSRozZoymTJniKv/cc8/piSee0Ny5c9WuXTs5nU45nU7l5+dLkvLz8/XII49ozZo12rFjh1JSUjR06FB16tRJcXFx52k3gerDZG8AUP08vodl5MiR2r9/v6ZOnSqn06levXopOTnZdSPurl275OX1vxw0e/ZsHT9+XLfeeqtbPYmJiZo2bZq8vb31448/asGCBcrJyVFYWJiuv/56PfXUU7Lb7ee4e0DNYLI3AKheHs/DYkXMwwIAQN3jyfd3jY4SAqyguMRwJgQA6hgCCxqU8/mbPwCAmsOvNaPBqI7f/AEA1AwCCxqE4hKjJz/dpPJu2Cpd9uSnm1RcUudv6QKAeonAggaB3/wBgLqNwIIGgd/8AYC6jcCCBoHf/AGAuo3AggaB3/wBgLqNwIIGgd/8AYC6jcCCBoPf/AGAuouJ49Cg8Js/AFA3EVjQ4Hh72RTTsWVtNwMA4AEuCQEAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMsjsAAAAMtjHhack+ISwyRsAIBqR2CBx0pDyvJNTi1N36tDBcdd60IdvkocEsk09wCA84rAAo8kZ2TpyU83KSu3sNz1ztxCTVq8gd/mAQCcV9zDgkpLzsjSpMUbKgwrkmT+/3+f/HSTiktMheUAAPAEgQWVUlxi9OSnm1SZCGIkZeUWam3moepuFgCggSCwoFLWZh4645mV8uw74ll5AAAqwj0sDUxVR/VUJXwENfOtShMBACiDwNKAlHfDbGVH9XgSPmySQhy/hyEAAM4HAks9durZlB0Hjmr6iv+WuQelsqN6erdvoVCHr5y5hWe8j6X0XE3ikEjmYwEAnDcEljrs1EAS2MQu2aQD+UUKauarwwVFeuqzX85634nR7yHjyU83aUBkSIUhw9vLpsQhkZq0eINsUoWhJYR5WAAA1YDAUkedbT4UT5w6qiemY8sKyw3sFqrZt19a5nVbNGmkm3q1VmxkCDPdAgCqBYHF4tzOojS1S0b6anO23vxmx3l/rcrcWDuwW6gGRIYwHT8AoEYRWCyquMRo5ldbNe+bHco5dqJGXrOyN9Z6e9nOeCYGAIDzjcBSi4pLjL7ddkAfrN+t3TlH1bp5Y90a1UZHCk/qL0t/Us7RmgkqjOoBAFgdgaUGVHRZZ/F3u1R0ssRVbt3OHH38w94abRujegAAdQGB5Twqb1K25Zuc5+3m2OrAqB4AQF1AYKmi08PJ4YLjeuoz92DS1H6B8otO1mIr3ZUOR34wtrPaBTbhhlkAQJ1BYDmDbc58xb2ySsX/f9KRxt5S34uClF9UrO93HFbhKZdzymOlsCJxNgUAUHcRWCrQ/vHPykyOdrRY+vznfbXSHk+FOnz1xOBIBTTxYfgxAKDOq9KvNc+aNUvt2rWTr6+voqOjtXbt2jOWf++993TRRRfJ19dX3bt317Jly9zWG2M0depUhYaGys/PT7Gxsdq6dWtVmnZelBdWrKg0evz5qnZ6a3y03poQrVdG9dLbE6/Q149dpxt6hCqmY0sN7dVaMR1bElYAAHWWx4HlnXfeUUJCghITE7Vhwwb17NlTcXFx2rev/DMP3377rUaPHq3x48dr48aNGjZsmIYNG6aMjAxXmeeff17//Oc/NWfOHH333Xdq0qSJ4uLiVFhY8zeqbnPm14mwIv1+iWfO7Zdq6pCuuqpzoK7qFEg4AQDUSzZjjEffz9HR0br88ss1c+ZMSVJJSYnCw8M1efJkPf7442XKjxw5UgUFBfrPf/7jWnbFFVeoV69emjNnjowxCgsL00MPPaSHH35YkpSbm6vg4GDNnz9fo0aNKlNnUVGRioqKXM/z8vIUHh6u3Nxc+fv7e7I7ZXT+y2c6ceZbU2pF88aNNDamnS5v10IHCoq4xAMAqPPy8vLkcDgq9f3t0T0sx48f1/r16zVlyhTXMi8vL8XGxiotLa3cbdLS0pSQkOC2LC4uTkuXLpUkZWZmyul0KjY21rXe4XAoOjpaaWlp5QaWpKQkPfnkk540vdKsEFaa+HjrhVt7KKCJnftPAACQh4HlwIEDKi4uVnBwsNvy4OBgbd68udxtnE5nueWdTqdrfemyisqcbsqUKW4hqPQMy/nQyKtmQ0uLJj7qFNhEJTKumW6v7BRIOAEA4BR1cpSQ3W6X3W6vlro/v6+vYqevOu/1ls59UjrTLZd1AACoPI8CS2BgoLy9vZWdne22PDs7WyEhIeVuExIScsbypf/Nzs5WaGioW5levXp50rzzolNIU9cEa1XRxO6tgqJi1/NQ5j4BAOCceTRKyMfHR1FRUUpJSXEtKykpUUpKimJiYsrdJiYmxq28JC1fvtxVvn379goJCXErk5eXp++++67COqtb5rOD5ek5jxB/u+bcfql+TIzT2xOvcBteTFgBAODceHxJKCEhQfHx8brsssvUu3dvTZ8+XQUFBRo3bpwkacyYMWrdurWSkpIkSffff7/69u2rf/zjHxo8eLCWLFmidevW6bXXXpMk2Ww2PfDAA3r66afVuXNntW/fXk888YTCwsI0bNiw87enHsp8dnCFM922aGKXbDZ52aRebZorLKCx26WdmI4ta63dAADURx4HlpEjR2r//v2aOnWqnE6nevXqpeTkZNdNs7t27ZKX1/9O3Fx55ZX697//rb/97W/6y1/+os6dO2vp0qXq1q2bq8yjjz6qgoIC3XHHHcrJyVGfPn2UnJwsX1/f87CLVdcppKm2Jw2u1TYAAIAqzMNiRZ6M4wYAANbgyfd3labmBwAAqEkEFgAAYHkEFgAAYHkEFgAAYHkEFgAAYHkEFgAAYHkEFgAAYHkEFgAAYHl18teaT1c6911eXl4ttwQAAFRW6fd2ZeawrReB5ciRI5Kk8PDwWm4JAADw1JEjR+RwOM5Ypl5MzV9SUqK9e/eqWbNmstk8/Z3lM8vLy1N4eLh+++03pv0/C/qqcuinyqOvKo++qjz6qnJqop+MMTpy5IjCwsLcfoewPPXiDIuXl5fatGlTra/h7+/PgV1J9FXl0E+VR19VHn1VefRV5VR3P53tzEopbroFAACWR2ABAACWR2A5C7vdrsTERNnt9tpuiuXRV5VDP1UefVV59FXl0VeVY7V+qhc33QIAgPqNMywAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCxnMWvWLLVr106+vr6Kjo7W2rVra7tJ1WbatGmy2Wxuj4suusi1vrCwUPfcc49atmyppk2b6pZbblF2drZbHbt27dLgwYPVuHFjBQUF6ZFHHtHJkyfdyqSmpurSSy+V3W5Xp06dNH/+/JrYvXOyevVqDRkyRGFhYbLZbFq6dKnbemOMpk6dqtDQUPn5+Sk2NlZbt251K3Po0CHddttt8vf3V/PmzTV+/Hjl5+e7lfnxxx919dVXy9fXV+Hh4Xr++efLtOW9997TRRddJF9fX3Xv3l3Lli077/t7Ls7WV2PHji1znA0cONCtTEPoq6SkJF1++eVq1qyZgoKCNGzYMG3ZssWtTE2+56z8WVeZvrr22mvLHFd33XWXW5n63lezZ89Wjx49XDPTxsTE6PPPP3etr/PHk0GFlixZYnx8fMzcuXPNzz//bCZOnGiaN29usrOza7tp1SIxMdF07drVZGVluR779+93rb/rrrtMeHi4SUlJMevWrTNXXHGFufLKK13rT548abp162ZiY2PNxo0bzbJly0xgYKCZMmWKq8yvv/5qGjdubBISEsymTZvMjBkzjLe3t0lOTq7RffXUsmXLzF//+lfz4YcfGknmo48+clv/7LPPGofDYZYuXWp++OEHc+ONN5r27dubY8eOucoMHDjQ9OzZ06xZs8b83//9n+nUqZMZPXq0a31ubq4JDg42t912m8nIyDBvv/228fPzM6+++qqrzDfffGO8vb3N888/bzZt2mT+9re/mUaNGpmffvqp2vugss7WV/Hx8WbgwIFux9mhQ4fcyjSEvoqLizPz5s0zGRkZJj093dxwww2mbdu2Jj8/31Wmpt5zVv+sq0xf9e3b10ycONHtuMrNzXWtbwh99cknn5jPPvvM/Pe//zVbtmwxf/nLX0yjRo1MRkaGMabuH08EljPo3bu3ueeee1zPi4uLTVhYmElKSqrFVlWfxMRE07Nnz3LX5eTkmEaNGpn33nvPteyXX34xkkxaWpox5vcvKi8vL+N0Ol1lZs+ebfz9/U1RUZExxphHH33UdO3a1a3ukSNHmri4uPO8N9Xn9C/hkpISExISYl544QXXspycHGO3283bb79tjDFm06ZNRpL5/vvvXWU+//xzY7PZzJ49e4wxxvzrX/8yAQEBrr4yxpjHHnvMdOnSxfV8xIgRZvDgwW7tiY6ONnfeeed53cfzpaLAMnTo0Aq3aah9tW/fPiPJrFq1yhhTs++5uvZZd3pfGfN7YLn//vsr3Kah9lVAQIB544036sXxxCWhChw/flzr169XbGysa5mXl5diY2OVlpZWiy2rXlu3blVYWJg6dOig2267Tbt27ZIkrV+/XidOnHDrj4suukht27Z19UdaWpq6d++u4OBgV5m4uDjl5eXp559/dpU5tY7SMnW5TzMzM+V0Ot32y+FwKDo62q1vmjdvrssuu8xVJjY2Vl5eXvruu+9cZa655hr5+Pi4ysTFxWnLli06fPiwq0x96L/U1FQFBQWpS5cumjRpkg4ePOha11D7Kjc3V5LUokULSTX3nquLn3Wn91Wpt956S4GBgerWrZumTJmio0ePutY1tL4qLi7WkiVLVFBQoJiYmHpxPNWLX2uuDgcOHFBxcbHbH06SgoODtXnz5lpqVfWKjo7W/Pnz1aVLF2VlZenJJ5/U1VdfrYyMDDmdTvn4+Kh58+Zu2wQHB8vpdEqSnE5nuf1Vuu5MZfLy8nTs2DH5+flV095Vn9J9K2+/Tt3voKAgt/UXXHCBWrRo4Vamffv2ZeooXRcQEFBh/5XWURcMHDhQN998s9q3b6/t27frL3/5iwYNGqS0tDR5e3s3yL4qKSnRAw88oKuuukrdunWTpBp7zx0+fLhOfdaV11eS9Mc//lEREREKCwvTjz/+qMcee0xbtmzRhx9+KKnh9NVPP/2kmJgYFRYWqmnTpvroo48UGRmp9PT0On88EVjgMmjQINf/9+jRQ9HR0YqIiNC7775bJ4MErGnUqFGu/+/evbt69Oihjh07KjU1Vf3796/FltWee+65RxkZGfr6669ruymWV1Ff3XHHHa7/7969u0JDQ9W/f39t375dHTt2rOlm1pouXbooPT1dubm5ev/99xUfH69Vq1bVdrPOCy4JVSAwMFDe3t5l7qDOzs5WSEhILbWqZjVv3lwXXnihtm3bppCQEB0/flw5OTluZU7tj5CQkHL7q3Tdmcr4+/vX2VBUum9nOlZCQkK0b98+t/UnT57UoUOHzkv/1eVjskOHDgoMDNS2bdskNby+uvfee/Wf//xHK1euVJs2bVzLa+o9V5c+6yrqq/JER0dLkttx1RD6ysfHR506dVJUVJSSkpLUs2dPvfLKK/XieCKwVMDHx0dRUVFKSUlxLSspKVFKSopiYmJqsWU1Jz8/X9u3b1doaKiioqLUqFEjt/7YsmWLdu3a5eqPmJgY/fTTT25fNsuXL5e/v78iIyNdZU6to7RMXe7T9u3bKyQkxG2/8vLy9N1337n1TU5OjtavX+8q89VXX6mkpMT1wRoTE6PVq1frxIkTrjLLly9Xly5dFBAQ4CpT3/pv9+7dOnjwoEJDQyU1nL4yxujee+/VRx99pK+++qrMJa6aes/Vhc+6s/VVedLT0yXJ7bhqCH11upKSEhUVFdWP4+mcbtmt55YsWWLsdruZP3++2bRpk7njjjtM8+bN3e6grk8eeughk5qaajIzM80333xjYmNjTWBgoNm3b58x5vchcW3btjVfffWVWbdunYmJiTExMTGu7UuHxF1//fUmPT3dJCcnm1atWpU7JO6RRx4xv/zyi5k1a1adGNZ85MgRs3HjRrNx40Yjybz00ktm48aNZufOncaY34c1N2/e3Hz88cfmxx9/NEOHDi13WPMll1xivvvuO/P111+bzp07uw3VzcnJMcHBweZPf/qTycjIMEuWLDGNGzcuM1T3ggsuMC+++KL55ZdfTGJioqWG6hpz5r46cuSIefjhh01aWprJzMw0K1asMJdeeqnp3LmzKSwsdNXREPpq0qRJxuFwmNTUVLehuEePHnWVqan3nNU/687WV9u2bTN///vfzbp160xmZqb5+OOPTYcOHcw111zjqqMh9NXjjz9uVq1aZTIzM82PP/5oHn/8cWOz2cyXX35pjKn7xxOB5SxmzJhh2rZta3x8fEzv3r3NmjVrartJ1WbkyJEmNDTU+Pj4mNatW5uRI0eabdu2udYfO3bM3H333SYgIMA0btzY3HTTTSYrK8utjh07dphBgwYZPz8/ExgYaB566CFz4sQJtzIrV640vXr1Mj4+PqZDhw5m3rx5NbF752TlypVGUplHfHy8Meb3oc1PPPGECQ4ONna73fTv399s2bLFrY6DBw+a0aNHm6ZNmxp/f38zbtw4c+TIEbcyP/zwg+nTp4+x2+2mdevW5tlnny3TlnfffddceOGFxsfHx3Tt2tV89tln1bbfVXGmvjp69Ki5/vrrTatWrUyjRo1MRESEmThxYpkPsobQV+X1kSS390NNvues/Fl3tr7atWuXueaaa0yLFi2M3W43nTp1Mo888ojbPCzG1P+++vOf/2wiIiKMj4+PadWqlenfv78rrBhT948nmzHGnNs5GgAAgOrFPSwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDyCCwAAMDy/h8OEkxyCMOlBQAAAABJRU5ErkJggg==",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "plt.scatter(xs, ys)\n",
    "plt.title('Time required for generating an output');"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We see that (1) the time needed to generate an output increases quadratically with the length of that output, and that (2) a large portion of the produced outputs are tens of thousands of characters long."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "To address these problems, we need a _smarter algorithm_ – one that is more efficient, that gets us better control over expansions, and that is able to foresee in `expr_grammar` that the `(expr)` alternative yields a potentially infinite expansion, in contrast to the other two."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Derivation Trees\n",
    "\n",
    "To both obtain a more efficient algorithm _and_ exercise better control over expansions, we will use a special representation for the strings that our grammar produces.  The general idea is to use a *tree* structure that will be subsequently expanded – a so-called *derivation tree*.  This representation allows us to always keep track of our expansion status – answering questions such as which elements have been expanded into which others, and which symbols still need to be expanded.  Furthermore, adding new elements to a tree is far more efficient than replacing strings again and again."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Like other trees used in programming, a derivation tree (also known as *parse tree* or *concrete syntax tree*) consists of *nodes* which have other nodes (called *child nodes*) as their *children*.  The tree starts with one node that has no parent; this is called the *root node*; a node without children is called a *leaf*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The grammar expansion process with derivation trees is illustrated in the following steps, using the arithmetic grammar [from\n",
    "the chapter on grammars](Grammars.ipynb).  We start with a single node as root of the tree, representing the *start symbol* – in our case `<start>`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:25.491734Z",
     "iopub.status.busy": "2025-01-16T09:40:25.491581Z",
     "iopub.status.idle": "2025-01-16T09:40:25.503172Z",
     "shell.execute_reply": "2025-01-16T09:40:25.502865Z"
    },
    "ipub": {
     "ignore": true
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "# ignore\n",
    "from graphviz import Digraph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:25.505032Z",
     "iopub.status.busy": "2025-01-16T09:40:25.504920Z",
     "iopub.status.idle": "2025-01-16T09:40:25.506811Z",
     "shell.execute_reply": "2025-01-16T09:40:25.506590Z"
    },
    "ipub": {
     "ignore": true
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "# ignore\n",
    "tree = Digraph(\"root\")\n",
    "tree.attr('node', shape='plain')\n",
    "tree.node(r\"\\<start\\>\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:25.508276Z",
     "iopub.status.busy": "2025-01-16T09:40:25.508160Z",
     "iopub.status.idle": "2025-01-16T09:40:25.976576Z",
     "shell.execute_reply": "2025-01-16T09:40:25.976220Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Title: root Pages: 1 -->\n",
       "<svg width=\"48pt\" height=\"22pt\"\n",
       " viewBox=\"0.00 0.00 47.75 22.25\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 18.25)\">\n",
       "<title>root</title>\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-18.25 43.75,-18.25 43.75,4 -4,4\"/>\n",
       "<!-- \\&lt;start\\&gt; -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>\\&lt;start\\&gt;</title>\n",
       "<text text-anchor=\"middle\" x=\"19.88\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x10660ba40>"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ignore\n",
    "tree"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "To expand the tree, we traverse it, searching for a nonterminal symbol $S$ without children.  $S$ thus is a symbol that still has to be expanded.  We then chose an expansion for $S$ from the grammar.  Then, we add the expansion as a new child of $S$.  For our start symbol `<start>`, the only expansion is `<expr>`, so we add it as a child."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:25.978652Z",
     "iopub.status.busy": "2025-01-16T09:40:25.978478Z",
     "iopub.status.idle": "2025-01-16T09:40:25.980492Z",
     "shell.execute_reply": "2025-01-16T09:40:25.980188Z"
    },
    "ipub": {
     "ignore": true
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "# ignore\n",
    "tree.edge(r\"\\<start\\>\", r\"\\<expr\\>\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:25.982116Z",
     "iopub.status.busy": "2025-01-16T09:40:25.981990Z",
     "iopub.status.idle": "2025-01-16T09:40:26.416814Z",
     "shell.execute_reply": "2025-01-16T09:40:26.416438Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Title: root Pages: 1 -->\n",
       "<svg width=\"49pt\" height=\"73pt\"\n",
       " viewBox=\"0.00 0.00 48.50 72.50\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 68.5)\">\n",
       "<title>root</title>\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-68.5 44.5,-68.5 44.5,4 -4,4\"/>\n",
       "<!-- \\&lt;start\\&gt; -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>\\&lt;start\\&gt;</title>\n",
       "<text text-anchor=\"middle\" x=\"20.25\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- \\&lt;expr\\&gt; -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>\\&lt;expr\\&gt;</title>\n",
       "<text text-anchor=\"middle\" x=\"20.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- \\&lt;start\\&gt;&#45;&gt;\\&lt;expr\\&gt; -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>\\&lt;start\\&gt;&#45;&gt;\\&lt;expr\\&gt;</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M20.25,-50.46C20.25,-44.24 20.25,-34.15 20.25,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"23.75,-25.6 20.25,-15.6 16.75,-25.6 23.75,-25.6\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x10660ba40>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ignore\n",
    "tree"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "To construct the produced string from a derivation tree, we traverse the tree in order and collect the symbols at the leaves of the tree.  In the case above, we obtain the string `\"<expr>\"`.\n",
    "\n",
    "To further expand the tree, we choose another symbol to expand, and add its expansion as new children.  This would get us the `<expr>` symbol, which gets expanded into `<expr> + <term>`, adding three children."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:26.418560Z",
     "iopub.status.busy": "2025-01-16T09:40:26.418441Z",
     "iopub.status.idle": "2025-01-16T09:40:26.420414Z",
     "shell.execute_reply": "2025-01-16T09:40:26.420164Z"
    },
    "ipub": {
     "ignore": true
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "# ignore\n",
    "tree.edge(r\"\\<expr\\>\", r\"\\<expr\\> \")\n",
    "tree.edge(r\"\\<expr\\>\", r\"+\")\n",
    "tree.edge(r\"\\<expr\\>\", r\"\\<term\\>\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:26.421999Z",
     "iopub.status.busy": "2025-01-16T09:40:26.421889Z",
     "iopub.status.idle": "2025-01-16T09:40:26.849585Z",
     "shell.execute_reply": "2025-01-16T09:40:26.849221Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Title: root Pages: 1 -->\n",
       "<svg width=\"138pt\" height=\"123pt\"\n",
       " viewBox=\"0.00 0.00 138.12 122.75\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 118.75)\">\n",
       "<title>root</title>\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-118.75 134.12,-118.75 134.12,4 -4,4\"/>\n",
       "<!-- \\&lt;start\\&gt; -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>\\&lt;start\\&gt;</title>\n",
       "<text text-anchor=\"middle\" x=\"66.12\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- \\&lt;expr\\&gt; -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>\\&lt;expr\\&gt;</title>\n",
       "<text text-anchor=\"middle\" x=\"66.12\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- \\&lt;start\\&gt;&#45;&gt;\\&lt;expr\\&gt; -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>\\&lt;start\\&gt;&#45;&gt;\\&lt;expr\\&gt;</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M66.12,-100.71C66.12,-94.49 66.12,-84.4 66.12,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"69.63,-75.85 66.13,-65.85 62.63,-75.85 69.63,-75.85\"/>\n",
       "</g>\n",
       "<!-- \\&lt;expr\\&gt;  -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>\\&lt;expr\\&gt; </title>\n",
       "<text text-anchor=\"middle\" x=\"22.12\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt; </text>\n",
       "</g>\n",
       "<!-- \\&lt;expr\\&gt;&#45;&gt;\\&lt;expr\\&gt;  -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>\\&lt;expr\\&gt;&#45;&gt;\\&lt;expr\\&gt; </title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M60.73,-50.46C54.4,-43.51 43.67,-31.75 35.13,-22.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"37.91,-20.24 28.58,-15.21 32.74,-24.96 37.91,-20.24\"/>\n",
       "</g>\n",
       "<!-- + -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>+</title>\n",
       "<text text-anchor=\"middle\" x=\"66.12\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">+</text>\n",
       "</g>\n",
       "<!-- \\&lt;expr\\&gt;&#45;&gt;+ -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>\\&lt;expr\\&gt;&#45;&gt;+</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M66.12,-50.46C66.12,-44.24 66.12,-34.15 66.12,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"69.63,-25.6 66.13,-15.6 62.63,-25.6 69.63,-25.6\"/>\n",
       "</g>\n",
       "<!-- \\&lt;term\\&gt; -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>\\&lt;term\\&gt;</title>\n",
       "<text text-anchor=\"middle\" x=\"109.12\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- \\&lt;expr\\&gt;&#45;&gt;\\&lt;term\\&gt; -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>\\&lt;expr\\&gt;&#45;&gt;\\&lt;term\\&gt;</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M71.4,-50.46C77.52,-43.59 87.85,-32 96.15,-22.68\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"98.76,-25.01 102.8,-15.22 93.54,-20.36 98.76,-25.01\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x10660ba40>"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ignore\n",
    "tree"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We repeat the expansion until there are no symbols left to expand:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:26.851401Z",
     "iopub.status.busy": "2025-01-16T09:40:26.851270Z",
     "iopub.status.idle": "2025-01-16T09:40:26.853702Z",
     "shell.execute_reply": "2025-01-16T09:40:26.853333Z"
    },
    "ipub": {
     "ignore": true
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "# ignore\n",
    "tree.edge(r\"\\<expr\\> \", r\"\\<term\\> \")\n",
    "tree.edge(r\"\\<term\\> \", r\"\\<factor\\> \")\n",
    "tree.edge(r\"\\<factor\\> \", r\"\\<integer\\> \")\n",
    "tree.edge(r\"\\<integer\\> \", r\"\\<digit\\> \")\n",
    "tree.edge(r\"\\<digit\\> \", r\"2 \")\n",
    "\n",
    "tree.edge(r\"\\<term\\>\", r\"\\<factor\\>\")\n",
    "tree.edge(r\"\\<factor\\>\", r\"\\<integer\\>\")\n",
    "tree.edge(r\"\\<integer\\>\", r\"\\<digit\\>\")\n",
    "tree.edge(r\"\\<digit\\>\", r\"2\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:26.855375Z",
     "iopub.status.busy": "2025-01-16T09:40:26.855248Z",
     "iopub.status.idle": "2025-01-16T09:40:27.318438Z",
     "shell.execute_reply": "2025-01-16T09:40:27.318056Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Title: root Pages: 1 -->\n",
       "<svg width=\"151pt\" height=\"374pt\"\n",
       " viewBox=\"0.00 0.00 150.88 374.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 370)\">\n",
       "<title>root</title>\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-370 146.88,-370 146.88,4 -4,4\"/>\n",
       "<!-- \\&lt;start\\&gt; -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>\\&lt;start\\&gt;</title>\n",
       "<text text-anchor=\"middle\" x=\"72.88\" y=\"-352.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- \\&lt;expr\\&gt; -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>\\&lt;expr\\&gt;</title>\n",
       "<text text-anchor=\"middle\" x=\"72.88\" y=\"-302.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- \\&lt;start\\&gt;&#45;&gt;\\&lt;expr\\&gt; -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>\\&lt;start\\&gt;&#45;&gt;\\&lt;expr\\&gt;</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M72.88,-351.96C72.88,-345.74 72.88,-335.65 72.88,-326.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"76.38,-327.1 72.88,-317.1 69.38,-327.1 76.38,-327.1\"/>\n",
       "</g>\n",
       "<!-- \\&lt;expr\\&gt;  -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>\\&lt;expr\\&gt; </title>\n",
       "<text text-anchor=\"middle\" x=\"28.88\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt; </text>\n",
       "</g>\n",
       "<!-- \\&lt;expr\\&gt;&#45;&gt;\\&lt;expr\\&gt;  -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>\\&lt;expr\\&gt;&#45;&gt;\\&lt;expr\\&gt; </title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M67.48,-301.71C61.15,-294.76 50.42,-283 41.88,-273.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"44.66,-271.49 35.33,-266.46 39.49,-276.21 44.66,-271.49\"/>\n",
       "</g>\n",
       "<!-- + -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>+</title>\n",
       "<text text-anchor=\"middle\" x=\"72.88\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\">+</text>\n",
       "</g>\n",
       "<!-- \\&lt;expr\\&gt;&#45;&gt;+ -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>\\&lt;expr\\&gt;&#45;&gt;+</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M72.88,-301.71C72.88,-295.49 72.88,-285.4 72.88,-276.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"76.38,-276.85 72.88,-266.85 69.38,-276.85 76.38,-276.85\"/>\n",
       "</g>\n",
       "<!-- \\&lt;term\\&gt; -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>\\&lt;term\\&gt;</title>\n",
       "<text text-anchor=\"middle\" x=\"115.88\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- \\&lt;expr\\&gt;&#45;&gt;\\&lt;term\\&gt; -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>\\&lt;expr\\&gt;&#45;&gt;\\&lt;term\\&gt;</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M78.15,-301.71C84.27,-294.84 94.6,-283.25 102.9,-273.93\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"105.51,-276.26 109.55,-266.47 100.29,-271.61 105.51,-276.26\"/>\n",
       "</g>\n",
       "<!-- \\&lt;term\\&gt;  -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>\\&lt;term\\&gt; </title>\n",
       "<text text-anchor=\"middle\" x=\"28.88\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt; </text>\n",
       "</g>\n",
       "<!-- \\&lt;expr\\&gt; &#45;&gt;\\&lt;term\\&gt;  -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>\\&lt;expr\\&gt; &#45;&gt;\\&lt;term\\&gt; </title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M28.88,-251.46C28.88,-245.24 28.88,-235.15 28.88,-226.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"32.38,-226.6 28.88,-216.6 25.38,-226.6 32.38,-226.6\"/>\n",
       "</g>\n",
       "<!-- \\&lt;factor\\&gt; -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>\\&lt;factor\\&gt;</title>\n",
       "<text text-anchor=\"middle\" x=\"115.88\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- \\&lt;term\\&gt;&#45;&gt;\\&lt;factor\\&gt; -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>\\&lt;term\\&gt;&#45;&gt;\\&lt;factor\\&gt;</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M115.88,-251.46C115.88,-245.24 115.88,-235.15 115.88,-226.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"119.38,-226.6 115.88,-216.6 112.38,-226.6 119.38,-226.6\"/>\n",
       "</g>\n",
       "<!-- \\&lt;factor\\&gt;  -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>\\&lt;factor\\&gt; </title>\n",
       "<text text-anchor=\"middle\" x=\"28.88\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt; </text>\n",
       "</g>\n",
       "<!-- \\&lt;term\\&gt; &#45;&gt;\\&lt;factor\\&gt;  -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>\\&lt;term\\&gt; &#45;&gt;\\&lt;factor\\&gt; </title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M28.88,-201.21C28.88,-194.99 28.88,-184.9 28.88,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"32.38,-176.35 28.88,-166.35 25.38,-176.35 32.38,-176.35\"/>\n",
       "</g>\n",
       "<!-- \\&lt;integer\\&gt;  -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>\\&lt;integer\\&gt; </title>\n",
       "<text text-anchor=\"middle\" x=\"28.88\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt; </text>\n",
       "</g>\n",
       "<!-- \\&lt;factor\\&gt; &#45;&gt;\\&lt;integer\\&gt;  -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>\\&lt;factor\\&gt; &#45;&gt;\\&lt;integer\\&gt; </title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M28.88,-150.96C28.88,-144.74 28.88,-134.65 28.88,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"32.38,-126.1 28.88,-116.1 25.38,-126.1 32.38,-126.1\"/>\n",
       "</g>\n",
       "<!-- \\&lt;digit\\&gt;  -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>\\&lt;digit\\&gt; </title>\n",
       "<text text-anchor=\"middle\" x=\"28.88\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt; </text>\n",
       "</g>\n",
       "<!-- \\&lt;integer\\&gt; &#45;&gt;\\&lt;digit\\&gt;  -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>\\&lt;integer\\&gt; &#45;&gt;\\&lt;digit\\&gt; </title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M28.88,-100.71C28.88,-94.49 28.88,-84.4 28.88,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"32.38,-75.85 28.88,-65.85 25.38,-75.85 32.38,-75.85\"/>\n",
       "</g>\n",
       "<!-- 2  -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>2 </title>\n",
       "<text text-anchor=\"middle\" x=\"28.88\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">2 </text>\n",
       "</g>\n",
       "<!-- \\&lt;digit\\&gt; &#45;&gt;2  -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>\\&lt;digit\\&gt; &#45;&gt;2 </title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M28.88,-50.46C28.88,-44.24 28.88,-34.15 28.88,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"32.38,-25.6 28.88,-15.6 25.38,-25.6 32.38,-25.6\"/>\n",
       "</g>\n",
       "<!-- \\&lt;integer\\&gt; -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>\\&lt;integer\\&gt;</title>\n",
       "<text text-anchor=\"middle\" x=\"115.88\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- \\&lt;factor\\&gt;&#45;&gt;\\&lt;integer\\&gt; -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>\\&lt;factor\\&gt;&#45;&gt;\\&lt;integer\\&gt;</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M115.88,-201.21C115.88,-194.99 115.88,-184.9 115.88,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"119.38,-176.35 115.88,-166.35 112.38,-176.35 119.38,-176.35\"/>\n",
       "</g>\n",
       "<!-- \\&lt;digit\\&gt; -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>\\&lt;digit\\&gt;</title>\n",
       "<text text-anchor=\"middle\" x=\"115.88\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- \\&lt;integer\\&gt;&#45;&gt;\\&lt;digit\\&gt; -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>\\&lt;integer\\&gt;&#45;&gt;\\&lt;digit\\&gt;</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M115.88,-150.96C115.88,-144.74 115.88,-134.65 115.88,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"119.38,-126.1 115.88,-116.1 112.38,-126.1 119.38,-126.1\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"115.88\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
       "</g>\n",
       "<!-- \\&lt;digit\\&gt;&#45;&gt;2 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>\\&lt;digit\\&gt;&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M115.88,-100.71C115.88,-94.49 115.88,-84.4 115.88,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"119.38,-75.85 115.88,-65.85 112.38,-75.85 119.38,-75.85\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x10660ba40>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ignore\n",
    "tree"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We now have a representation for the string `2 + 2`.  In contrast to the string alone, though, the derivation tree records _the entire structure_ (and production history, or _derivation_ history) of the produced string.  It also allows for simple comparison and manipulation – say, replacing one subtree (substructure) against another."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Representing Derivation Trees\n",
    "\n",
    "To represent a derivation tree in Python, we use the following format.  A node is a pair\n",
    "\n",
    "```python\n",
    "(SYMBOL_NAME, CHILDREN)\n",
    "```\n",
    "\n",
    "where `SYMBOL_NAME` is a string representing the node (i.e. `\"<start>\"` or `\"+\"`) and `CHILDREN` is a list of children nodes.\n",
    "\n",
    "`CHILDREN` can take some special values:\n",
    "\n",
    "1. `None` as a placeholder for future expansion.  This means that the node is a *nonterminal symbol* that should be expanded further.\n",
    "2. `[]` (i.e., the empty list) to indicate _no_ children.  This means that the node is a *terminal symbol* that can no longer be expanded."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The type `DerivationTree` captures this very structure. (`Any` should actually read `DerivationTree`, but the Python static type checker cannot handle recursive types well.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.321136Z",
     "iopub.status.busy": "2025-01-16T09:40:27.320973Z",
     "iopub.status.idle": "2025-01-16T09:40:27.323333Z",
     "shell.execute_reply": "2025-01-16T09:40:27.322884Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "DerivationTree = Tuple[str, Optional[List[Any]]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Let us take a very simple derivation tree, representing the intermediate step `<expr> + <term>`, above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.325455Z",
     "iopub.status.busy": "2025-01-16T09:40:27.325310Z",
     "iopub.status.idle": "2025-01-16T09:40:27.327621Z",
     "shell.execute_reply": "2025-01-16T09:40:27.327227Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "derivation_tree: DerivationTree = (\"<start>\",\n",
    "                   [(\"<expr>\",\n",
    "                     [(\"<expr>\", None),\n",
    "                      (\" + \", []),\n",
    "                         (\"<term>\", None)]\n",
    "                     )])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "To better understand the structure of this tree, let us introduce a function `display_tree()` that visualizes this tree."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Excursion: Implementing `display_tree()`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We use the `dot` drawing program from the `graphviz` package algorithmically, traversing the above structure.  (Unless you're deeply interested in tree visualization, you can directly skip to the example below.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.329783Z",
     "iopub.status.busy": "2025-01-16T09:40:27.329654Z",
     "iopub.status.idle": "2025-01-16T09:40:27.331604Z",
     "shell.execute_reply": "2025-01-16T09:40:27.331325Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from graphviz import Digraph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.333270Z",
     "iopub.status.busy": "2025-01-16T09:40:27.333159Z",
     "iopub.status.idle": "2025-01-16T09:40:27.334662Z",
     "shell.execute_reply": "2025-01-16T09:40:27.334413Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from IPython.display import display"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.336210Z",
     "iopub.status.busy": "2025-01-16T09:40:27.336084Z",
     "iopub.status.idle": "2025-01-16T09:40:27.337957Z",
     "shell.execute_reply": "2025-01-16T09:40:27.337633Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "import re\n",
    "import string"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.339764Z",
     "iopub.status.busy": "2025-01-16T09:40:27.339646Z",
     "iopub.status.idle": "2025-01-16T09:40:27.342462Z",
     "shell.execute_reply": "2025-01-16T09:40:27.342110Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "def dot_escape(s: str, show_ascii=None) -> str:\n",
    "    \"\"\"Return s in a form suitable for dot.\n",
    "    If `show_ascii` is True or length of `s` is 1, also append ascii value.\"\"\"\n",
    "    escaped_s = ''\n",
    "    if show_ascii is None:\n",
    "        show_ascii = (len(s) == 1)  # Default: Single chars only\n",
    "\n",
    "    if show_ascii and s == '\\n':\n",
    "        return '\\\\\\\\n (10)'\n",
    "\n",
    "    s = s.replace('\\n', '\\\\n')\n",
    "    for c in s:\n",
    "        if re.match('[,<>\\\\\\\\\"]', c):\n",
    "            escaped_s += '\\\\' + c\n",
    "        elif c in string.printable and 31 < ord(c) < 127:\n",
    "            escaped_s += c\n",
    "        else:\n",
    "            escaped_s += '\\\\\\\\x' + format(ord(c), '02x')\n",
    "\n",
    "        if show_ascii:\n",
    "            escaped_s += f' ({ord(c)})'\n",
    "\n",
    "    return escaped_s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.344317Z",
     "iopub.status.busy": "2025-01-16T09:40:27.344176Z",
     "iopub.status.idle": "2025-01-16T09:40:27.346185Z",
     "shell.execute_reply": "2025-01-16T09:40:27.345876Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "assert dot_escape(\"hello\") == \"hello\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.347864Z",
     "iopub.status.busy": "2025-01-16T09:40:27.347733Z",
     "iopub.status.idle": "2025-01-16T09:40:27.349702Z",
     "shell.execute_reply": "2025-01-16T09:40:27.349393Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "assert dot_escape(\"<hello>, world\") == \"\\\\<hello\\\\>\\\\, world\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.351445Z",
     "iopub.status.busy": "2025-01-16T09:40:27.351312Z",
     "iopub.status.idle": "2025-01-16T09:40:27.353246Z",
     "shell.execute_reply": "2025-01-16T09:40:27.352960Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "assert dot_escape(\"\\\\n\") == \"\\\\\\\\n\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.355008Z",
     "iopub.status.busy": "2025-01-16T09:40:27.354893Z",
     "iopub.status.idle": "2025-01-16T09:40:27.356602Z",
     "shell.execute_reply": "2025-01-16T09:40:27.356339Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "assert dot_escape(\"\\n\", show_ascii=False) == \"\\\\\\\\n\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.358064Z",
     "iopub.status.busy": "2025-01-16T09:40:27.357962Z",
     "iopub.status.idle": "2025-01-16T09:40:27.359595Z",
     "shell.execute_reply": "2025-01-16T09:40:27.359345Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "assert dot_escape(\"\\n\", show_ascii=True) == \"\\\\\\\\n (10)\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.361085Z",
     "iopub.status.busy": "2025-01-16T09:40:27.360987Z",
     "iopub.status.idle": "2025-01-16T09:40:27.362526Z",
     "shell.execute_reply": "2025-01-16T09:40:27.362288Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "assert dot_escape(\"\\n\", show_ascii=True) == \"\\\\\\\\n (10)\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.364019Z",
     "iopub.status.busy": "2025-01-16T09:40:27.363935Z",
     "iopub.status.idle": "2025-01-16T09:40:27.365531Z",
     "shell.execute_reply": "2025-01-16T09:40:27.365288Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "assert dot_escape('\\x01', show_ascii=False) == \"\\\\\\\\x01\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.367043Z",
     "iopub.status.busy": "2025-01-16T09:40:27.366955Z",
     "iopub.status.idle": "2025-01-16T09:40:27.368688Z",
     "shell.execute_reply": "2025-01-16T09:40:27.368433Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "assert dot_escape('\\x01') == \"\\\\\\\\x01 (1)\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "While we are interested at present in visualizing a `derivation_tree`, it is in our interest to generalize the visualization procedure. In particular, it would be helpful if our method `display_tree()` can display *any* tree    like data structure. To enable this, we define a helper method `extract_node()` that extract the current symbol and  children from a given data structure. The default implementation simply extracts the symbol, children, and           annotation from any `derivation_tree` node."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.370296Z",
     "iopub.status.busy": "2025-01-16T09:40:27.370182Z",
     "iopub.status.idle": "2025-01-16T09:40:27.372057Z",
     "shell.execute_reply": "2025-01-16T09:40:27.371809Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def extract_node(node, id):\n",
    "    symbol, children, *annotation = node\n",
    "    return symbol, children, ''.join(str(a) for a in annotation)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "While visualizing a tree, it is often useful to display certain nodes differently. For example, it is sometimes useful to distinguish between non-processed nodes and processed nodes. We define a helper procedure `default_node_attr()` that provides the basic display, which can be customized by the user."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.373687Z",
     "iopub.status.busy": "2025-01-16T09:40:27.373573Z",
     "iopub.status.idle": "2025-01-16T09:40:27.375210Z",
     "shell.execute_reply": "2025-01-16T09:40:27.374959Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def default_node_attr(dot, nid, symbol, ann):\n",
    "    dot.node(repr(nid), dot_escape(symbol))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Similar to nodes, the edges may also require modifications. We define `default_edge_attr()` as a helper        procedure that can be customized by the user."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.376836Z",
     "iopub.status.busy": "2025-01-16T09:40:27.376717Z",
     "iopub.status.idle": "2025-01-16T09:40:27.378584Z",
     "shell.execute_reply": "2025-01-16T09:40:27.378313Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def default_edge_attr(dot, start_node, stop_node):\n",
    "    dot.edge(repr(start_node), repr(stop_node))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "While visualizing a tree, one may sometimes wish to change the appearance of the tree. For example, it is      sometimes easier to view the tree if it was laid out left to right rather than top to bottom. We define another      helper procedure `default_graph_attr()` for that."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.380195Z",
     "iopub.status.busy": "2025-01-16T09:40:27.380077Z",
     "iopub.status.idle": "2025-01-16T09:40:27.381745Z",
     "shell.execute_reply": "2025-01-16T09:40:27.381515Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def default_graph_attr(dot):\n",
    "    dot.attr('node', shape='plain')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Finally, we define a method `display_tree()` that accepts these four functions `extract_node()`,               `default_edge_attr()`, `default_node_attr()` and `default_graph_attr()` and uses them to display the tree."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.383331Z",
     "iopub.status.busy": "2025-01-16T09:40:27.383216Z",
     "iopub.status.idle": "2025-01-16T09:40:27.386033Z",
     "shell.execute_reply": "2025-01-16T09:40:27.385700Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def display_tree(derivation_tree: DerivationTree,\n",
    "                 log: bool = False,\n",
    "                 extract_node: Callable = extract_node,\n",
    "                 node_attr: Callable = default_node_attr,\n",
    "                 edge_attr: Callable = default_edge_attr,\n",
    "                 graph_attr: Callable = default_graph_attr) -> Any:\n",
    "\n",
    "    # If we import display_tree, we also have to import its functions\n",
    "    from graphviz import Digraph\n",
    "\n",
    "    counter = 0\n",
    "\n",
    "    def traverse_tree(dot, tree, id=0):\n",
    "        (symbol, children, annotation) = extract_node(tree, id)\n",
    "        node_attr(dot, id, symbol, annotation)\n",
    "\n",
    "        if children:\n",
    "            for child in children:\n",
    "                nonlocal counter\n",
    "                counter += 1\n",
    "                child_id = counter\n",
    "                edge_attr(dot, id, child_id)\n",
    "                traverse_tree(dot, child, child_id)\n",
    "\n",
    "    dot = Digraph(comment=\"Derivation Tree\")\n",
    "    graph_attr(dot)\n",
    "    traverse_tree(dot, derivation_tree)\n",
    "    if log:\n",
    "        print(dot)\n",
    "    return dot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### End of Excursion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "This is what our tree visualizes into:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.387742Z",
     "iopub.status.busy": "2025-01-16T09:40:27.387626Z",
     "iopub.status.idle": "2025-01-16T09:40:27.835070Z",
     "shell.execute_reply": "2025-01-16T09:40:27.834645Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"142pt\" height=\"123pt\"\n",
       " viewBox=\"0.00 0.00 142.25 122.75\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 118.75)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-118.75 138.25,-118.75 138.25,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M66.25,-100.71C66.25,-94.49 66.25,-84.4 66.25,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"69.75,-75.85 66.25,-65.85 62.75,-75.85 69.75,-75.85\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"20.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M60.61,-50.46C53.99,-43.51 42.78,-31.75 33.85,-22.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"36.41,-20.01 26.98,-15.19 31.35,-24.84 36.41,-20.01\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>1&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M66.25,-50.46C66.25,-44.24 66.25,-34.15 66.25,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"69.75,-25.6 66.25,-15.6 62.75,-25.6 69.75,-25.6\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"113.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>1&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M72.01,-50.46C78.85,-43.44 90.47,-31.51 99.64,-22.1\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"101.91,-24.78 106.38,-15.17 96.9,-19.9 101.91,-24.78\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089809b0>"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display_tree(derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.837386Z",
     "iopub.status.busy": "2025-01-16T09:40:27.837249Z",
     "iopub.status.idle": "2025-01-16T09:40:27.842026Z",
     "shell.execute_reply": "2025-01-16T09:40:27.841719Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "    \n",
       "    <script>\n",
       "    var bad_answers = new Map();\n",
       "\n",
       "    function answer(quiz_id) {\n",
       "        ans = 0;\n",
       "        for (i = 1;; i++) {\n",
       "            checkbox = document.getElementById(quiz_id + \"-\" + i.toString());\n",
       "            if (!checkbox)\n",
       "                break;\n",
       "            if (checkbox.checked)\n",
       "                ans |= (1 << i);\n",
       "        }\n",
       "        return ans;\n",
       "    }\n",
       "    function check_selection(quiz_id, correct_answer, multiple_choice, hint) {\n",
       "        given_answer = answer(quiz_id);\n",
       "        if (given_answer == correct_answer)\n",
       "        {\n",
       "            document.getElementById(quiz_id + \"-submit\").value = \"Correct!\";\n",
       "            document.getElementById(quiz_id + \"-hint\").innerHTML = \"\";\n",
       "\n",
       "            for (i = 1;; i++) {\n",
       "                checkbox = document.getElementById(quiz_id + \"-\" + i.toString());\n",
       "                label = document.getElementById(quiz_id + \"-\" + i.toString() + \"-label\")\n",
       "                if (!checkbox)\n",
       "                    break;\n",
       "\n",
       "                if (checkbox.checked) {\n",
       "                    label.style.fontWeight = \"bold\";\n",
       "                }\n",
       "                else {\n",
       "                    label.style.textDecoration = \"line-through\";\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "        else \n",
       "        {\n",
       "            document.getElementById(quiz_id + \"-submit\").value = \"Try again\";\n",
       "\n",
       "            if (!bad_answers.has(quiz_id)) {\n",
       "                bad_answers.set(quiz_id, 1);\n",
       "            }\n",
       "            else {\n",
       "                bad_answers.set(quiz_id, bad_answers.get(quiz_id) + 1);\n",
       "            }\n",
       "\n",
       "            if (bad_answers.get(quiz_id) >= 2 && hint.length > 0) {\n",
       "                document.getElementById(quiz_id + \"-hint\").innerHTML = \n",
       "                    \"&nbsp;&nbsp;(Hint: <code>\" + hint + \"</code>)\";\n",
       "            }\n",
       "\n",
       "            if (!multiple_choice) {\n",
       "                for (i = 1;; i++) {\n",
       "                    checkbox = document.getElementById(quiz_id + \"-\" + i.toString());\n",
       "                    label = document.getElementById(quiz_id + \"-\" + i.toString() + \"-label\")\n",
       "\n",
       "                    if (!checkbox)\n",
       "                        break;\n",
       "                    if (checkbox.checked) {\n",
       "                        label.style.textDecoration = \"line-through\";\n",
       "                    }\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "    function clear_selection(quiz_id) {\n",
       "        document.getElementById(quiz_id + \"-submit\").value = \"Submit\";\n",
       "        document.getElementById(quiz_id + \"-hint\").innerHTML = \"\";\n",
       "    }\n",
       "    </script>\n",
       "    \n",
       "    <div class=\"quiz\">\n",
       "    <h3 class=\"quiz_title\">Quiz</h3>\n",
       "    <p>\n",
       "    <div class=\"quiz_question\">And which of these is the internal representation of <code>derivation_tree</code>?</div>\n",
       "    </p>\n",
       "    <p>\n",
       "    <div class=\"quiz_options\" title=\"Pick a choice.\">\n",
       "    \n",
       "        <input type=\"radio\" name=\"eb319efa-d3ed-11ef-8343-6298cf1a5790\" id=\"eb319efa-d3ed-11ef-8343-6298cf1a5790-1\" onclick=\"clear_selection('eb319efa-d3ed-11ef-8343-6298cf1a5790')\">\n",
       "        <label id=\"eb319efa-d3ed-11ef-8343-6298cf1a5790-1-label\" for=\"eb319efa-d3ed-11ef-8343-6298cf1a5790-1\"><code>('&lt;start&gt;', [('&lt;expr&gt;', (['&lt;expr&gt; + &lt;term&gt;']))])</code></label><br>\n",
       "    \n",
       "        <input type=\"radio\" name=\"eb319efa-d3ed-11ef-8343-6298cf1a5790\" id=\"eb319efa-d3ed-11ef-8343-6298cf1a5790-2\" onclick=\"clear_selection('eb319efa-d3ed-11ef-8343-6298cf1a5790')\">\n",
       "        <label id=\"eb319efa-d3ed-11ef-8343-6298cf1a5790-2-label\" for=\"eb319efa-d3ed-11ef-8343-6298cf1a5790-2\"><code>('&lt;start&gt;', [('&lt;expr&gt;', (['&lt;expr&gt;', ' + ', &lt;term&gt;']))])</code></label><br>\n",
       "    \n",
       "        <input type=\"radio\" name=\"eb319efa-d3ed-11ef-8343-6298cf1a5790\" id=\"eb319efa-d3ed-11ef-8343-6298cf1a5790-3\" onclick=\"clear_selection('eb319efa-d3ed-11ef-8343-6298cf1a5790')\">\n",
       "        <label id=\"eb319efa-d3ed-11ef-8343-6298cf1a5790-3-label\" for=\"eb319efa-d3ed-11ef-8343-6298cf1a5790-3\"><code>('&lt;start&gt;', [('&lt;expr&gt;', [('&lt;expr&gt;', None), (' + ', []), ('&lt;term&gt;', None)])])</code></label><br>\n",
       "    \n",
       "        <input type=\"radio\" name=\"eb319efa-d3ed-11ef-8343-6298cf1a5790\" id=\"eb319efa-d3ed-11ef-8343-6298cf1a5790-4\" onclick=\"clear_selection('eb319efa-d3ed-11ef-8343-6298cf1a5790')\">\n",
       "        <label id=\"eb319efa-d3ed-11ef-8343-6298cf1a5790-4-label\" for=\"eb319efa-d3ed-11ef-8343-6298cf1a5790-4\"><code>(('&lt;start&gt;', [('&lt;expr&gt;', [('&lt;expr&gt;', None), (' + ', []), ('&lt;term&gt;', None)])]), None)</code></label><br>\n",
       "    \n",
       "    </div>\n",
       "    </p>\n",
       "    <input id=\"eb319efa-d3ed-11ef-8343-6298cf1a5790-submit\" type=\"submit\" value=\"Submit\" onclick=\"check_selection('eb319efa-d3ed-11ef-8343-6298cf1a5790', 8, 0, '')\">\n",
       "    <span class=\"quiz_hint\" id=\"eb319efa-d3ed-11ef-8343-6298cf1a5790-hint\"></span>\n",
       "    </div>\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "quiz(\"And which of these is the internal representation of `derivation_tree`?\",\n",
    "    [\n",
    "        \"`('<start>', [('<expr>', (['<expr> + <term>']))])`\",\n",
    "        \"`('<start>', [('<expr>', (['<expr>', ' + ', <term>']))])`\",\n",
    "        \"`\" + repr(derivation_tree) + \"`\",\n",
    "        \"`(\" + repr(derivation_tree) + \", None)`\"\n",
    "    ], len(\"eleven\") - len(\"one\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "You can check it out yourself:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.843981Z",
     "iopub.status.busy": "2025-01-16T09:40:27.843854Z",
     "iopub.status.idle": "2025-01-16T09:40:27.846250Z",
     "shell.execute_reply": "2025-01-16T09:40:27.845967Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "('<start>', [('<expr>', [('<expr>', None), (' + ', []), ('<term>', None)])])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "derivation_tree"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Within this book, we also occasionally use a function `display_annotated_tree()` which allows adding annotations to individual nodes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Excursion: Source code and example for `display_annotated_tree()`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "`display_annotated_tree()` displays an annotated tree structure, and lays out the graph left to right."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.848147Z",
     "iopub.status.busy": "2025-01-16T09:40:27.848026Z",
     "iopub.status.idle": "2025-01-16T09:40:27.851280Z",
     "shell.execute_reply": "2025-01-16T09:40:27.850964Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def display_annotated_tree(tree: DerivationTree,\n",
    "                           a_nodes: Dict[int, str],\n",
    "                           a_edges: Dict[Tuple[int, int], str],\n",
    "                           log: bool = False):\n",
    "    def graph_attr(dot):\n",
    "        dot.attr('node', shape='plain')\n",
    "        dot.graph_attr['rankdir'] = 'LR'\n",
    "\n",
    "    def annotate_node(dot, nid, symbol, ann):\n",
    "        if nid in a_nodes:\n",
    "            dot.node(repr(nid), \n",
    "                     \"%s (%s)\" % (dot_escape(unicode_escape(symbol)),\n",
    "                                  a_nodes[nid]))\n",
    "        else:\n",
    "            dot.node(repr(nid), dot_escape(unicode_escape(symbol)))\n",
    "\n",
    "    def annotate_edge(dot, start_node, stop_node):\n",
    "        if (start_node, stop_node) in a_edges:\n",
    "            dot.edge(repr(start_node), repr(stop_node),\n",
    "                     a_edges[(start_node, stop_node)])\n",
    "        else:\n",
    "            dot.edge(repr(start_node), repr(stop_node))\n",
    "\n",
    "    return display_tree(tree, log=log,\n",
    "                        node_attr=annotate_node,\n",
    "                        edge_attr=annotate_edge,\n",
    "                        graph_attr=graph_attr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:27.853028Z",
     "iopub.status.busy": "2025-01-16T09:40:27.852889Z",
     "iopub.status.idle": "2025-01-16T09:40:28.296387Z",
     "shell.execute_reply": "2025-01-16T09:40:28.296030Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"226pt\" height=\"87pt\"\n",
       " viewBox=\"0.00 0.00 225.75 87.25\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 83.25)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-83.25 221.75,-83.25 221.75,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"19.88\" y=\"-33.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"97\" y=\"-33.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M39.45,-40.12C47.27,-40.12 56.59,-40.12 65.38,-40.12\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"65.35,-43.63 75.35,-40.13 65.35,-36.63 65.35,-43.63\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"192.25\" y=\"-65.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M110.82,-47.13C117.88,-50.77 126.88,-55.09 135.25,-58.12 143.48,-61.11 152.61,-63.68 161.07,-65.78\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"160.03,-69.13 170.56,-68 161.62,-62.32 160.03,-69.13\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"192.25\" y=\"-33.95\" font-family=\"Times,serif\" font-size=\"14.00\"> + &#160;(plus)</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>1&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M116.87,-40.12C128.07,-40.12 142.62,-40.12 155.88,-40.12\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"155.57,-43.63 165.57,-40.13 155.57,-36.63 155.57,-43.63\"/>\n",
       "<text text-anchor=\"middle\" x=\"142\" y=\"-41.08\" font-family=\"Times,serif\" font-size=\"14.00\">op</text>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"192.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>1&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M116.87,-33.45C129.51,-28.97 146.42,-22.99 160.93,-17.86\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"161.76,-21.28 170.02,-14.64 159.42,-14.68 161.76,-21.28\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x108982d80>"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display_annotated_tree(derivation_tree, {3: 'plus'}, {(1, 3): 'op'}, log=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### End of Excursion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "If we want to see all the leaf nodes in a tree as a string, the following `all_terminals()` function comes in handy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.298409Z",
     "iopub.status.busy": "2025-01-16T09:40:28.298267Z",
     "iopub.status.idle": "2025-01-16T09:40:28.300679Z",
     "shell.execute_reply": "2025-01-16T09:40:28.300404Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def all_terminals(tree: DerivationTree) -> str:\n",
    "    (symbol, children) = tree\n",
    "    if children is None:\n",
    "        # This is a nonterminal symbol not expanded yet\n",
    "        return symbol\n",
    "\n",
    "    if len(children) == 0:\n",
    "        # This is a terminal symbol\n",
    "        return symbol\n",
    "\n",
    "    # This is an expanded symbol:\n",
    "    # Concatenate all terminal symbols from all children\n",
    "    return ''.join([all_terminals(c) for c in children])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.302196Z",
     "iopub.status.busy": "2025-01-16T09:40:28.302079Z",
     "iopub.status.idle": "2025-01-16T09:40:28.304255Z",
     "shell.execute_reply": "2025-01-16T09:40:28.303989Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'<expr> + <term>'"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "all_terminals(derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The alternative `tree_to_string()` function also converts the tree to a string; however, it replaces nonterminal symbols by empty strings."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.305820Z",
     "iopub.status.busy": "2025-01-16T09:40:28.305702Z",
     "iopub.status.idle": "2025-01-16T09:40:28.307579Z",
     "shell.execute_reply": "2025-01-16T09:40:28.307307Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def tree_to_string(tree: DerivationTree) -> str:\n",
    "    symbol, children, *_ = tree\n",
    "    if children:\n",
    "        return ''.join(tree_to_string(c) for c in children)\n",
    "    else:\n",
    "        return '' if is_nonterminal(symbol) else symbol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.309108Z",
     "iopub.status.busy": "2025-01-16T09:40:28.308996Z",
     "iopub.status.idle": "2025-01-16T09:40:28.311028Z",
     "shell.execute_reply": "2025-01-16T09:40:28.310805Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "' + '"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tree_to_string(derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Expanding a Node"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Let us now develop an algorithm that takes a tree with non-expanded symbols (say, `derivation_tree`, above), and expands all these symbols one after the other.  As with earlier fuzzers, we create a special subclass of `Fuzzer` – in this case, `GrammarFuzzer`.  A `GrammarFuzzer` gets a grammar and a start symbol; the other parameters will be used later to further control creation and to support debugging."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.312648Z",
     "iopub.status.busy": "2025-01-16T09:40:28.312520Z",
     "iopub.status.idle": "2025-01-16T09:40:28.314150Z",
     "shell.execute_reply": "2025-01-16T09:40:28.313920Z"
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from Fuzzer import Fuzzer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.315671Z",
     "iopub.status.busy": "2025-01-16T09:40:28.315555Z",
     "iopub.status.idle": "2025-01-16T09:40:28.317877Z",
     "shell.execute_reply": "2025-01-16T09:40:28.317574Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(Fuzzer):\n",
    "    \"\"\"Produce strings from grammars efficiently, using derivation trees.\"\"\"\n",
    "\n",
    "    def __init__(self,\n",
    "                 grammar: Grammar,\n",
    "                 start_symbol: str = START_SYMBOL,\n",
    "                 min_nonterminals: int = 0,\n",
    "                 max_nonterminals: int = 10,\n",
    "                 disp: bool = False,\n",
    "                 log: Union[bool, int] = False) -> None:\n",
    "        \"\"\"Produce strings from `grammar`, starting with `start_symbol`.\n",
    "        If `min_nonterminals` or `max_nonterminals` is given, use them as limits \n",
    "        for the number of nonterminals produced.  \n",
    "        If `disp` is set, display the intermediate derivation trees.\n",
    "        If `log` is set, show intermediate steps as text on standard output.\"\"\"\n",
    "\n",
    "        self.grammar = grammar\n",
    "        self.start_symbol = start_symbol\n",
    "        self.min_nonterminals = min_nonterminals\n",
    "        self.max_nonterminals = max_nonterminals\n",
    "        self.disp = disp\n",
    "        self.log = log\n",
    "        self.check_grammar()  # Invokes is_valid_grammar()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "To add further methods to `GrammarFuzzer`, we use the hack already introduced for [the `MutationFuzzer` class](MutationFuzzer.ipynb).  The construct\n",
    "\n",
    "```python\n",
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def new_method(self, args):\n",
    "        pass\n",
    "```\n",
    "\n",
    "allows us to add a new method `new_method()` to the `GrammarFuzzer` class. (Actually, we get a new `GrammarFuzzer` class that extends the old one, but for all our purposes, this does not matter.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Excursion: `check_grammar()` implementation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We can use the above hack to define the helper method `check_grammar()`, which checks the given grammar for consistency:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.319580Z",
     "iopub.status.busy": "2025-01-16T09:40:28.319459Z",
     "iopub.status.idle": "2025-01-16T09:40:28.321494Z",
     "shell.execute_reply": "2025-01-16T09:40:28.321281Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def check_grammar(self) -> None:\n",
    "        \"\"\"Check the grammar passed\"\"\"\n",
    "        assert self.start_symbol in self.grammar\n",
    "        assert is_valid_grammar(\n",
    "            self.grammar,\n",
    "            start_symbol=self.start_symbol,\n",
    "            supported_opts=self.supported_opts())\n",
    "\n",
    "    def supported_opts(self) -> Set[str]:\n",
    "        \"\"\"Set of supported options. To be overloaded in subclasses.\"\"\"\n",
    "        return set()  # We don't support specific options"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### End of Excursion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Let us now define a helper method `init_tree()` that constructs a tree with just the start symbol:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.322980Z",
     "iopub.status.busy": "2025-01-16T09:40:28.322886Z",
     "iopub.status.idle": "2025-01-16T09:40:28.324557Z",
     "shell.execute_reply": "2025-01-16T09:40:28.324321Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def init_tree(self) -> DerivationTree:\n",
    "        return (self.start_symbol, None)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.325941Z",
     "iopub.status.busy": "2025-01-16T09:40:28.325856Z",
     "iopub.status.idle": "2025-01-16T09:40:28.759909Z",
     "shell.execute_reply": "2025-01-16T09:40:28.759572Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"48pt\" height=\"22pt\"\n",
       " viewBox=\"0.00 0.00 47.75 22.25\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 18.25)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-18.25 43.75,-18.25 43.75,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"19.88\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x108982b10>"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = GrammarFuzzer(EXPR_GRAMMAR)\n",
    "display_tree(f.init_tree())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "This is the tree we want to expand."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Picking a Children Alternative to be Expanded"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "One of the central methods in `GrammarFuzzer` is `choose_node_expansion()`. This method gets a node (say, the `<start>` node) and a list of possible lists of children to be expanded (one for every possible expansion from the grammar), chooses one of them, and returns its index in the possible children list."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "By overloading this method (notably in later chapters), we can implement different strategies – for now, it simply randomly picks one of the given lists of children (which in turn are lists of derivation trees)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.761802Z",
     "iopub.status.busy": "2025-01-16T09:40:28.761671Z",
     "iopub.status.idle": "2025-01-16T09:40:28.763904Z",
     "shell.execute_reply": "2025-01-16T09:40:28.763664Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def choose_node_expansion(self, node: DerivationTree,\n",
    "                              children_alternatives: List[List[DerivationTree]]) -> int:\n",
    "        \"\"\"Return index of expansion in `children_alternatives` to be selected.\n",
    "           'children_alternatives`: a list of possible children for `node`.\n",
    "           Defaults to random. To be overloaded in subclasses.\"\"\"\n",
    "        return random.randrange(0, len(children_alternatives))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Getting a List of Possible Expansions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "To actually obtain the list of possible children, we will need a helper function `expansion_to_children()` that takes an expansion string and decomposes it into a list of derivation trees – one for each symbol (terminal or nonterminal) in the string."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Excursion: Implementing `expansion_to_children()`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The function `expansion_to_children()` uses the `re.split()` method to split an expansion string into a list of children nodes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.765577Z",
     "iopub.status.busy": "2025-01-16T09:40:28.765469Z",
     "iopub.status.idle": "2025-01-16T09:40:28.767637Z",
     "shell.execute_reply": "2025-01-16T09:40:28.767345Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def expansion_to_children(expansion: Expansion) -> List[DerivationTree]:\n",
    "    # print(\"Converting \" + repr(expansion))\n",
    "    # strings contains all substrings -- both terminals and nonterminals such\n",
    "    # that ''.join(strings) == expansion\n",
    "\n",
    "    expansion = exp_string(expansion)\n",
    "    assert isinstance(expansion, str)\n",
    "\n",
    "    if expansion == \"\":  # Special case: epsilon expansion\n",
    "        return [(\"\", [])]\n",
    "\n",
    "    strings = re.split(RE_NONTERMINAL, expansion)\n",
    "    return [(s, None) if is_nonterminal(s) else (s, [])\n",
    "            for s in strings if len(s) > 0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### End of Excursion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.769184Z",
     "iopub.status.busy": "2025-01-16T09:40:28.769075Z",
     "iopub.status.idle": "2025-01-16T09:40:28.771219Z",
     "shell.execute_reply": "2025-01-16T09:40:28.770974Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('<term>', None), (' + ', []), ('<expr>', None)]"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expansion_to_children(\"<term> + <expr>\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The case of an *epsilon expansion*, i.e. expanding into an empty string as in `<symbol> ::=` needs special treatment:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.773043Z",
     "iopub.status.busy": "2025-01-16T09:40:28.772937Z",
     "iopub.status.idle": "2025-01-16T09:40:28.775074Z",
     "shell.execute_reply": "2025-01-16T09:40:28.774804Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('', [])]"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expansion_to_children(\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Just like `nonterminals()` in the [chapter on Grammars](Grammars.ipynb), we provide for future extensions, allowing the expansion to be a tuple with extra data (which will be ignored)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.776568Z",
     "iopub.status.busy": "2025-01-16T09:40:28.776456Z",
     "iopub.status.idle": "2025-01-16T09:40:28.778595Z",
     "shell.execute_reply": "2025-01-16T09:40:28.778361Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('+', []), ('<term>', None)]"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expansion_to_children((\"+<term>\", {\"extra_data\": 1234}))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We realize this helper as a method in `GrammarFuzzer` such that it can be overloaded by subclasses:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.780093Z",
     "iopub.status.busy": "2025-01-16T09:40:28.779988Z",
     "iopub.status.idle": "2025-01-16T09:40:28.781726Z",
     "shell.execute_reply": "2025-01-16T09:40:28.781493Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def expansion_to_children(self, expansion: Expansion) -> List[DerivationTree]:\n",
    "        return expansion_to_children(expansion)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Putting Things Together"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "With this, we can now take\n",
    "\n",
    "1. some non-expanded node in the tree, \n",
    "2. choose a random expansion, and\n",
    "3. return the new tree."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "This is what the method `expand_node_randomly()` does."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Excursion: `expand_node_randomly()` implementation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The function `expand_node_randomly()` uses a helper function `choose_node_expansion()` to randomly pick an index from an array of possible children.  (`choose_node_expansion()` can be overloaded in subclasses.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.783394Z",
     "iopub.status.busy": "2025-01-16T09:40:28.783288Z",
     "iopub.status.idle": "2025-01-16T09:40:28.784879Z",
     "shell.execute_reply": "2025-01-16T09:40:28.784536Z"
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "import random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.786528Z",
     "iopub.status.busy": "2025-01-16T09:40:28.786402Z",
     "iopub.status.idle": "2025-01-16T09:40:28.788969Z",
     "shell.execute_reply": "2025-01-16T09:40:28.788699Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def expand_node_randomly(self, node: DerivationTree) -> DerivationTree:\n",
    "        \"\"\"Choose a random expansion for `node` and return it\"\"\"\n",
    "        (symbol, children) = node\n",
    "        assert children is None\n",
    "\n",
    "        if self.log:\n",
    "            print(\"Expanding\", all_terminals(node), \"randomly\")\n",
    "\n",
    "        # Fetch the possible expansions from grammar...\n",
    "        expansions = self.grammar[symbol]\n",
    "        children_alternatives: List[List[DerivationTree]] = [\n",
    "            self.expansion_to_children(expansion) for expansion in expansions\n",
    "        ]\n",
    "\n",
    "        # ... and select a random expansion\n",
    "        index = self.choose_node_expansion(node, children_alternatives)\n",
    "        chosen_children = children_alternatives[index]\n",
    "\n",
    "        # Process children (for subclasses)\n",
    "        chosen_children = self.process_chosen_children(chosen_children,\n",
    "                                                       expansions[index])\n",
    "\n",
    "        # Return with new children\n",
    "        return (symbol, chosen_children)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The generic `expand_node()` method can later be used to select different expansion strategies; as of now, it only uses `expand_node_randomly()`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.790439Z",
     "iopub.status.busy": "2025-01-16T09:40:28.790342Z",
     "iopub.status.idle": "2025-01-16T09:40:28.792132Z",
     "shell.execute_reply": "2025-01-16T09:40:28.791850Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def expand_node(self, node: DerivationTree) -> DerivationTree:\n",
    "        return self.expand_node_randomly(node)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The helper function `process_chosen_children()` does nothing; it can be overloaded by subclasses to process the children once chosen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.793652Z",
     "iopub.status.busy": "2025-01-16T09:40:28.793540Z",
     "iopub.status.idle": "2025-01-16T09:40:28.795385Z",
     "shell.execute_reply": "2025-01-16T09:40:28.795166Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def process_chosen_children(self,\n",
    "                                chosen_children: List[DerivationTree],\n",
    "                                expansion: Expansion) -> List[DerivationTree]:\n",
    "        \"\"\"Process children after selection.  By default, does nothing.\"\"\"\n",
    "        return chosen_children"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### End of Excursion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "This is how `expand_node_randomly()` works:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:28.796876Z",
     "iopub.status.busy": "2025-01-16T09:40:28.796770Z",
     "iopub.status.idle": "2025-01-16T09:40:29.231328Z",
     "shell.execute_reply": "2025-01-16T09:40:29.230940Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Before expand_node_randomly():\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"62pt\" height=\"22pt\"\n",
       " viewBox=\"0.00 0.00 62.00 22.25\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 18.25)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-18.25 58,-18.25 58,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089833b0>"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = GrammarFuzzer(EXPR_GRAMMAR, log=True)\n",
    "\n",
    "print(\"Before expand_node_randomly():\")\n",
    "expr_tree = (\"<integer>\", None)\n",
    "display_tree(expr_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:29.233321Z",
     "iopub.status.busy": "2025-01-16T09:40:29.233189Z",
     "iopub.status.idle": "2025-01-16T09:40:29.690961Z",
     "shell.execute_reply": "2025-01-16T09:40:29.690442Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "After expand_node_randomly():\n",
      "Expanding <integer> randomly\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"62pt\" height=\"73pt\"\n",
       " viewBox=\"0.00 0.00 62.00 72.50\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 68.5)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-68.5 58,-68.5 58,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M27,-50.46C27,-44.24 27,-34.15 27,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"30.5,-25.6 27,-15.6 23.5,-25.6 30.5,-25.6\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x108982d50>"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(\"After expand_node_randomly():\")\n",
    "expr_tree = f.expand_node_randomly(expr_tree)\n",
    "display_tree(expr_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:29.693095Z",
     "iopub.status.busy": "2025-01-16T09:40:29.692909Z",
     "iopub.status.idle": "2025-01-16T09:40:29.695007Z",
     "shell.execute_reply": "2025-01-16T09:40:29.694678Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "# docassert\n",
    "assert expr_tree[1][0][0] == '<digit>'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:29.696902Z",
     "iopub.status.busy": "2025-01-16T09:40:29.696776Z",
     "iopub.status.idle": "2025-01-16T09:40:29.701251Z",
     "shell.execute_reply": "2025-01-16T09:40:29.700989Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "    \n",
       "    <script>\n",
       "    var bad_answers = new Map();\n",
       "\n",
       "    function answer(quiz_id) {\n",
       "        ans = 0;\n",
       "        for (i = 1;; i++) {\n",
       "            checkbox = document.getElementById(quiz_id + \"-\" + i.toString());\n",
       "            if (!checkbox)\n",
       "                break;\n",
       "            if (checkbox.checked)\n",
       "                ans |= (1 << i);\n",
       "        }\n",
       "        return ans;\n",
       "    }\n",
       "    function check_selection(quiz_id, correct_answer, multiple_choice, hint) {\n",
       "        given_answer = answer(quiz_id);\n",
       "        if (given_answer == correct_answer)\n",
       "        {\n",
       "            document.getElementById(quiz_id + \"-submit\").value = \"Correct!\";\n",
       "            document.getElementById(quiz_id + \"-hint\").innerHTML = \"\";\n",
       "\n",
       "            for (i = 1;; i++) {\n",
       "                checkbox = document.getElementById(quiz_id + \"-\" + i.toString());\n",
       "                label = document.getElementById(quiz_id + \"-\" + i.toString() + \"-label\")\n",
       "                if (!checkbox)\n",
       "                    break;\n",
       "\n",
       "                if (checkbox.checked) {\n",
       "                    label.style.fontWeight = \"bold\";\n",
       "                }\n",
       "                else {\n",
       "                    label.style.textDecoration = \"line-through\";\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "        else \n",
       "        {\n",
       "            document.getElementById(quiz_id + \"-submit\").value = \"Try again\";\n",
       "\n",
       "            if (!bad_answers.has(quiz_id)) {\n",
       "                bad_answers.set(quiz_id, 1);\n",
       "            }\n",
       "            else {\n",
       "                bad_answers.set(quiz_id, bad_answers.get(quiz_id) + 1);\n",
       "            }\n",
       "\n",
       "            if (bad_answers.get(quiz_id) >= 2 && hint.length > 0) {\n",
       "                document.getElementById(quiz_id + \"-hint\").innerHTML = \n",
       "                    \"&nbsp;&nbsp;(Hint: <code>\" + hint + \"</code>)\";\n",
       "            }\n",
       "\n",
       "            if (!multiple_choice) {\n",
       "                for (i = 1;; i++) {\n",
       "                    checkbox = document.getElementById(quiz_id + \"-\" + i.toString());\n",
       "                    label = document.getElementById(quiz_id + \"-\" + i.toString() + \"-label\")\n",
       "\n",
       "                    if (!checkbox)\n",
       "                        break;\n",
       "                    if (checkbox.checked) {\n",
       "                        label.style.textDecoration = \"line-through\";\n",
       "                    }\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "    function clear_selection(quiz_id) {\n",
       "        document.getElementById(quiz_id + \"-submit\").value = \"Submit\";\n",
       "        document.getElementById(quiz_id + \"-hint\").innerHTML = \"\";\n",
       "    }\n",
       "    </script>\n",
       "    \n",
       "    <div class=\"quiz\">\n",
       "    <h3 class=\"quiz_title\">Quiz</h3>\n",
       "    <p>\n",
       "    <div class=\"quiz_question\">What tree do we get if we expand the <code>&lt;digit&gt;</code> subtree?</div>\n",
       "    </p>\n",
       "    <p>\n",
       "    <div class=\"quiz_options\" title=\"Pick a choice.\">\n",
       "    \n",
       "        <input type=\"radio\" name=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790\" id=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790-1\" onclick=\"clear_selection('ec4d5270-d3ed-11ef-8343-6298cf1a5790')\">\n",
       "        <label id=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790-1-label\" for=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790-1\">We get another <code>&lt;digit&gt;</code> as new child of <code>&lt;digit&gt;</code></label><br>\n",
       "    \n",
       "        <input type=\"radio\" name=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790\" id=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790-2\" onclick=\"clear_selection('ec4d5270-d3ed-11ef-8343-6298cf1a5790')\">\n",
       "        <label id=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790-2-label\" for=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790-2\">We get some digit as child of <code>&lt;digit&gt;</code></label><br>\n",
       "    \n",
       "        <input type=\"radio\" name=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790\" id=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790-3\" onclick=\"clear_selection('ec4d5270-d3ed-11ef-8343-6298cf1a5790')\">\n",
       "        <label id=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790-3-label\" for=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790-3\">We get another <code>&lt;digit&gt;</code> as second child of <code>&lt;integer&gt;</code></label><br>\n",
       "    \n",
       "        <input type=\"radio\" name=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790\" id=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790-4\" onclick=\"clear_selection('ec4d5270-d3ed-11ef-8343-6298cf1a5790')\">\n",
       "        <label id=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790-4-label\" for=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790-4\">The entire tree becomes a single node with a digit</label><br>\n",
       "    \n",
       "    </div>\n",
       "    </p>\n",
       "    <input id=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790-submit\" type=\"submit\" value=\"Submit\" onclick=\"check_selection('ec4d5270-d3ed-11ef-8343-6298cf1a5790', 4, 0, 'len(&quot;2&quot;) + len(&quot;2&quot;)')\">\n",
       "    <span class=\"quiz_hint\" id=\"ec4d5270-d3ed-11ef-8343-6298cf1a5790-hint\"></span>\n",
       "    </div>\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "quiz(\"What tree do we get if we expand the `<digit>` subtree?\",\n",
    "     [\n",
    "         \"We get another `<digit>` as new child of `<digit>`\",\n",
    "         \"We get some digit as child of `<digit>`\",\n",
    "         \"We get another `<digit>` as second child of `<integer>`\",\n",
    "         \"The entire tree becomes a single node with a digit\"\n",
    "     ], 'len(\"2\") + len(\"2\")')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We can surely put this to the test, right? Here we go:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:29.703200Z",
     "iopub.status.busy": "2025-01-16T09:40:29.703068Z",
     "iopub.status.idle": "2025-01-16T09:40:30.129548Z",
     "shell.execute_reply": "2025-01-16T09:40:30.129114Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"49pt\" height=\"22pt\"\n",
       " viewBox=\"0.00 0.00 49.25 22.25\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 18.25)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-18.25 45.25,-18.25 45.25,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"20.62\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089a4fe0>"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "digit_subtree = expr_tree[1][0]  # type: ignore\n",
    "display_tree(digit_subtree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:30.131564Z",
     "iopub.status.busy": "2025-01-16T09:40:30.131403Z",
     "iopub.status.idle": "2025-01-16T09:40:30.561787Z",
     "shell.execute_reply": "2025-01-16T09:40:30.561453Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "After expanding the <digit> subtree:\n",
      "Expanding <digit> randomly\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"49pt\" height=\"73pt\"\n",
       " viewBox=\"0.00 0.00 49.25 72.50\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 68.5)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-68.5 45.25,-68.5 45.25,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"20.62\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"20.62\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">7 (55)</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M20.62,-50.46C20.62,-44.24 20.62,-34.15 20.62,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"24.13,-25.6 20.63,-15.6 17.13,-25.6 24.13,-25.6\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089a4440>"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(\"After expanding the <digit> subtree:\")\n",
    "digit_subtree = f.expand_node_randomly(digit_subtree)\n",
    "display_tree(digit_subtree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We see that `<digit>` gets expanded again according to the grammar rules – namely, into a single digit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:30.563620Z",
     "iopub.status.busy": "2025-01-16T09:40:30.563493Z",
     "iopub.status.idle": "2025-01-16T09:40:30.566956Z",
     "shell.execute_reply": "2025-01-16T09:40:30.566705Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "    \n",
       "    <script>\n",
       "    var bad_answers = new Map();\n",
       "\n",
       "    function answer(quiz_id) {\n",
       "        ans = 0;\n",
       "        for (i = 1;; i++) {\n",
       "            checkbox = document.getElementById(quiz_id + \"-\" + i.toString());\n",
       "            if (!checkbox)\n",
       "                break;\n",
       "            if (checkbox.checked)\n",
       "                ans |= (1 << i);\n",
       "        }\n",
       "        return ans;\n",
       "    }\n",
       "    function check_selection(quiz_id, correct_answer, multiple_choice, hint) {\n",
       "        given_answer = answer(quiz_id);\n",
       "        if (given_answer == correct_answer)\n",
       "        {\n",
       "            document.getElementById(quiz_id + \"-submit\").value = \"Correct!\";\n",
       "            document.getElementById(quiz_id + \"-hint\").innerHTML = \"\";\n",
       "\n",
       "            for (i = 1;; i++) {\n",
       "                checkbox = document.getElementById(quiz_id + \"-\" + i.toString());\n",
       "                label = document.getElementById(quiz_id + \"-\" + i.toString() + \"-label\")\n",
       "                if (!checkbox)\n",
       "                    break;\n",
       "\n",
       "                if (checkbox.checked) {\n",
       "                    label.style.fontWeight = \"bold\";\n",
       "                }\n",
       "                else {\n",
       "                    label.style.textDecoration = \"line-through\";\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "        else \n",
       "        {\n",
       "            document.getElementById(quiz_id + \"-submit\").value = \"Try again\";\n",
       "\n",
       "            if (!bad_answers.has(quiz_id)) {\n",
       "                bad_answers.set(quiz_id, 1);\n",
       "            }\n",
       "            else {\n",
       "                bad_answers.set(quiz_id, bad_answers.get(quiz_id) + 1);\n",
       "            }\n",
       "\n",
       "            if (bad_answers.get(quiz_id) >= 2 && hint.length > 0) {\n",
       "                document.getElementById(quiz_id + \"-hint\").innerHTML = \n",
       "                    \"&nbsp;&nbsp;(Hint: <code>\" + hint + \"</code>)\";\n",
       "            }\n",
       "\n",
       "            if (!multiple_choice) {\n",
       "                for (i = 1;; i++) {\n",
       "                    checkbox = document.getElementById(quiz_id + \"-\" + i.toString());\n",
       "                    label = document.getElementById(quiz_id + \"-\" + i.toString() + \"-label\")\n",
       "\n",
       "                    if (!checkbox)\n",
       "                        break;\n",
       "                    if (checkbox.checked) {\n",
       "                        label.style.textDecoration = \"line-through\";\n",
       "                    }\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "    function clear_selection(quiz_id) {\n",
       "        document.getElementById(quiz_id + \"-submit\").value = \"Submit\";\n",
       "        document.getElementById(quiz_id + \"-hint\").innerHTML = \"\";\n",
       "    }\n",
       "    </script>\n",
       "    \n",
       "    <div class=\"quiz\">\n",
       "    <h3 class=\"quiz_title\">Quiz</h3>\n",
       "    <p>\n",
       "    <div class=\"quiz_question\">Is the original <code>expr_tree</code> affected by this change?</div>\n",
       "    </p>\n",
       "    <p>\n",
       "    <div class=\"quiz_options\" title=\"Pick a choice.\">\n",
       "    \n",
       "        <input type=\"radio\" name=\"ecd191ca-d3ed-11ef-8343-6298cf1a5790\" id=\"ecd191ca-d3ed-11ef-8343-6298cf1a5790-1\" onclick=\"clear_selection('ecd191ca-d3ed-11ef-8343-6298cf1a5790')\">\n",
       "        <label id=\"ecd191ca-d3ed-11ef-8343-6298cf1a5790-1-label\" for=\"ecd191ca-d3ed-11ef-8343-6298cf1a5790-1\">No, it is unchanged</label><br>\n",
       "    \n",
       "        <input type=\"radio\" name=\"ecd191ca-d3ed-11ef-8343-6298cf1a5790\" id=\"ecd191ca-d3ed-11ef-8343-6298cf1a5790-2\" onclick=\"clear_selection('ecd191ca-d3ed-11ef-8343-6298cf1a5790')\">\n",
       "        <label id=\"ecd191ca-d3ed-11ef-8343-6298cf1a5790-2-label\" for=\"ecd191ca-d3ed-11ef-8343-6298cf1a5790-2\">Yes, it has also gained a new child</label><br>\n",
       "    \n",
       "    </div>\n",
       "    </p>\n",
       "    <input id=\"ecd191ca-d3ed-11ef-8343-6298cf1a5790-submit\" type=\"submit\" value=\"Submit\" onclick=\"check_selection('ecd191ca-d3ed-11ef-8343-6298cf1a5790', 2, 0, '1 ** (1 - 1)')\">\n",
       "    <span class=\"quiz_hint\" id=\"ecd191ca-d3ed-11ef-8343-6298cf1a5790-hint\"></span>\n",
       "    </div>\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "quiz(\"Is the original `expr_tree` affected by this change?\",\n",
    "     [\n",
    "         \"No, it is unchanged\",\n",
    "         \"Yes, it has also gained a new child\"\n",
    "     ], \"1 ** (1 - 1)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Although we have changed one of the subtrees, the original `expr_tree` is unaffected:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:30.568610Z",
     "iopub.status.busy": "2025-01-16T09:40:30.568521Z",
     "iopub.status.idle": "2025-01-16T09:40:31.021758Z",
     "shell.execute_reply": "2025-01-16T09:40:31.020970Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"62pt\" height=\"73pt\"\n",
       " viewBox=\"0.00 0.00 62.00 72.50\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 68.5)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-68.5 58,-68.5 58,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M27,-50.46C27,-44.24 27,-34.15 27,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"30.5,-25.6 27,-15.6 23.5,-25.6 30.5,-25.6\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089a4e30>"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display_tree(expr_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "That is because `expand_node_randomly()` returns a new (expanded) tree and does not change the tree passed as argument."
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Expanding a Tree\n",
    "\n",
    "Let us now apply our functions for expanding a single node to some node in the tree.  To this end, we first need to _search the tree for non-expanded nodes_.  `possible_expansions()` counts how many unexpanded symbols there are in a tree:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:31.023766Z",
     "iopub.status.busy": "2025-01-16T09:40:31.023604Z",
     "iopub.status.idle": "2025-01-16T09:40:31.026383Z",
     "shell.execute_reply": "2025-01-16T09:40:31.025967Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def possible_expansions(self, node: DerivationTree) -> int:\n",
    "        (symbol, children) = node\n",
    "        if children is None:\n",
    "            return 1\n",
    "\n",
    "        return sum(self.possible_expansions(c) for c in children)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:31.028169Z",
     "iopub.status.busy": "2025-01-16T09:40:31.028049Z",
     "iopub.status.idle": "2025-01-16T09:40:31.030336Z",
     "shell.execute_reply": "2025-01-16T09:40:31.029927Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2\n"
     ]
    }
   ],
   "source": [
    "f = GrammarFuzzer(EXPR_GRAMMAR)\n",
    "print(f.possible_expansions(derivation_tree))"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The method `any_possible_expansions()` returns True if the tree has any non-expanded nodes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:31.032431Z",
     "iopub.status.busy": "2025-01-16T09:40:31.032271Z",
     "iopub.status.idle": "2025-01-16T09:40:31.034700Z",
     "shell.execute_reply": "2025-01-16T09:40:31.034301Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def any_possible_expansions(self, node: DerivationTree) -> bool:\n",
    "        (symbol, children) = node\n",
    "        if children is None:\n",
    "            return True\n",
    "\n",
    "        return any(self.any_possible_expansions(c) for c in children)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:31.036583Z",
     "iopub.status.busy": "2025-01-16T09:40:31.036473Z",
     "iopub.status.idle": "2025-01-16T09:40:31.038958Z",
     "shell.execute_reply": "2025-01-16T09:40:31.038593Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = GrammarFuzzer(EXPR_GRAMMAR)\n",
    "f.any_possible_expansions(derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Here comes `expand_tree_once()`, the core method of our tree expansion algorithm.  It first checks whether it is currently being applied on a nonterminal symbol without expansion; if so, it invokes `expand_node()` on it, as discussed above.  "
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "If the node is already expanded (i.e. has children), it checks the subset of children which still have non-expanded symbols, randomly selects one of them, and applies itself recursively on that child."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Excursion: `expand_tree_once()` implementation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The `expand_tree_once()` method replaces the child _in place_, meaning that it actually mutates the tree being passed as an argument rather than returning a new tree.  This in-place mutation is what makes this function particularly efficient.  Again, we use a helper method (`choose_tree_expansion()`) to return the chosen index from a list of children that can be expanded."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:31.041222Z",
     "iopub.status.busy": "2025-01-16T09:40:31.041105Z",
     "iopub.status.idle": "2025-01-16T09:40:31.044074Z",
     "shell.execute_reply": "2025-01-16T09:40:31.043774Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def choose_tree_expansion(self,\n",
    "                              tree: DerivationTree,\n",
    "                              children: List[DerivationTree]) -> int:\n",
    "        \"\"\"Return index of subtree in `children` to be selected for expansion.\n",
    "           Defaults to random.\"\"\"\n",
    "        return random.randrange(0, len(children))\n",
    "\n",
    "    def expand_tree_once(self, tree: DerivationTree) -> DerivationTree:\n",
    "        \"\"\"Choose an unexpanded symbol in tree; expand it.\n",
    "           Can be overloaded in subclasses.\"\"\"\n",
    "        (symbol, children) = tree\n",
    "        if children is None:\n",
    "            # Expand this node\n",
    "            return self.expand_node(tree)\n",
    "\n",
    "        # Find all children with possible expansions\n",
    "        expandable_children = [\n",
    "            c for c in children if self.any_possible_expansions(c)]\n",
    "\n",
    "        # `index_map` translates an index in `expandable_children`\n",
    "        # back into the original index in `children`\n",
    "        index_map = [i for (i, c) in enumerate(children)\n",
    "                     if c in expandable_children]\n",
    "\n",
    "        # Select a random child\n",
    "        child_to_be_expanded = \\\n",
    "            self.choose_tree_expansion(tree, expandable_children)\n",
    "\n",
    "        # Expand in place\n",
    "        children[index_map[child_to_be_expanded]] = \\\n",
    "            self.expand_tree_once(expandable_children[child_to_be_expanded])\n",
    "\n",
    "        return tree"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### End of Excursion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Let us illustrate how `expand_tree_once()` works. We start with our derivation tree from above..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:31.045943Z",
     "iopub.status.busy": "2025-01-16T09:40:31.045783Z",
     "iopub.status.idle": "2025-01-16T09:40:31.489367Z",
     "shell.execute_reply": "2025-01-16T09:40:31.488895Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"142pt\" height=\"123pt\"\n",
       " viewBox=\"0.00 0.00 142.25 122.75\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 118.75)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-118.75 138.25,-118.75 138.25,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M66.25,-100.71C66.25,-94.49 66.25,-84.4 66.25,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"69.75,-75.85 66.25,-65.85 62.75,-75.85 69.75,-75.85\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"20.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M60.61,-50.46C53.99,-43.51 42.78,-31.75 33.85,-22.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"36.41,-20.01 26.98,-15.19 31.35,-24.84 36.41,-20.01\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>1&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M66.25,-50.46C66.25,-44.24 66.25,-34.15 66.25,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"69.75,-25.6 66.25,-15.6 62.75,-25.6 69.75,-25.6\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"113.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>1&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M72.01,-50.46C78.85,-43.44 90.47,-31.51 99.64,-22.1\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"101.91,-24.78 106.38,-15.17 96.9,-19.9 101.91,-24.78\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x108982e40>"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "derivation_tree = (\"<start>\",\n",
    "                   [(\"<expr>\",\n",
    "                     [(\"<expr>\", None),\n",
    "                      (\" + \", []),\n",
    "                         (\"<term>\", None)]\n",
    "                     )])\n",
    "display_tree(derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "... and now expand it twice:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:31.491606Z",
     "iopub.status.busy": "2025-01-16T09:40:31.491452Z",
     "iopub.status.idle": "2025-01-16T09:40:31.910416Z",
     "shell.execute_reply": "2025-01-16T09:40:31.909997Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expanding <expr> randomly\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"143pt\" height=\"173pt\"\n",
       " viewBox=\"0.00 0.00 143.00 173.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 169)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-169 139,-169 139,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"67\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"67\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M67,-150.96C67,-144.74 67,-134.65 67,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"70.5,-126.1 67,-116.1 63.5,-126.1 70.5,-126.1\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"21\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M61.36,-100.71C54.74,-93.76 43.53,-82 34.6,-72.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"37.16,-70.26 27.73,-65.44 32.1,-75.09 37.16,-70.26\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"67\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>1&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M67,-100.71C67,-94.49 67,-84.4 67,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"70.5,-75.85 67,-65.85 63.5,-75.85 70.5,-75.85\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"114\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>1&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M72.76,-100.71C79.6,-93.69 91.22,-81.76 100.39,-72.35\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"102.66,-75.03 107.13,-65.42 97.65,-70.15 102.66,-75.03\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"21\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>2&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M21,-50.46C21,-44.24 21,-34.15 21,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"24.5,-25.6 21,-15.6 17.5,-25.6 24.5,-25.6\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x108983500>"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = GrammarFuzzer(EXPR_GRAMMAR, log=True)\n",
    "derivation_tree = f.expand_tree_once(derivation_tree)\n",
    "display_tree(derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:31.912211Z",
     "iopub.status.busy": "2025-01-16T09:40:31.912084Z",
     "iopub.status.idle": "2025-01-16T09:40:32.338671Z",
     "shell.execute_reply": "2025-01-16T09:40:32.338170Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expanding <term> randomly\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"208pt\" height=\"173pt\"\n",
       " viewBox=\"0.00 0.00 208.00 173.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 169)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-169 204,-169 204,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"81\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"81\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M81,-150.96C81,-144.74 81,-134.65 81,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"84.5,-126.1 81,-116.1 77.5,-126.1 84.5,-126.1\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"30\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M74.75,-100.71C67.25,-93.62 54.46,-81.51 44.46,-72.06\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"47.07,-69.71 37.4,-65.38 42.26,-74.79 47.07,-69.71\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"81\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>1&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M81,-100.71C81,-94.49 81,-84.4 81,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"84.5,-75.85 81,-65.85 77.5,-75.85 84.5,-75.85\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"130\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>1&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M87.01,-100.71C94.14,-93.69 106.25,-81.76 115.81,-72.35\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"118.19,-74.91 122.86,-65.4 113.28,-69.92 118.19,-74.91\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"21\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>2&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M28.9,-50.46C27.74,-44.24 25.85,-34.15 24.22,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"27.66,-24.77 22.39,-15.58 20.78,-26.05 27.66,-24.77\"/>\n",
       "</g>\n",
       "<!-- 6 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>6</title>\n",
       "<text text-anchor=\"middle\" x=\"84\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;6 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>5&#45;&gt;6</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M124.36,-50.46C117.74,-43.51 106.53,-31.75 97.6,-22.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"100.16,-20.01 90.73,-15.19 95.1,-24.84 100.16,-20.01\"/>\n",
       "</g>\n",
       "<!-- 7 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>7</title>\n",
       "<text text-anchor=\"middle\" x=\"133\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\"> * </text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;7 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>5&#45;&gt;7</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M130.37,-50.46C130.75,-44.24 131.38,-34.15 131.93,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"135.41,-25.8 132.54,-15.6 128.42,-25.36 135.41,-25.8\"/>\n",
       "</g>\n",
       "<!-- 8 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>8</title>\n",
       "<text text-anchor=\"middle\" x=\"179\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;8 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>5&#45;&gt;8</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M136.01,-50.46C143.14,-43.44 155.25,-31.51 164.81,-22.1\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"167.19,-24.66 171.86,-15.15 162.28,-19.67 167.19,-24.66\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x108983110>"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "derivation_tree = f.expand_tree_once(derivation_tree)\n",
    "display_tree(derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We see that with each step, one more symbol is expanded.  Now all it takes is to apply this again and again, expanding the tree further and further."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Closing the Expansion\n",
    "\n",
    "With `expand_tree_once()`, we can keep on expanding the tree – but how do we actually stop?  The key idea here, introduced by Luke in \\cite{Luke2000}, is that after inflating the derivation tree to some maximum size, we _only want to apply expansions that increase the size of the tree by a minimum_.  For `<factor>`, for instance, we would prefer an expansion into `<integer>`, as this will not introduce further recursion (and potential size inflation); for `<integer>`, likewise, an expansion into `<digit>` is preferred, as it will less increase tree size than `<digit><integer>`."
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "To identify the _cost_ of expanding a symbol, we introduce two functions that mutually rely on each other:\n",
    "\n",
    "* `symbol_cost()` returns the minimum cost of all expansions of a symbol, using `expansion_cost()` to compute the cost for each expansion.\n",
    "* `expansion_cost()` returns the sum of all expansions in `expansions`. If a nonterminal is encountered again during traversal, the cost of the expansion is $\\infty$, indicating (potentially infinite) recursion."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Excursion: Implementing Cost Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:32.340934Z",
     "iopub.status.busy": "2025-01-16T09:40:32.340705Z",
     "iopub.status.idle": "2025-01-16T09:40:32.344116Z",
     "shell.execute_reply": "2025-01-16T09:40:32.343816Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def symbol_cost(self, symbol: str, seen: Set[str] = set()) \\\n",
    "            -> Union[int, float]:\n",
    "        expansions = self.grammar[symbol]\n",
    "        return min(self.expansion_cost(e, seen | {symbol}) for e in expansions)\n",
    "\n",
    "    def expansion_cost(self, expansion: Expansion,\n",
    "                       seen: Set[str] = set()) -> Union[int, float]:\n",
    "        symbols = nonterminals(expansion)\n",
    "        if len(symbols) == 0:\n",
    "            return 1  # no symbol\n",
    "\n",
    "        if any(s in seen for s in symbols):\n",
    "            return float('inf')\n",
    "\n",
    "        # the value of a expansion is the sum of all expandable variables\n",
    "        # inside + 1\n",
    "        return sum(self.symbol_cost(s, seen) for s in symbols) + 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### End of Excursion"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Here are two examples: The minimum cost of expanding a digit is 1, since we have to choose from one of its expansions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:32.345858Z",
     "iopub.status.busy": "2025-01-16T09:40:32.345761Z",
     "iopub.status.idle": "2025-01-16T09:40:32.347850Z",
     "shell.execute_reply": "2025-01-16T09:40:32.347479Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "f = GrammarFuzzer(EXPR_GRAMMAR)\n",
    "assert f.symbol_cost(\"<digit>\") == 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The minimum cost of expanding `<expr>`, though, is five, as this is the minimum number of expansions required.  (`<expr>` $\\rightarrow$ `<term>` $\\rightarrow$ `<factor>` $\\rightarrow$ `<integer>` $\\rightarrow$ `<digit>` $\\rightarrow$ 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:32.349755Z",
     "iopub.status.busy": "2025-01-16T09:40:32.349659Z",
     "iopub.status.idle": "2025-01-16T09:40:32.351521Z",
     "shell.execute_reply": "2025-01-16T09:40:32.351243Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "assert f.symbol_cost(\"<expr>\") == 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We define `expand_node_by_cost(self, node, choose)`, a variant of `expand_node()` that takes the above cost into account.  It determines the minimum cost `cost` across all children and then chooses a child from the list using the `choose` function, which by default is the minimum cost.  If multiple children all have the same minimum cost, it chooses randomly between these."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Excursion: `expand_node_by_cost()` implementation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:32.353254Z",
     "iopub.status.busy": "2025-01-16T09:40:32.353150Z",
     "iopub.status.idle": "2025-01-16T09:40:32.356484Z",
     "shell.execute_reply": "2025-01-16T09:40:32.356197Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def expand_node_by_cost(self, node: DerivationTree, \n",
    "                            choose: Callable = min) -> DerivationTree:\n",
    "        (symbol, children) = node\n",
    "        assert children is None\n",
    "\n",
    "        # Fetch the possible expansions from grammar...\n",
    "        expansions = self.grammar[symbol]\n",
    "\n",
    "        children_alternatives_with_cost = [(self.expansion_to_children(expansion),\n",
    "                                            self.expansion_cost(expansion, {symbol}),\n",
    "                                            expansion)\n",
    "                                           for expansion in expansions]\n",
    "\n",
    "        costs = [cost for (child, cost, expansion)\n",
    "                 in children_alternatives_with_cost]\n",
    "        chosen_cost = choose(costs)\n",
    "        children_with_chosen_cost = [child for (child, child_cost, _) \n",
    "                                     in children_alternatives_with_cost\n",
    "                                     if child_cost == chosen_cost]\n",
    "        expansion_with_chosen_cost = [expansion for (_, child_cost, expansion)\n",
    "                                      in children_alternatives_with_cost\n",
    "                                      if child_cost == chosen_cost]\n",
    "\n",
    "        index = self.choose_node_expansion(node, children_with_chosen_cost)\n",
    "\n",
    "        chosen_children = children_with_chosen_cost[index]\n",
    "        chosen_expansion = expansion_with_chosen_cost[index]\n",
    "        chosen_children = self.process_chosen_children(\n",
    "            chosen_children, chosen_expansion)\n",
    "\n",
    "        # Return with a new list\n",
    "        return (symbol, chosen_children)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### End of Excursion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The shortcut `expand_node_min_cost()` passes `min()` as the `choose` function, which makes it expand nodes at minimum cost."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:32.358440Z",
     "iopub.status.busy": "2025-01-16T09:40:32.358325Z",
     "iopub.status.idle": "2025-01-16T09:40:32.360443Z",
     "shell.execute_reply": "2025-01-16T09:40:32.360177Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def expand_node_min_cost(self, node: DerivationTree) -> DerivationTree:\n",
    "        if self.log:\n",
    "            print(\"Expanding\", all_terminals(node), \"at minimum cost\")\n",
    "\n",
    "        return self.expand_node_by_cost(node, min)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We can now apply this function to close the expansion of our derivation tree, using `expand_tree_once()` with the above `expand_node_min_cost()` as expansion function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:32.362140Z",
     "iopub.status.busy": "2025-01-16T09:40:32.362015Z",
     "iopub.status.idle": "2025-01-16T09:40:32.363929Z",
     "shell.execute_reply": "2025-01-16T09:40:32.363656Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def expand_node(self, node: DerivationTree) -> DerivationTree:\n",
    "        return self.expand_node_min_cost(node)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:32.365524Z",
     "iopub.status.busy": "2025-01-16T09:40:32.365417Z",
     "iopub.status.idle": "2025-01-16T09:40:32.772765Z",
     "shell.execute_reply": "2025-01-16T09:40:32.772378Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"208pt\" height=\"173pt\"\n",
       " viewBox=\"0.00 0.00 208.00 173.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 169)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-169 204,-169 204,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"81\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"81\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M81,-150.96C81,-144.74 81,-134.65 81,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"84.5,-126.1 81,-116.1 77.5,-126.1 84.5,-126.1\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"30\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M74.75,-100.71C67.25,-93.62 54.46,-81.51 44.46,-72.06\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"47.07,-69.71 37.4,-65.38 42.26,-74.79 47.07,-69.71\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"81\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>1&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M81,-100.71C81,-94.49 81,-84.4 81,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"84.5,-75.85 81,-65.85 77.5,-75.85 84.5,-75.85\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"130\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>1&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M87.01,-100.71C94.14,-93.69 106.25,-81.76 115.81,-72.35\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"118.19,-74.91 122.86,-65.4 113.28,-69.92 118.19,-74.91\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"21\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>2&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M28.9,-50.46C27.74,-44.24 25.85,-34.15 24.22,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"27.66,-24.77 22.39,-15.58 20.78,-26.05 27.66,-24.77\"/>\n",
       "</g>\n",
       "<!-- 6 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>6</title>\n",
       "<text text-anchor=\"middle\" x=\"84\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;6 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>5&#45;&gt;6</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M124.36,-50.46C117.74,-43.51 106.53,-31.75 97.6,-22.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"100.16,-20.01 90.73,-15.19 95.1,-24.84 100.16,-20.01\"/>\n",
       "</g>\n",
       "<!-- 7 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>7</title>\n",
       "<text text-anchor=\"middle\" x=\"133\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\"> * </text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;7 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>5&#45;&gt;7</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M130.37,-50.46C130.75,-44.24 131.38,-34.15 131.93,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"135.41,-25.8 132.54,-15.6 128.42,-25.36 135.41,-25.8\"/>\n",
       "</g>\n",
       "<!-- 8 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>8</title>\n",
       "<text text-anchor=\"middle\" x=\"179\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;8 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>5&#45;&gt;8</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M136.01,-50.46C143.14,-43.44 155.25,-31.51 164.81,-22.1\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"167.19,-24.66 171.86,-15.15 162.28,-19.67 167.19,-24.66\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x108982660>"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = GrammarFuzzer(EXPR_GRAMMAR, log=True)\n",
    "display_tree(derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:32.774418Z",
     "iopub.status.busy": "2025-01-16T09:40:32.774297Z",
     "iopub.status.idle": "2025-01-16T09:40:32.776076Z",
     "shell.execute_reply": "2025-01-16T09:40:32.775837Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "# docassert\n",
    "assert f.any_possible_expansions(derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:32.777619Z",
     "iopub.status.busy": "2025-01-16T09:40:32.777519Z",
     "iopub.status.idle": "2025-01-16T09:40:33.172180Z",
     "shell.execute_reply": "2025-01-16T09:40:33.171852Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expanding <term> at minimum cost\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"211pt\" height=\"223pt\"\n",
       " viewBox=\"0.00 0.00 211.00 223.25\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 219.25)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-219.25 207,-219.25 207,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"84\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"84\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M84,-201.21C84,-194.99 84,-184.9 84,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"87.5,-176.35 84,-166.35 80.5,-176.35 87.5,-176.35\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"33\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M77.75,-150.96C70.25,-143.87 57.46,-131.76 47.46,-122.31\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"50.07,-119.96 40.4,-115.63 45.26,-125.04 50.07,-119.96\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"84\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>1&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M84,-150.96C84,-144.74 84,-134.65 84,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"87.5,-126.1 84,-116.1 80.5,-126.1 87.5,-126.1\"/>\n",
       "</g>\n",
       "<!-- 6 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>6</title>\n",
       "<text text-anchor=\"middle\" x=\"133\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;6 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>1&#45;&gt;6</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M90.01,-150.96C97.14,-143.94 109.25,-132.01 118.81,-122.6\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"121.19,-125.16 125.86,-115.65 116.28,-120.17 121.19,-125.16\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"24\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>2&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M31.9,-100.71C30.74,-94.49 28.85,-84.4 27.22,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"30.66,-75.02 25.39,-65.83 23.78,-76.3 30.66,-75.02\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"24\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 3&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>3&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M24,-50.46C24,-44.24 24,-34.15 24,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"27.5,-25.6 24,-15.6 20.5,-25.6 27.5,-25.6\"/>\n",
       "</g>\n",
       "<!-- 7 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>7</title>\n",
       "<text text-anchor=\"middle\" x=\"87\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 6&#45;&gt;7 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>6&#45;&gt;7</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M127.36,-100.71C120.74,-93.76 109.53,-82 100.6,-72.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"103.16,-70.26 93.73,-65.44 98.1,-75.09 103.16,-70.26\"/>\n",
       "</g>\n",
       "<!-- 8 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>8</title>\n",
       "<text text-anchor=\"middle\" x=\"136\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\"> * </text>\n",
       "</g>\n",
       "<!-- 6&#45;&gt;8 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>6&#45;&gt;8</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M133.37,-100.71C133.75,-94.49 134.38,-84.4 134.93,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"138.41,-76.05 135.54,-65.85 131.42,-75.61 138.41,-76.05\"/>\n",
       "</g>\n",
       "<!-- 9 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>9</title>\n",
       "<text text-anchor=\"middle\" x=\"182\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 6&#45;&gt;9 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>6&#45;&gt;9</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M139.01,-100.71C146.14,-93.69 158.25,-81.76 167.81,-72.35\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"170.19,-74.91 174.86,-65.4 165.28,-69.92 170.19,-74.91\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089bc050>"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "if f.any_possible_expansions(derivation_tree):\n",
    "    derivation_tree = f.expand_tree_once(derivation_tree)\n",
    "display_tree(derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:33.173847Z",
     "iopub.status.busy": "2025-01-16T09:40:33.173725Z",
     "iopub.status.idle": "2025-01-16T09:40:33.175536Z",
     "shell.execute_reply": "2025-01-16T09:40:33.175294Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "# docassert\n",
    "assert f.any_possible_expansions(derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:33.177111Z",
     "iopub.status.busy": "2025-01-16T09:40:33.177003Z",
     "iopub.status.idle": "2025-01-16T09:40:33.584656Z",
     "shell.execute_reply": "2025-01-16T09:40:33.584091Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expanding <factor> at minimum cost\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"215pt\" height=\"223pt\"\n",
       " viewBox=\"0.00 0.00 215.00 223.25\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 219.25)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-219.25 211,-219.25 211,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"87\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"87\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M87,-201.21C87,-194.99 87,-184.9 87,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"90.5,-176.35 87,-166.35 83.5,-176.35 90.5,-176.35\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"36\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M80.75,-150.96C73.25,-143.87 60.46,-131.76 50.46,-122.31\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"53.07,-119.96 43.4,-115.63 48.26,-125.04 53.07,-119.96\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"87\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>1&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M87,-150.96C87,-144.74 87,-134.65 87,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"90.5,-126.1 87,-116.1 83.5,-126.1 90.5,-126.1\"/>\n",
       "</g>\n",
       "<!-- 6 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>6</title>\n",
       "<text text-anchor=\"middle\" x=\"136\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;6 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>1&#45;&gt;6</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M93.01,-150.96C100.14,-143.94 112.25,-132.01 121.81,-122.6\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"124.19,-125.16 128.86,-115.65 119.28,-120.17 124.19,-125.16\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>2&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.9,-100.71C33.74,-94.49 31.85,-84.4 30.22,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"33.66,-75.02 28.39,-65.83 26.78,-76.3 33.66,-75.02\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"24\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 3&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>3&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M26.63,-50.46C26.25,-44.24 25.62,-34.15 25.07,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"28.58,-25.36 24.46,-15.6 21.59,-25.8 28.58,-25.36\"/>\n",
       "</g>\n",
       "<!-- 7 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>7</title>\n",
       "<text text-anchor=\"middle\" x=\"91\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 6&#45;&gt;7 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>6&#45;&gt;7</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M130.48,-100.71C124,-93.76 113.04,-82 104.3,-72.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"106.98,-70.37 97.6,-65.45 101.86,-75.15 106.98,-70.37\"/>\n",
       "</g>\n",
       "<!-- 9 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>9</title>\n",
       "<text text-anchor=\"middle\" x=\"140\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\"> * </text>\n",
       "</g>\n",
       "<!-- 6&#45;&gt;9 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>6&#45;&gt;9</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M136.49,-100.71C137.01,-94.49 137.84,-84.4 138.57,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"142.04,-76.1 139.38,-65.85 135.07,-75.52 142.04,-76.1\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>10</title>\n",
       "<text text-anchor=\"middle\" x=\"186\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 6&#45;&gt;10 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>6&#45;&gt;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M142.13,-100.71C149.48,-93.62 162.02,-81.51 171.82,-72.06\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"173.96,-74.85 178.73,-65.39 169.1,-69.82 173.96,-74.85\"/>\n",
       "</g>\n",
       "<!-- 8 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>8</title>\n",
       "<text text-anchor=\"middle\" x=\"93\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 7&#45;&gt;8 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>7&#45;&gt;8</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M91.25,-50.46C91.5,-44.24 91.92,-34.15 92.28,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"95.77,-25.74 92.69,-15.6 88.78,-25.45 95.77,-25.74\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089bd1f0>"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "if f.any_possible_expansions(derivation_tree):\n",
    "    derivation_tree = f.expand_tree_once(derivation_tree)\n",
    "display_tree(derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:33.586724Z",
     "iopub.status.busy": "2025-01-16T09:40:33.586566Z",
     "iopub.status.idle": "2025-01-16T09:40:33.588649Z",
     "shell.execute_reply": "2025-01-16T09:40:33.588359Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "# docassert\n",
    "assert f.any_possible_expansions(derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:33.590317Z",
     "iopub.status.busy": "2025-01-16T09:40:33.590199Z",
     "iopub.status.idle": "2025-01-16T09:40:34.009892Z",
     "shell.execute_reply": "2025-01-16T09:40:34.009537Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expanding <term> at minimum cost\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"218pt\" height=\"223pt\"\n",
       " viewBox=\"0.00 0.00 218.00 223.25\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 219.25)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-219.25 214,-219.25 214,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"87\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"87\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M87,-201.21C87,-194.99 87,-184.9 87,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"90.5,-176.35 87,-166.35 83.5,-176.35 90.5,-176.35\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"36\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M80.75,-150.96C73.25,-143.87 60.46,-131.76 50.46,-122.31\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"53.07,-119.96 43.4,-115.63 48.26,-125.04 53.07,-119.96\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"87\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>1&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M87,-150.96C87,-144.74 87,-134.65 87,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"90.5,-126.1 87,-116.1 83.5,-126.1 90.5,-126.1\"/>\n",
       "</g>\n",
       "<!-- 6 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>6</title>\n",
       "<text text-anchor=\"middle\" x=\"136\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;6 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>1&#45;&gt;6</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M93.01,-150.96C100.14,-143.94 112.25,-132.01 121.81,-122.6\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"124.19,-125.16 128.86,-115.65 119.28,-120.17 124.19,-125.16\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>2&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.9,-100.71C33.74,-94.49 31.85,-84.4 30.22,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"33.66,-75.02 28.39,-65.83 26.78,-76.3 33.66,-75.02\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"24\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 3&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>3&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M26.63,-50.46C26.25,-44.24 25.62,-34.15 25.07,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"28.58,-25.36 24.46,-15.6 21.59,-25.8 28.58,-25.36\"/>\n",
       "</g>\n",
       "<!-- 7 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>7</title>\n",
       "<text text-anchor=\"middle\" x=\"91\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 6&#45;&gt;7 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>6&#45;&gt;7</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M130.48,-100.71C124,-93.76 113.04,-82 104.3,-72.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"106.98,-70.37 97.6,-65.45 101.86,-75.15 106.98,-70.37\"/>\n",
       "</g>\n",
       "<!-- 9 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>9</title>\n",
       "<text text-anchor=\"middle\" x=\"140\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\"> * </text>\n",
       "</g>\n",
       "<!-- 6&#45;&gt;9 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>6&#45;&gt;9</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M136.49,-100.71C137.01,-94.49 137.84,-84.4 138.57,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"142.04,-76.1 139.38,-65.85 135.07,-75.52 142.04,-76.1\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>10</title>\n",
       "<text text-anchor=\"middle\" x=\"186\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 6&#45;&gt;10 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>6&#45;&gt;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M142.13,-100.71C149.48,-93.62 162.02,-81.51 171.82,-72.06\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"173.96,-74.85 178.73,-65.39 169.1,-69.82 173.96,-74.85\"/>\n",
       "</g>\n",
       "<!-- 8 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>8</title>\n",
       "<text text-anchor=\"middle\" x=\"93\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 7&#45;&gt;8 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>7&#45;&gt;8</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M91.25,-50.46C91.5,-44.24 91.92,-34.15 92.28,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"95.77,-25.74 92.69,-15.6 88.78,-25.45 95.77,-25.74\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>11</title>\n",
       "<text text-anchor=\"middle\" x=\"186\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 10&#45;&gt;11 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>10&#45;&gt;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M186,-50.46C186,-44.24 186,-34.15 186,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"189.5,-25.6 186,-15.6 182.5,-25.6 189.5,-25.6\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089bd370>"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "if f.any_possible_expansions(derivation_tree):\n",
    "    derivation_tree = f.expand_tree_once(derivation_tree)\n",
    "display_tree(derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We keep on expanding until all nonterminals are expanded."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:34.011644Z",
     "iopub.status.busy": "2025-01-16T09:40:34.011519Z",
     "iopub.status.idle": "2025-01-16T09:40:34.014190Z",
     "shell.execute_reply": "2025-01-16T09:40:34.013811Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expanding <integer> at minimum cost\n",
      "Expanding <digit> at minimum cost\n",
      "Expanding <factor> at minimum cost\n",
      "Expanding <integer> at minimum cost\n",
      "Expanding <factor> at minimum cost\n",
      "Expanding <integer> at minimum cost\n",
      "Expanding <digit> at minimum cost\n",
      "Expanding <digit> at minimum cost\n"
     ]
    }
   ],
   "source": [
    "while f.any_possible_expansions(derivation_tree):\n",
    "    derivation_tree = f.expand_tree_once(derivation_tree)    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Here is the final tree:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:34.015729Z",
     "iopub.status.busy": "2025-01-16T09:40:34.015616Z",
     "iopub.status.idle": "2025-01-16T09:40:34.415938Z",
     "shell.execute_reply": "2025-01-16T09:40:34.414882Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"224pt\" height=\"374pt\"\n",
       " viewBox=\"0.00 0.00 224.00 374.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 370)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-370 220,-370 220,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"90\" y=\"-352.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"90\" y=\"-302.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M90,-351.96C90,-345.74 90,-335.65 90,-326.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"93.5,-327.1 90,-317.1 86.5,-327.1 93.5,-327.1\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"39\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M83.75,-301.71C76.25,-294.62 63.46,-282.51 53.46,-273.06\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"56.07,-270.71 46.4,-266.38 51.26,-275.79 56.07,-270.71\"/>\n",
       "</g>\n",
       "<!-- 8 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>8</title>\n",
       "<text text-anchor=\"middle\" x=\"90\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;8 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>1&#45;&gt;8</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M90,-301.71C90,-295.49 90,-285.4 90,-276.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"93.5,-276.85 90,-266.85 86.5,-276.85 93.5,-276.85\"/>\n",
       "</g>\n",
       "<!-- 9 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>9</title>\n",
       "<text text-anchor=\"middle\" x=\"139\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;9 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>1&#45;&gt;9</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M96.01,-301.71C103.14,-294.69 115.25,-282.76 124.81,-273.35\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"127.19,-275.91 131.86,-266.4 122.28,-270.92 127.19,-275.91\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"30\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>2&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M37.9,-251.46C36.74,-245.24 34.85,-235.15 33.22,-226.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"36.66,-225.77 31.39,-216.58 29.78,-227.05 36.66,-225.77\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 3&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>3&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M29.63,-201.21C29.25,-194.99 28.62,-184.9 28.07,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"31.58,-176.11 27.46,-166.35 24.59,-176.55 31.58,-176.11\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 4&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>4&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M27,-150.96C27,-144.74 27,-134.65 27,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"30.5,-126.1 27,-116.1 23.5,-126.1 30.5,-126.1\"/>\n",
       "</g>\n",
       "<!-- 6 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>6</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;6 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>5&#45;&gt;6</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M27,-100.71C27,-94.49 27,-84.4 27,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"30.5,-75.85 27,-65.85 23.5,-75.85 30.5,-75.85\"/>\n",
       "</g>\n",
       "<!-- 7 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>7</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">7 (55)</text>\n",
       "</g>\n",
       "<!-- 6&#45;&gt;7 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>6&#45;&gt;7</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M27,-50.46C27,-44.24 27,-34.15 27,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"30.5,-25.6 27,-15.6 23.5,-25.6 30.5,-25.6\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>10</title>\n",
       "<text text-anchor=\"middle\" x=\"94\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 9&#45;&gt;10 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>9&#45;&gt;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M133.48,-251.46C127,-244.51 116.04,-232.75 107.3,-223.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"109.98,-221.12 100.6,-216.2 104.86,-225.9 109.98,-221.12\"/>\n",
       "</g>\n",
       "<!-- 14 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>14</title>\n",
       "<text text-anchor=\"middle\" x=\"143\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\"> * </text>\n",
       "</g>\n",
       "<!-- 9&#45;&gt;14 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>9&#45;&gt;14</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M139.49,-251.46C140.01,-245.24 140.84,-235.15 141.57,-226.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"145.04,-226.85 142.38,-216.6 138.07,-226.27 145.04,-226.85\"/>\n",
       "</g>\n",
       "<!-- 15 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>15</title>\n",
       "<text text-anchor=\"middle\" x=\"189\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 9&#45;&gt;15 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>9&#45;&gt;15</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M145.13,-251.46C152.48,-244.37 165.02,-232.26 174.82,-222.81\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"176.96,-225.6 181.73,-216.14 172.1,-220.57 176.96,-225.6\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>11</title>\n",
       "<text text-anchor=\"middle\" x=\"96\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 10&#45;&gt;11 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>10&#45;&gt;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M94.25,-201.21C94.5,-194.99 94.92,-184.9 95.28,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"98.77,-176.49 95.69,-166.35 91.78,-176.2 98.77,-176.49\"/>\n",
       "</g>\n",
       "<!-- 12 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>12</title>\n",
       "<text text-anchor=\"middle\" x=\"96\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 11&#45;&gt;12 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>11&#45;&gt;12</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M96,-150.96C96,-144.74 96,-134.65 96,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"99.5,-126.1 96,-116.1 92.5,-126.1 99.5,-126.1\"/>\n",
       "</g>\n",
       "<!-- 13 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>13</title>\n",
       "<text text-anchor=\"middle\" x=\"96\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">5 (53)</text>\n",
       "</g>\n",
       "<!-- 12&#45;&gt;13 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>12&#45;&gt;13</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M96,-100.71C96,-94.49 96,-84.4 96,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"99.5,-75.85 96,-65.85 92.5,-75.85 99.5,-75.85\"/>\n",
       "</g>\n",
       "<!-- 16 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>16</title>\n",
       "<text text-anchor=\"middle\" x=\"189\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 15&#45;&gt;16 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>15&#45;&gt;16</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M189,-201.21C189,-194.99 189,-184.9 189,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"192.5,-176.35 189,-166.35 185.5,-176.35 192.5,-176.35\"/>\n",
       "</g>\n",
       "<!-- 17 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>17</title>\n",
       "<text text-anchor=\"middle\" x=\"189\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 16&#45;&gt;17 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>16&#45;&gt;17</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M189,-150.96C189,-144.74 189,-134.65 189,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"192.5,-126.1 189,-116.1 185.5,-126.1 192.5,-126.1\"/>\n",
       "</g>\n",
       "<!-- 18 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>18</title>\n",
       "<text text-anchor=\"middle\" x=\"189\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 17&#45;&gt;18 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>17&#45;&gt;18</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M189,-100.71C189,-94.49 189,-84.4 189,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"192.5,-75.85 189,-65.85 185.5,-75.85 192.5,-75.85\"/>\n",
       "</g>\n",
       "<!-- 19 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>19</title>\n",
       "<text text-anchor=\"middle\" x=\"189\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">8 (56)</text>\n",
       "</g>\n",
       "<!-- 18&#45;&gt;19 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>18&#45;&gt;19</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M189,-50.46C189,-44.24 189,-34.15 189,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"192.5,-25.6 189,-15.6 185.5,-25.6 192.5,-25.6\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x108982720>"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display_tree(derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We see that in each step, `expand_node_min_cost()` chooses an expansion that does not increase the number of symbols, eventually closing all open expansions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Node Inflation\n",
    "\n",
    "Especially at the beginning of an expansion, we may be interested in getting _as many nodes as possible_ – that is, we'd like to prefer expansions that give us _more_ nonterminals to expand.  This is actually the exact opposite of what `expand_node_min_cost()` gives us, and we can implement a method `expand_node_max_cost()` that will always choose among the nodes with the _highest_ cost:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:34.418979Z",
     "iopub.status.busy": "2025-01-16T09:40:34.418752Z",
     "iopub.status.idle": "2025-01-16T09:40:34.423719Z",
     "shell.execute_reply": "2025-01-16T09:40:34.422340Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def expand_node_max_cost(self, node: DerivationTree) -> DerivationTree:\n",
    "        if self.log:\n",
    "            print(\"Expanding\", all_terminals(node), \"at maximum cost\")\n",
    "\n",
    "        return self.expand_node_by_cost(node, max)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "To illustrate `expand_node_max_cost()`, we can again redefine `expand_node()` to use it, and then use `expand_tree_once()` to show a few expansion steps:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:34.429339Z",
     "iopub.status.busy": "2025-01-16T09:40:34.429145Z",
     "iopub.status.idle": "2025-01-16T09:40:34.431737Z",
     "shell.execute_reply": "2025-01-16T09:40:34.431282Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def expand_node(self, node: DerivationTree) -> DerivationTree:\n",
    "        return self.expand_node_max_cost(node)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:34.433772Z",
     "iopub.status.busy": "2025-01-16T09:40:34.433606Z",
     "iopub.status.idle": "2025-01-16T09:40:34.435533Z",
     "shell.execute_reply": "2025-01-16T09:40:34.435202Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "derivation_tree = (\"<start>\",\n",
    "                   [(\"<expr>\",\n",
    "                     [(\"<expr>\", None),\n",
    "                      (\" + \", []),\n",
    "                         (\"<term>\", None)]\n",
    "                     )])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:34.437219Z",
     "iopub.status.busy": "2025-01-16T09:40:34.437096Z",
     "iopub.status.idle": "2025-01-16T09:40:34.849269Z",
     "shell.execute_reply": "2025-01-16T09:40:34.848916Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"142pt\" height=\"123pt\"\n",
       " viewBox=\"0.00 0.00 142.25 122.75\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 118.75)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-118.75 138.25,-118.75 138.25,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M66.25,-100.71C66.25,-94.49 66.25,-84.4 66.25,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"69.75,-75.85 66.25,-65.85 62.75,-75.85 69.75,-75.85\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"20.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M60.61,-50.46C53.99,-43.51 42.78,-31.75 33.85,-22.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"36.41,-20.01 26.98,-15.19 31.35,-24.84 36.41,-20.01\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>1&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M66.25,-50.46C66.25,-44.24 66.25,-34.15 66.25,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"69.75,-25.6 66.25,-15.6 62.75,-25.6 69.75,-25.6\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"113.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>1&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M72.01,-50.46C78.85,-43.44 90.47,-31.51 99.64,-22.1\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"101.91,-24.78 106.38,-15.17 96.9,-19.9 101.91,-24.78\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089bd490>"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = GrammarFuzzer(EXPR_GRAMMAR, log=True)\n",
    "display_tree(derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:34.851178Z",
     "iopub.status.busy": "2025-01-16T09:40:34.851033Z",
     "iopub.status.idle": "2025-01-16T09:40:34.852793Z",
     "shell.execute_reply": "2025-01-16T09:40:34.852509Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# docassert\n",
    "assert f.any_possible_expansions(derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:34.854390Z",
     "iopub.status.busy": "2025-01-16T09:40:34.854286Z",
     "iopub.status.idle": "2025-01-16T09:40:35.294921Z",
     "shell.execute_reply": "2025-01-16T09:40:35.294578Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expanding <term> at maximum cost\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"188pt\" height=\"173pt\"\n",
       " viewBox=\"0.00 0.00 188.25 173.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 169)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-169 184.25,-169 184.25,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M66.25,-150.96C66.25,-144.74 66.25,-134.65 66.25,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"69.75,-126.1 66.25,-116.1 62.75,-126.1 69.75,-126.1\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"20.25\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M60.61,-100.71C53.99,-93.76 42.78,-82 33.85,-72.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"36.41,-70.26 26.98,-65.44 31.35,-75.09 36.41,-70.26\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>1&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M66.25,-100.71C66.25,-94.49 66.25,-84.4 66.25,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"69.75,-75.85 66.25,-65.85 62.75,-75.85 69.75,-75.85\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"113.25\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>1&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M72.01,-100.71C78.85,-93.69 90.47,-81.76 99.64,-72.35\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"101.91,-75.03 106.38,-65.42 96.9,-70.15 101.91,-75.03\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"64.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 4&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>4&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M107.24,-50.46C100.11,-43.44 88,-31.51 78.44,-22.1\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"80.97,-19.67 71.39,-15.15 76.06,-24.66 80.97,-19.67\"/>\n",
       "</g>\n",
       "<!-- 6 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>6</title>\n",
       "<text text-anchor=\"middle\" x=\"113.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\"> * </text>\n",
       "</g>\n",
       "<!-- 4&#45;&gt;6 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>4&#45;&gt;6</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M113.25,-50.46C113.25,-44.24 113.25,-34.15 113.25,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"116.75,-25.6 113.25,-15.6 109.75,-25.6 116.75,-25.6\"/>\n",
       "</g>\n",
       "<!-- 7 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>7</title>\n",
       "<text text-anchor=\"middle\" x=\"159.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 4&#45;&gt;7 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>4&#45;&gt;7</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M118.89,-50.46C125.51,-43.51 136.72,-31.75 145.65,-22.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"148.15,-24.84 152.52,-15.19 143.09,-20.01 148.15,-24.84\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089bd250>"
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "if f.any_possible_expansions(derivation_tree):\n",
    "    derivation_tree = f.expand_tree_once(derivation_tree)\n",
    "display_tree(derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:35.296941Z",
     "iopub.status.busy": "2025-01-16T09:40:35.296796Z",
     "iopub.status.idle": "2025-01-16T09:40:35.298826Z",
     "shell.execute_reply": "2025-01-16T09:40:35.298514Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "# docassert\n",
    "assert f.any_possible_expansions(derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:35.300355Z",
     "iopub.status.busy": "2025-01-16T09:40:35.300247Z",
     "iopub.status.idle": "2025-01-16T09:40:35.724760Z",
     "shell.execute_reply": "2025-01-16T09:40:35.724398Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expanding <factor> at maximum cost\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"188pt\" height=\"223pt\"\n",
       " viewBox=\"0.00 0.00 188.25 223.25\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 219.25)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-219.25 184.25,-219.25 184.25,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M66.25,-201.21C66.25,-194.99 66.25,-184.9 66.25,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"69.75,-176.35 66.25,-166.35 62.75,-176.35 69.75,-176.35\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"20.25\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M60.61,-150.96C53.99,-144.01 42.78,-132.25 33.85,-122.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"36.41,-120.51 26.98,-115.69 31.35,-125.34 36.41,-120.51\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>1&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M66.25,-150.96C66.25,-144.74 66.25,-134.65 66.25,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"69.75,-126.1 66.25,-116.1 62.75,-126.1 69.75,-126.1\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"113.25\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>1&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M72.01,-150.96C78.85,-143.94 90.47,-132.01 99.64,-122.6\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"101.91,-125.28 106.38,-115.67 96.9,-120.4 101.91,-125.28\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"64.25\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 4&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>4&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M107.24,-100.71C100.11,-93.69 88,-81.76 78.44,-72.35\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"80.97,-69.92 71.39,-65.4 76.06,-74.91 80.97,-69.92\"/>\n",
       "</g>\n",
       "<!-- 8 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>8</title>\n",
       "<text text-anchor=\"middle\" x=\"113.25\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\"> * </text>\n",
       "</g>\n",
       "<!-- 4&#45;&gt;8 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>4&#45;&gt;8</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M113.25,-100.71C113.25,-94.49 113.25,-84.4 113.25,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"116.75,-75.85 113.25,-65.85 109.75,-75.85 116.75,-75.85\"/>\n",
       "</g>\n",
       "<!-- 9 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>9</title>\n",
       "<text text-anchor=\"middle\" x=\"159.25\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 4&#45;&gt;9 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>4&#45;&gt;9</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M118.89,-100.71C125.51,-93.76 136.72,-82 145.65,-72.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"148.15,-75.09 152.52,-65.44 143.09,-70.26 148.15,-75.09\"/>\n",
       "</g>\n",
       "<!-- 6 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>6</title>\n",
       "<text text-anchor=\"middle\" x=\"36.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&#45; (45)</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;6 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>5&#45;&gt;6</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M60.82,-50.46C57,-43.87 50.67,-32.97 45.39,-23.87\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"48.52,-22.29 40.47,-15.4 42.46,-25.8 48.52,-22.29\"/>\n",
       "</g>\n",
       "<!-- 7 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>7</title>\n",
       "<text text-anchor=\"middle\" x=\"93.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;7 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>5&#45;&gt;7</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M67.81,-50.46C71.76,-43.87 78.32,-32.97 83.79,-23.87\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"86.73,-25.76 88.89,-15.39 80.73,-22.16 86.73,-25.76\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089bdd30>"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "if f.any_possible_expansions(derivation_tree):\n",
    "    derivation_tree = f.expand_tree_once(derivation_tree)\n",
    "display_tree(derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:35.726414Z",
     "iopub.status.busy": "2025-01-16T09:40:35.726296Z",
     "iopub.status.idle": "2025-01-16T09:40:35.728210Z",
     "shell.execute_reply": "2025-01-16T09:40:35.727944Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "# docassert\n",
    "assert f.any_possible_expansions(derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:35.729912Z",
     "iopub.status.busy": "2025-01-16T09:40:35.729793Z",
     "iopub.status.idle": "2025-01-16T09:40:36.137560Z",
     "shell.execute_reply": "2025-01-16T09:40:36.137097Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expanding <expr> at maximum cost\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"300pt\" height=\"223pt\"\n",
       " viewBox=\"0.00 0.00 300.00 223.25\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 219.25)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-219.25 296,-219.25 296,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"146\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"146\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M146,-201.21C146,-194.99 146,-184.9 146,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"149.5,-176.35 146,-166.35 142.5,-176.35 149.5,-176.35\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"93\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M139.5,-150.96C131.71,-143.87 118.41,-131.76 108.03,-122.31\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"110.42,-119.75 100.67,-115.61 105.71,-124.93 110.42,-119.75\"/>\n",
       "</g>\n",
       "<!-- 6 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>6</title>\n",
       "<text text-anchor=\"middle\" x=\"146\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;6 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>1&#45;&gt;6</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M146,-150.96C146,-144.74 146,-134.65 146,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"149.5,-126.1 146,-116.1 142.5,-126.1 149.5,-126.1\"/>\n",
       "</g>\n",
       "<!-- 7 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>7</title>\n",
       "<text text-anchor=\"middle\" x=\"197\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;7 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>1&#45;&gt;7</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M152.25,-150.96C159.75,-143.87 172.54,-131.76 182.54,-122.31\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"184.74,-125.04 189.6,-115.63 179.93,-119.96 184.74,-125.04\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"21\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>2&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M83.88,-100.51C72.68,-93 53.4,-80.09 39.13,-70.53\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"41.33,-67.79 31.08,-65.13 37.44,-73.6 41.33,-67.79\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"68\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>2&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M89.93,-100.71C86.56,-94.2 80.99,-83.46 76.31,-74.42\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"79.5,-72.95 71.79,-65.68 73.28,-76.17 79.5,-72.95\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"114\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>2&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M95.58,-100.71C98.38,-94.27 102.98,-83.69 106.88,-74.72\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"110.02,-76.29 110.8,-65.73 103.6,-73.5 110.02,-76.29\"/>\n",
       "</g>\n",
       "<!-- 8 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>8</title>\n",
       "<text text-anchor=\"middle\" x=\"176\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 7&#45;&gt;8 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>7&#45;&gt;8</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M194.42,-100.71C191.62,-94.27 187.02,-83.69 183.12,-74.72\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"186.4,-73.5 179.2,-65.73 179.98,-76.29 186.4,-73.5\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>11</title>\n",
       "<text text-anchor=\"middle\" x=\"225\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\"> * </text>\n",
       "</g>\n",
       "<!-- 7&#45;&gt;11 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>7&#45;&gt;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M200.43,-100.71C204.25,-94.12 210.58,-83.22 215.86,-74.12\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"218.79,-76.05 220.78,-65.65 212.73,-72.54 218.79,-76.05\"/>\n",
       "</g>\n",
       "<!-- 12 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>12</title>\n",
       "<text text-anchor=\"middle\" x=\"271\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 7&#45;&gt;12 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>7&#45;&gt;12</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M206.37,-100.51C217.89,-93 237.7,-80.09 252.36,-70.53\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"254.2,-73.51 260.67,-65.11 250.38,-67.64 254.2,-73.51\"/>\n",
       "</g>\n",
       "<!-- 9 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>9</title>\n",
       "<text text-anchor=\"middle\" x=\"147\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&#45; (45)</text>\n",
       "</g>\n",
       "<!-- 8&#45;&gt;9 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>8&#45;&gt;9</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M172.44,-50.46C168.49,-43.87 161.93,-32.97 156.46,-23.87\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"159.52,-22.16 151.36,-15.39 153.52,-25.76 159.52,-22.16\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>10</title>\n",
       "<text text-anchor=\"middle\" x=\"204\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 8&#45;&gt;10 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>8&#45;&gt;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M179.43,-50.46C183.25,-43.87 189.58,-32.97 194.86,-23.87\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"197.79,-25.8 199.78,-15.4 191.73,-22.29 197.79,-25.8\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089bdb50>"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "if f.any_possible_expansions(derivation_tree):\n",
    "    derivation_tree = f.expand_tree_once(derivation_tree)\n",
    "display_tree(derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We see that with each step, the number of nonterminals increases.  Obviously, we have to put a limit on this number."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Three Expansion Phases\n",
    "\n",
    "We can now put all three phases together in a single function `expand_tree()` which will work as follows:\n",
    "\n",
    "1. **Max cost expansion.** Expand the tree using expansions with maximum cost until we have at least `min_nonterminals` nonterminals.  This phase can be easily skipped by setting `min_nonterminals` to zero.\n",
    "2. **Random expansion.**  Keep on expanding the tree randomly until we reach `max_nonterminals` nonterminals.\n",
    "3. **Min cost expansion.** Close the expansion with minimum cost.\n",
    "\n",
    "We implement these three phases by having `expand_node` reference the expansion method to apply.  This is controlled by setting `expand_node` (the method reference) to first `expand_node_max_cost` (i.e., calling `expand_node()` invokes `expand_node_max_cost()`), then `expand_node_randomly`, and finally `expand_node_min_cost`.  In the first two phases, we also set a maximum limit of `min_nonterminals` and `max_nonterminals`, respectively."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Excursion: Implementation of three-phase `expand_tree()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:36.139722Z",
     "iopub.status.busy": "2025-01-16T09:40:36.139569Z",
     "iopub.status.idle": "2025-01-16T09:40:36.143384Z",
     "shell.execute_reply": "2025-01-16T09:40:36.143054Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def log_tree(self, tree: DerivationTree) -> None:\n",
    "        \"\"\"Output a tree if self.log is set; if self.display is also set, show the tree structure\"\"\"\n",
    "        if self.log:\n",
    "            print(\"Tree:\", all_terminals(tree))\n",
    "            if self.disp:\n",
    "                display(display_tree(tree))\n",
    "            # print(self.possible_expansions(tree), \"possible expansion(s) left\")\n",
    "\n",
    "    def expand_tree_with_strategy(self, tree: DerivationTree,\n",
    "                                  expand_node_method: Callable,\n",
    "                                  limit: Optional[int] = None):\n",
    "        \"\"\"Expand tree using `expand_node_method` as node expansion function\n",
    "        until the number of possible expansions reaches `limit`.\"\"\"\n",
    "        self.expand_node = expand_node_method  # type: ignore\n",
    "        while ((limit is None\n",
    "                or self.possible_expansions(tree) < limit)\n",
    "               and self.any_possible_expansions(tree)):\n",
    "            tree = self.expand_tree_once(tree)\n",
    "            self.log_tree(tree)\n",
    "        return tree\n",
    "\n",
    "    def expand_tree(self, tree: DerivationTree) -> DerivationTree:\n",
    "        \"\"\"Expand `tree` in a three-phase strategy until all expansions are complete.\"\"\"\n",
    "        self.log_tree(tree)\n",
    "        tree = self.expand_tree_with_strategy(\n",
    "            tree, self.expand_node_max_cost, self.min_nonterminals)\n",
    "        tree = self.expand_tree_with_strategy(\n",
    "            tree, self.expand_node_randomly, self.max_nonterminals)\n",
    "        tree = self.expand_tree_with_strategy(\n",
    "            tree, self.expand_node_min_cost)\n",
    "\n",
    "        assert self.possible_expansions(tree) == 0\n",
    "\n",
    "        return tree"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### End of Excursion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Let us try this out on our example. We start with a half-expanded derivation tree:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:36.145326Z",
     "iopub.status.busy": "2025-01-16T09:40:36.145197Z",
     "iopub.status.idle": "2025-01-16T09:40:36.147169Z",
     "shell.execute_reply": "2025-01-16T09:40:36.146844Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "initial_derivation_tree: DerivationTree = (\"<start>\",\n",
    "                   [(\"<expr>\",\n",
    "                     [(\"<expr>\", None),\n",
    "                      (\" + \", []),\n",
    "                         (\"<term>\", None)]\n",
    "                     )])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:36.149030Z",
     "iopub.status.busy": "2025-01-16T09:40:36.148915Z",
     "iopub.status.idle": "2025-01-16T09:40:36.589233Z",
     "shell.execute_reply": "2025-01-16T09:40:36.588576Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"142pt\" height=\"123pt\"\n",
       " viewBox=\"0.00 0.00 142.25 122.75\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 118.75)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-118.75 138.25,-118.75 138.25,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M66.25,-100.71C66.25,-94.49 66.25,-84.4 66.25,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"69.75,-75.85 66.25,-65.85 62.75,-75.85 69.75,-75.85\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"20.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M60.61,-50.46C53.99,-43.51 42.78,-31.75 33.85,-22.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"36.41,-20.01 26.98,-15.19 31.35,-24.84 36.41,-20.01\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"66.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>1&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M66.25,-50.46C66.25,-44.24 66.25,-34.15 66.25,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"69.75,-25.6 66.25,-15.6 62.75,-25.6 69.75,-25.6\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"113.25\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>1&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M72.01,-50.46C78.85,-43.44 90.47,-31.51 99.64,-22.1\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"101.91,-24.78 106.38,-15.17 96.9,-19.9 101.91,-24.78\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089821b0>"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display_tree(initial_derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We now apply our expansion strategy on this tree.  We see that initially, nodes are expanded at maximum cost, then randomly, and then closing the expansion at minimum cost."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:36.591321Z",
     "iopub.status.busy": "2025-01-16T09:40:36.591186Z",
     "iopub.status.idle": "2025-01-16T09:40:36.595096Z",
     "shell.execute_reply": "2025-01-16T09:40:36.594687Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tree: <expr> + <term>\n",
      "Expanding <expr> at maximum cost\n",
      "Tree: <term> + <expr> + <term>\n",
      "Expanding <expr> randomly\n",
      "Tree: <term> + <term> + <term>\n",
      "Expanding <term> randomly\n",
      "Tree: <factor> / <term> + <term> + <term>\n",
      "Expanding <term> randomly\n",
      "Tree: <factor> / <factor> + <term> + <term>\n",
      "Expanding <factor> randomly\n",
      "Tree: <integer> / <factor> + <term> + <term>\n",
      "Expanding <term> randomly\n",
      "Tree: <integer> / <factor> + <factor> * <term> + <term>\n",
      "Expanding <factor> at minimum cost\n",
      "Tree: <integer> / <integer> + <factor> * <term> + <term>\n",
      "Expanding <integer> at minimum cost\n",
      "Tree: <integer> / <digit> + <factor> * <term> + <term>\n",
      "Expanding <factor> at minimum cost\n",
      "Tree: <integer> / <digit> + <integer> * <term> + <term>\n",
      "Expanding <integer> at minimum cost\n",
      "Tree: <digit> / <digit> + <integer> * <term> + <term>\n",
      "Expanding <term> at minimum cost\n",
      "Tree: <digit> / <digit> + <integer> * <term> + <factor>\n",
      "Expanding <digit> at minimum cost\n",
      "Tree: <digit> / 5 + <integer> * <term> + <factor>\n",
      "Expanding <factor> at minimum cost\n",
      "Tree: <digit> / 5 + <integer> * <term> + <integer>\n",
      "Expanding <integer> at minimum cost\n",
      "Tree: <digit> / 5 + <integer> * <term> + <digit>\n",
      "Expanding <integer> at minimum cost\n",
      "Tree: <digit> / 5 + <digit> * <term> + <digit>\n",
      "Expanding <digit> at minimum cost\n",
      "Tree: 7 / 5 + <digit> * <term> + <digit>\n",
      "Expanding <digit> at minimum cost\n",
      "Tree: 7 / 5 + <digit> * <term> + 0\n",
      "Expanding <term> at minimum cost\n",
      "Tree: 7 / 5 + <digit> * <factor> + 0\n",
      "Expanding <factor> at minimum cost\n",
      "Tree: 7 / 5 + <digit> * <integer> + 0\n",
      "Expanding <integer> at minimum cost\n",
      "Tree: 7 / 5 + <digit> * <digit> + 0\n",
      "Expanding <digit> at minimum cost\n",
      "Tree: 7 / 5 + 4 * <digit> + 0\n",
      "Expanding <digit> at minimum cost\n",
      "Tree: 7 / 5 + 4 * 2 + 0\n"
     ]
    }
   ],
   "source": [
    "f = GrammarFuzzer(\n",
    "    EXPR_GRAMMAR,\n",
    "    min_nonterminals=3,\n",
    "    max_nonterminals=5,\n",
    "    log=True)\n",
    "derivation_tree = f.expand_tree(initial_derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "This is the final derivation tree:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:36.597440Z",
     "iopub.status.busy": "2025-01-16T09:40:36.597334Z",
     "iopub.status.idle": "2025-01-16T09:40:37.251147Z",
     "shell.execute_reply": "2025-01-16T09:40:37.250540Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"360pt\" height=\"475pt\"\n",
       " viewBox=\"0.00 0.00 359.62 474.50\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 470.5)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-470.5 355.62,-470.5 355.62,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"236\" y=\"-453.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"236\" y=\"-402.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M236,-452.46C236,-446.24 236,-436.15 236,-427.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"239.5,-427.6 236,-417.6 232.5,-427.6 239.5,-427.6\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"186\" y=\"-352.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M229.87,-402.21C222.52,-395.12 209.98,-383.01 200.18,-373.56\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"202.9,-371.32 193.27,-366.89 198.04,-376.35 202.9,-371.32\"/>\n",
       "</g>\n",
       "<!-- 27 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>27</title>\n",
       "<text text-anchor=\"middle\" x=\"236\" y=\"-352.7\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;27 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>1&#45;&gt;27</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M236,-402.21C236,-395.99 236,-385.9 236,-377.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"239.5,-377.35 236,-367.35 232.5,-377.35 239.5,-377.35\"/>\n",
       "</g>\n",
       "<!-- 28 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>28</title>\n",
       "<text text-anchor=\"middle\" x=\"285\" y=\"-352.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;28 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>1&#45;&gt;28</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M242.01,-402.21C249.14,-395.19 261.25,-383.26 270.81,-373.85\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"273.19,-376.41 277.86,-366.9 268.28,-371.42 273.19,-376.41\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"111\" y=\"-302.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>2&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M176.5,-351.76C164.72,-344.18 144.36,-331.08 129.46,-321.5\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"131.76,-318.82 121.46,-316.35 127.98,-324.71 131.76,-318.82\"/>\n",
       "</g>\n",
       "<!-- 14 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>14</title>\n",
       "<text text-anchor=\"middle\" x=\"179\" y=\"-302.45\" font-family=\"Times,serif\" font-size=\"14.00\"> + </text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;14 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>2&#45;&gt;14</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M185.14,-351.96C184.24,-345.74 182.78,-335.65 181.5,-326.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"184.98,-326.48 180.08,-317.09 178.05,-327.49 184.98,-326.48\"/>\n",
       "</g>\n",
       "<!-- 15 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>15</title>\n",
       "<text text-anchor=\"middle\" x=\"225\" y=\"-302.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;15 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>2&#45;&gt;15</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M190.78,-351.96C196.28,-345.16 205.51,-333.74 212.99,-324.48\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"215.66,-326.74 219.23,-316.77 210.22,-322.34 215.66,-326.74\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 3&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>3&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M100.36,-301.51C87.11,-293.9 64.17,-280.73 47.47,-271.13\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"49.28,-268.14 38.86,-266.19 45.79,-274.21 49.28,-268.14\"/>\n",
       "</g>\n",
       "<!-- 8 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>8</title>\n",
       "<text text-anchor=\"middle\" x=\"75\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\"> / </text>\n",
       "</g>\n",
       "<!-- 3&#45;&gt;8 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>3&#45;&gt;8</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M106.59,-301.71C101.57,-294.98 93.17,-283.73 86.3,-274.52\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"89.14,-272.47 80.36,-266.55 83.53,-276.66 89.14,-272.47\"/>\n",
       "</g>\n",
       "<!-- 9 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>9</title>\n",
       "<text text-anchor=\"middle\" x=\"120\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 3&#45;&gt;9 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>3&#45;&gt;9</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M112.1,-301.71C113.26,-295.49 115.15,-285.4 116.78,-276.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"120.22,-277.3 118.61,-266.83 113.34,-276.02 120.22,-277.3\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 4&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>4&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M27,-251.46C27,-245.24 27,-235.15 27,-226.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"30.5,-226.6 27,-216.6 23.5,-226.6 30.5,-226.6\"/>\n",
       "</g>\n",
       "<!-- 6 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>6</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;6 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>5&#45;&gt;6</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M27,-201.21C27,-194.99 27,-184.9 27,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"30.5,-176.35 27,-166.35 23.5,-176.35 30.5,-176.35\"/>\n",
       "</g>\n",
       "<!-- 7 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>7</title>\n",
       "<text text-anchor=\"middle\" x=\"27\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">7 (55)</text>\n",
       "</g>\n",
       "<!-- 6&#45;&gt;7 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>6&#45;&gt;7</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M27,-150.96C27,-144.74 27,-134.65 27,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"30.5,-126.1 27,-116.1 23.5,-126.1 30.5,-126.1\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>10</title>\n",
       "<text text-anchor=\"middle\" x=\"107\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 9&#45;&gt;10 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>9&#45;&gt;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M118.41,-251.46C116.71,-245.16 113.95,-234.92 111.57,-226.08\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"114.98,-225.3 109,-216.55 108.22,-227.12 114.98,-225.3\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>11</title>\n",
       "<text text-anchor=\"middle\" x=\"106\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 10&#45;&gt;11 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>10&#45;&gt;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M106.88,-201.21C106.75,-194.99 106.54,-184.9 106.36,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"109.86,-176.28 106.15,-166.35 102.86,-176.42 109.86,-176.28\"/>\n",
       "</g>\n",
       "<!-- 12 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>12</title>\n",
       "<text text-anchor=\"middle\" x=\"106\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 11&#45;&gt;12 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>11&#45;&gt;12</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M106,-150.96C106,-144.74 106,-134.65 106,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"109.5,-126.1 106,-116.1 102.5,-126.1 109.5,-126.1\"/>\n",
       "</g>\n",
       "<!-- 13 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>13</title>\n",
       "<text text-anchor=\"middle\" x=\"106\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">5 (53)</text>\n",
       "</g>\n",
       "<!-- 12&#45;&gt;13 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>12&#45;&gt;13</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M106,-100.71C106,-94.49 106,-84.4 106,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"109.5,-75.85 106,-65.85 102.5,-75.85 109.5,-75.85\"/>\n",
       "</g>\n",
       "<!-- 16 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>16</title>\n",
       "<text text-anchor=\"middle\" x=\"225\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 15&#45;&gt;16 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>15&#45;&gt;16</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M225,-301.71C225,-295.49 225,-285.4 225,-276.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"228.5,-276.85 225,-266.85 221.5,-276.85 228.5,-276.85\"/>\n",
       "</g>\n",
       "<!-- 17 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>17</title>\n",
       "<text text-anchor=\"middle\" x=\"176\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 16&#45;&gt;17 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>16&#45;&gt;17</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M218.99,-251.46C211.86,-244.44 199.75,-232.51 190.19,-223.1\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"192.72,-220.67 183.14,-216.15 187.81,-225.66 192.72,-220.67\"/>\n",
       "</g>\n",
       "<!-- 21 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>21</title>\n",
       "<text text-anchor=\"middle\" x=\"225\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\"> * </text>\n",
       "</g>\n",
       "<!-- 16&#45;&gt;21 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>16&#45;&gt;21</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M225,-251.46C225,-245.24 225,-235.15 225,-226.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"228.5,-226.6 225,-216.6 221.5,-226.6 228.5,-226.6\"/>\n",
       "</g>\n",
       "<!-- 22 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>22</title>\n",
       "<text text-anchor=\"middle\" x=\"271\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 16&#45;&gt;22 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>16&#45;&gt;22</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M230.64,-251.46C237.26,-244.51 248.47,-232.75 257.4,-223.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"259.9,-225.84 264.27,-216.19 254.84,-221.01 259.9,-225.84\"/>\n",
       "</g>\n",
       "<!-- 18 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>18</title>\n",
       "<text text-anchor=\"middle\" x=\"178\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 17&#45;&gt;18 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>17&#45;&gt;18</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M176.25,-201.21C176.5,-194.99 176.92,-184.9 177.28,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"180.77,-176.49 177.69,-166.35 173.78,-176.2 180.77,-176.49\"/>\n",
       "</g>\n",
       "<!-- 19 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>19</title>\n",
       "<text text-anchor=\"middle\" x=\"178\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 18&#45;&gt;19 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>18&#45;&gt;19</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M178,-150.96C178,-144.74 178,-134.65 178,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"181.5,-126.1 178,-116.1 174.5,-126.1 181.5,-126.1\"/>\n",
       "</g>\n",
       "<!-- 20 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>20</title>\n",
       "<text text-anchor=\"middle\" x=\"178\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">4 (52)</text>\n",
       "</g>\n",
       "<!-- 19&#45;&gt;20 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>19&#45;&gt;20</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M178,-100.71C178,-94.49 178,-84.4 178,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"181.5,-75.85 178,-65.85 174.5,-75.85 181.5,-75.85\"/>\n",
       "</g>\n",
       "<!-- 23 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>23</title>\n",
       "<text text-anchor=\"middle\" x=\"271\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 22&#45;&gt;23 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>22&#45;&gt;23</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M271,-201.21C271,-194.99 271,-184.9 271,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"274.5,-176.35 271,-166.35 267.5,-176.35 274.5,-176.35\"/>\n",
       "</g>\n",
       "<!-- 24 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>24</title>\n",
       "<text text-anchor=\"middle\" x=\"271\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 23&#45;&gt;24 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>23&#45;&gt;24</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M271,-150.96C271,-144.74 271,-134.65 271,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"274.5,-126.1 271,-116.1 267.5,-126.1 274.5,-126.1\"/>\n",
       "</g>\n",
       "<!-- 25 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>25</title>\n",
       "<text text-anchor=\"middle\" x=\"271\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 24&#45;&gt;25 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>24&#45;&gt;25</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M271,-100.71C271,-94.49 271,-84.4 271,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"274.5,-75.85 271,-65.85 267.5,-75.85 274.5,-75.85\"/>\n",
       "</g>\n",
       "<!-- 26 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>26</title>\n",
       "<text text-anchor=\"middle\" x=\"271\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">2 (50)</text>\n",
       "</g>\n",
       "<!-- 25&#45;&gt;26 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>25&#45;&gt;26</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M271,-50.46C271,-44.24 271,-34.15 271,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"274.5,-25.6 271,-15.6 267.5,-25.6 274.5,-25.6\"/>\n",
       "</g>\n",
       "<!-- 29 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>29</title>\n",
       "<text text-anchor=\"middle\" x=\"289\" y=\"-302.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 28&#45;&gt;29 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>28&#45;&gt;29</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M285.49,-351.96C286.01,-345.74 286.84,-335.65 287.57,-326.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"291.04,-327.35 288.38,-317.1 284.07,-326.77 291.04,-327.35\"/>\n",
       "</g>\n",
       "<!-- 30 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>30</title>\n",
       "<text text-anchor=\"middle\" x=\"311\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 29&#45;&gt;30 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>29&#45;&gt;30</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M291.7,-301.71C294.63,-295.27 299.46,-284.69 303.55,-275.72\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"306.69,-277.27 307.65,-266.72 300.32,-274.36 306.69,-277.27\"/>\n",
       "</g>\n",
       "<!-- 31 -->\n",
       "<g id=\"node32\" class=\"node\">\n",
       "<title>31</title>\n",
       "<text text-anchor=\"middle\" x=\"331\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 30&#45;&gt;31 -->\n",
       "<g id=\"edge31\" class=\"edge\">\n",
       "<title>30&#45;&gt;31</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M313.45,-251.46C316.12,-245.02 320.51,-234.44 324.22,-225.47\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"327.35,-227.07 327.95,-216.49 320.89,-224.39 327.35,-227.07\"/>\n",
       "</g>\n",
       "<!-- 32 -->\n",
       "<g id=\"node33\" class=\"node\">\n",
       "<title>32</title>\n",
       "<text text-anchor=\"middle\" x=\"331\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">0 (48)</text>\n",
       "</g>\n",
       "<!-- 31&#45;&gt;32 -->\n",
       "<g id=\"edge32\" class=\"edge\">\n",
       "<title>31&#45;&gt;32</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M331,-201.21C331,-194.99 331,-184.9 331,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"334.5,-176.35 331,-166.35 327.5,-176.35 334.5,-176.35\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089a4110>"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display_tree(derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "And this is the resulting string:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:37.253732Z",
     "iopub.status.busy": "2025-01-16T09:40:37.253575Z",
     "iopub.status.idle": "2025-01-16T09:40:37.256392Z",
     "shell.execute_reply": "2025-01-16T09:40:37.256133Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'7 / 5 + 4 * 2 + 0'"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "all_terminals(derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Putting it all Together"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Based on this, we can now define a function `fuzz()` that – like `simple_grammar_fuzzer()` – simply takes a grammar and produces a string from it.  It thus no longer exposes the complexity of derivation trees."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:37.258441Z",
     "iopub.status.busy": "2025-01-16T09:40:37.258295Z",
     "iopub.status.idle": "2025-01-16T09:40:37.261072Z",
     "shell.execute_reply": "2025-01-16T09:40:37.260747Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "class GrammarFuzzer(GrammarFuzzer):\n",
    "    def fuzz_tree(self) -> DerivationTree:\n",
    "        \"\"\"Produce a derivation tree from the grammar.\"\"\"\n",
    "        tree = self.init_tree()\n",
    "        # print(tree)\n",
    "\n",
    "        # Expand all nonterminals\n",
    "        tree = self.expand_tree(tree)\n",
    "        if self.log:\n",
    "            print(repr(all_terminals(tree)))\n",
    "        if self.disp:\n",
    "            display(display_tree(tree))\n",
    "        return tree\n",
    "\n",
    "    def fuzz(self) -> str:\n",
    "        \"\"\"Produce a string from the grammar.\"\"\"\n",
    "        self.derivation_tree = self.fuzz_tree()\n",
    "        return all_terminals(self.derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We can now apply this on all our defined grammars (and visualize the derivation tree along)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:37.262853Z",
     "iopub.status.busy": "2025-01-16T09:40:37.262731Z",
     "iopub.status.idle": "2025-01-16T09:40:37.266272Z",
     "shell.execute_reply": "2025-01-16T09:40:37.265915Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'18.3 * 21.95 / 0'"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = GrammarFuzzer(EXPR_GRAMMAR)\n",
    "f.fuzz()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "After calling `fuzz()`, the produced derivation tree is accessible in the `derivation_tree` attribute:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:37.268481Z",
     "iopub.status.busy": "2025-01-16T09:40:37.268353Z",
     "iopub.status.idle": "2025-01-16T09:40:37.722507Z",
     "shell.execute_reply": "2025-01-16T09:40:37.722124Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"518pt\" height=\"424pt\"\n",
       " viewBox=\"0.00 0.00 517.62 424.25\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 420.25)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-420.25 513.62,-420.25 513.62,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"274.62\" y=\"-402.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"274.62\" y=\"-352.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;expr&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M274.62,-402.21C274.62,-395.99 274.62,-385.9 274.62,-377.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"278.13,-377.35 274.63,-367.35 271.13,-377.35 278.13,-377.35\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"274.62\" y=\"-302.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M274.62,-351.96C274.62,-345.74 274.62,-335.65 274.62,-326.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"278.13,-327.1 274.63,-317.1 271.13,-327.1 278.13,-327.1\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"160.62\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>2&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M260.19,-301.51C241.43,-293.57 208.38,-279.59 185.53,-269.92\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"186.91,-266.7 176.34,-266.03 184.18,-273.15 186.91,-266.7\"/>\n",
       "</g>\n",
       "<!-- 14 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>14</title>\n",
       "<text text-anchor=\"middle\" x=\"274.62\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\"> * </text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;14 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>2&#45;&gt;14</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M274.62,-301.71C274.62,-295.49 274.62,-285.4 274.62,-276.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"278.13,-276.85 274.63,-266.85 271.13,-276.85 278.13,-276.85\"/>\n",
       "</g>\n",
       "<!-- 15 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>15</title>\n",
       "<text text-anchor=\"middle\" x=\"335.62\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;15 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>2&#45;&gt;15</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M282.35,-301.51C291.57,-294.22 307.23,-281.83 319.2,-272.37\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"321.29,-275.17 326.97,-266.22 316.95,-269.68 321.29,-275.17\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"70.62\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 3&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>3&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M149.23,-251.26C134.89,-243.58 109.98,-230.22 92.04,-220.61\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"93.73,-217.54 83.26,-215.9 90.42,-223.71 93.73,-217.54\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>10</title>\n",
       "<text text-anchor=\"middle\" x=\"130.62\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">. (46)</text>\n",
       "</g>\n",
       "<!-- 3&#45;&gt;10 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>3&#45;&gt;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M156.95,-251.46C152.81,-244.8 145.92,-233.73 140.23,-224.57\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"143.39,-223.02 135.13,-216.38 137.44,-226.72 143.39,-223.02\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>11</title>\n",
       "<text text-anchor=\"middle\" x=\"190.62\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 3&#45;&gt;11 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>3&#45;&gt;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M164.3,-251.46C168.44,-244.8 175.33,-233.73 181.02,-224.57\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"183.81,-226.72 186.12,-216.38 177.86,-223.02 183.81,-226.72\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"20.62\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 4&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>4&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M64.49,-201.21C57.15,-194.12 44.6,-182.01 34.8,-172.56\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"37.52,-170.32 27.9,-165.89 32.66,-175.35 37.52,-170.32\"/>\n",
       "</g>\n",
       "<!-- 7 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>7</title>\n",
       "<text text-anchor=\"middle\" x=\"86.62\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 4&#45;&gt;7 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>4&#45;&gt;7</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M72.59,-201.21C74.7,-194.84 78.15,-184.43 81.1,-175.53\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"84.35,-176.87 84.17,-166.28 77.7,-174.67 84.35,-176.87\"/>\n",
       "</g>\n",
       "<!-- 6 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>6</title>\n",
       "<text text-anchor=\"middle\" x=\"20.62\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">1 (49)</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;6 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>5&#45;&gt;6</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M20.62,-150.96C20.62,-144.74 20.62,-134.65 20.62,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"24.13,-126.1 20.63,-116.1 17.13,-126.1 24.13,-126.1\"/>\n",
       "</g>\n",
       "<!-- 8 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>8</title>\n",
       "<text text-anchor=\"middle\" x=\"86.62\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 7&#45;&gt;8 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>7&#45;&gt;8</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M86.62,-150.96C86.62,-144.74 86.62,-134.65 86.62,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"90.13,-126.1 86.63,-116.1 83.13,-126.1 90.13,-126.1\"/>\n",
       "</g>\n",
       "<!-- 9 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>9</title>\n",
       "<text text-anchor=\"middle\" x=\"86.62\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">8 (56)</text>\n",
       "</g>\n",
       "<!-- 8&#45;&gt;9 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>8&#45;&gt;9</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M86.62,-100.71C86.62,-94.49 86.62,-84.4 86.62,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"90.13,-75.85 86.63,-65.85 83.13,-75.85 90.13,-75.85\"/>\n",
       "</g>\n",
       "<!-- 12 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>12</title>\n",
       "<text text-anchor=\"middle\" x=\"174.62\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 11&#45;&gt;12 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>11&#45;&gt;12</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M188.66,-201.21C186.55,-194.84 183.1,-184.43 180.15,-175.53\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"183.55,-174.67 177.08,-166.28 176.9,-176.87 183.55,-174.67\"/>\n",
       "</g>\n",
       "<!-- 13 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>13</title>\n",
       "<text text-anchor=\"middle\" x=\"157.62\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">3 (51)</text>\n",
       "</g>\n",
       "<!-- 12&#45;&gt;13 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>12&#45;&gt;13</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M172.54,-150.96C170.3,-144.59 166.63,-134.18 163.49,-125.28\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"166.85,-124.29 160.23,-116.02 160.25,-126.61 166.85,-124.29\"/>\n",
       "</g>\n",
       "<!-- 16 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>16</title>\n",
       "<text text-anchor=\"middle\" x=\"311.62\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 15&#45;&gt;16 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>15&#45;&gt;16</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M332.68,-251.46C329.44,-244.95 324.1,-234.21 319.61,-225.17\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"322.85,-223.84 315.27,-216.44 316.59,-226.96 322.85,-223.84\"/>\n",
       "</g>\n",
       "<!-- 30 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>30</title>\n",
       "<text text-anchor=\"middle\" x=\"359.62\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\"> / </text>\n",
       "</g>\n",
       "<!-- 15&#45;&gt;30 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>15&#45;&gt;30</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M338.57,-251.46C341.81,-244.95 347.15,-234.21 351.64,-225.17\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"354.66,-226.96 355.98,-216.44 348.4,-223.84 354.66,-226.96\"/>\n",
       "</g>\n",
       "<!-- 31 -->\n",
       "<g id=\"node32\" class=\"node\">\n",
       "<title>31</title>\n",
       "<text text-anchor=\"middle\" x=\"422.62\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;term&gt;</text>\n",
       "</g>\n",
       "<!-- 15&#45;&gt;31 -->\n",
       "<g id=\"edge31\" class=\"edge\">\n",
       "<title>15&#45;&gt;31</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M346.64,-251.26C360.5,-243.58 384.58,-230.22 401.92,-220.61\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"403.32,-223.83 410.37,-215.92 399.93,-217.71 403.32,-223.83\"/>\n",
       "</g>\n",
       "<!-- 17 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>17</title>\n",
       "<text text-anchor=\"middle\" x=\"251.62\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 16&#45;&gt;17 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>16&#45;&gt;17</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M304.27,-201.21C295.27,-193.97 279.78,-181.52 267.92,-171.98\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"270.21,-169.33 260.22,-165.79 265.82,-174.78 270.21,-169.33\"/>\n",
       "</g>\n",
       "<!-- 23 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>23</title>\n",
       "<text text-anchor=\"middle\" x=\"311.62\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">. (46)</text>\n",
       "</g>\n",
       "<!-- 16&#45;&gt;23 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>16&#45;&gt;23</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M311.62,-201.21C311.62,-194.99 311.62,-184.9 311.62,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"315.13,-176.35 311.63,-166.35 308.13,-176.35 315.13,-176.35\"/>\n",
       "</g>\n",
       "<!-- 24 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>24</title>\n",
       "<text text-anchor=\"middle\" x=\"371.62\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 16&#45;&gt;24 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>16&#45;&gt;24</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M318.98,-201.21C327.98,-193.97 343.47,-181.52 355.33,-171.98\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"357.43,-174.78 363.03,-165.79 353.04,-169.33 357.43,-174.78\"/>\n",
       "</g>\n",
       "<!-- 18 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>18</title>\n",
       "<text text-anchor=\"middle\" x=\"212.62\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 17&#45;&gt;18 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>17&#45;&gt;18</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M246.84,-150.96C241.35,-144.16 232.12,-132.74 224.63,-123.48\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"227.41,-121.34 218.4,-115.77 221.96,-125.74 227.41,-121.34\"/>\n",
       "</g>\n",
       "<!-- 20 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>20</title>\n",
       "<text text-anchor=\"middle\" x=\"278.62\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 17&#45;&gt;20 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>17&#45;&gt;20</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M254.94,-150.96C258.62,-144.37 264.72,-133.47 269.81,-124.37\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"272.72,-126.35 274.55,-115.91 266.61,-122.93 272.72,-126.35\"/>\n",
       "</g>\n",
       "<!-- 19 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>19</title>\n",
       "<text text-anchor=\"middle\" x=\"212.62\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">2 (50)</text>\n",
       "</g>\n",
       "<!-- 18&#45;&gt;19 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>18&#45;&gt;19</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M212.62,-100.71C212.62,-94.49 212.62,-84.4 212.62,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"216.13,-75.85 212.63,-65.85 209.13,-75.85 216.13,-75.85\"/>\n",
       "</g>\n",
       "<!-- 21 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>21</title>\n",
       "<text text-anchor=\"middle\" x=\"278.62\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 20&#45;&gt;21 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>20&#45;&gt;21</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M278.62,-100.71C278.62,-94.49 278.62,-84.4 278.62,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"282.13,-75.85 278.63,-65.85 275.13,-75.85 282.13,-75.85\"/>\n",
       "</g>\n",
       "<!-- 22 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>22</title>\n",
       "<text text-anchor=\"middle\" x=\"278.62\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">1 (49)</text>\n",
       "</g>\n",
       "<!-- 21&#45;&gt;22 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>21&#45;&gt;22</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M278.62,-50.46C278.62,-44.24 278.62,-34.15 278.62,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"282.13,-25.6 278.63,-15.6 275.13,-25.6 282.13,-25.6\"/>\n",
       "</g>\n",
       "<!-- 25 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>25</title>\n",
       "<text text-anchor=\"middle\" x=\"344.62\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 24&#45;&gt;25 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>24&#45;&gt;25</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M368.31,-150.96C364.63,-144.37 358.53,-133.47 353.44,-124.37\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"356.64,-122.93 348.7,-115.91 350.53,-126.35 356.64,-122.93\"/>\n",
       "</g>\n",
       "<!-- 27 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>27</title>\n",
       "<text text-anchor=\"middle\" x=\"410.62\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 24&#45;&gt;27 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>24&#45;&gt;27</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M376.41,-150.96C381.9,-144.16 391.13,-132.74 398.62,-123.48\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"401.29,-125.74 404.85,-115.77 395.84,-121.34 401.29,-125.74\"/>\n",
       "</g>\n",
       "<!-- 26 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>26</title>\n",
       "<text text-anchor=\"middle\" x=\"344.62\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">9 (57)</text>\n",
       "</g>\n",
       "<!-- 25&#45;&gt;26 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>25&#45;&gt;26</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M344.62,-100.71C344.62,-94.49 344.62,-84.4 344.62,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"348.13,-75.85 344.63,-65.85 341.13,-75.85 348.13,-75.85\"/>\n",
       "</g>\n",
       "<!-- 28 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>28</title>\n",
       "<text text-anchor=\"middle\" x=\"410.62\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 27&#45;&gt;28 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>27&#45;&gt;28</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M410.62,-100.71C410.62,-94.49 410.62,-84.4 410.62,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"414.13,-75.85 410.63,-65.85 407.13,-75.85 414.13,-75.85\"/>\n",
       "</g>\n",
       "<!-- 29 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>29</title>\n",
       "<text text-anchor=\"middle\" x=\"410.62\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">5 (53)</text>\n",
       "</g>\n",
       "<!-- 28&#45;&gt;29 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>28&#45;&gt;29</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M410.62,-50.46C410.62,-44.24 410.62,-34.15 410.62,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"414.13,-25.6 410.63,-15.6 407.13,-25.6 414.13,-25.6\"/>\n",
       "</g>\n",
       "<!-- 32 -->\n",
       "<g id=\"node33\" class=\"node\">\n",
       "<title>32</title>\n",
       "<text text-anchor=\"middle\" x=\"453.62\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;factor&gt;</text>\n",
       "</g>\n",
       "<!-- 31&#45;&gt;32 -->\n",
       "<g id=\"edge32\" class=\"edge\">\n",
       "<title>31&#45;&gt;32</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M426.43,-201.21C430.7,-194.55 437.82,-183.48 443.7,-174.32\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"446.51,-176.42 448.97,-166.11 440.62,-172.63 446.51,-176.42\"/>\n",
       "</g>\n",
       "<!-- 33 -->\n",
       "<g id=\"node34\" class=\"node\">\n",
       "<title>33</title>\n",
       "<text text-anchor=\"middle\" x=\"482.62\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;integer&gt;</text>\n",
       "</g>\n",
       "<!-- 32&#45;&gt;33 -->\n",
       "<g id=\"edge33\" class=\"edge\">\n",
       "<title>32&#45;&gt;33</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M457.18,-150.96C461.14,-144.37 467.69,-133.47 473.16,-124.37\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"476.11,-126.26 478.26,-115.89 470.11,-122.66 476.11,-126.26\"/>\n",
       "</g>\n",
       "<!-- 34 -->\n",
       "<g id=\"node35\" class=\"node\">\n",
       "<title>34</title>\n",
       "<text text-anchor=\"middle\" x=\"482.62\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 33&#45;&gt;34 -->\n",
       "<g id=\"edge34\" class=\"edge\">\n",
       "<title>33&#45;&gt;34</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M482.62,-100.71C482.62,-94.49 482.62,-84.4 482.62,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"486.13,-75.85 482.63,-65.85 479.13,-75.85 486.13,-75.85\"/>\n",
       "</g>\n",
       "<!-- 35 -->\n",
       "<g id=\"node36\" class=\"node\">\n",
       "<title>35</title>\n",
       "<text text-anchor=\"middle\" x=\"482.62\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">0 (48)</text>\n",
       "</g>\n",
       "<!-- 34&#45;&gt;35 -->\n",
       "<g id=\"edge35\" class=\"edge\">\n",
       "<title>34&#45;&gt;35</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M482.62,-50.46C482.62,-44.24 482.62,-34.15 482.62,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"486.13,-25.6 482.63,-15.6 479.13,-25.6 486.13,-25.6\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089a4c20>"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display_tree(f.derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Let us try out the grammar fuzzer (and its trees) on other grammar formats."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:37.724499Z",
     "iopub.status.busy": "2025-01-16T09:40:37.724370Z",
     "iopub.status.idle": "2025-01-16T09:40:37.727149Z",
     "shell.execute_reply": "2025-01-16T09:40:37.726932Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'ftp://user:password@www.google.com:53/def?abc=def'"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = GrammarFuzzer(URL_GRAMMAR)\n",
    "f.fuzz()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:37.728815Z",
     "iopub.status.busy": "2025-01-16T09:40:37.728707Z",
     "iopub.status.idle": "2025-01-16T09:40:38.180410Z",
     "shell.execute_reply": "2025-01-16T09:40:38.180048Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"573pt\" height=\"324pt\"\n",
       " viewBox=\"0.00 0.00 573.00 323.75\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 319.75)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-319.75 569,-319.75 569,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"214.5\" y=\"-302.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"214.5\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;url&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M214.5,-301.71C214.5,-295.49 214.5,-285.4 214.5,-276.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"218,-276.85 214.5,-266.85 211,-276.85 218,-276.85\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"69.5\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;scheme&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M198.98,-252.21C175.01,-244.24 128.59,-228.79 98.34,-218.72\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"99.65,-215.47 89.05,-215.63 97.44,-222.11 99.65,-215.47\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"158.5\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">://</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>1&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M207.63,-251.46C198.96,-243.98 183.82,-230.94 172.63,-221.3\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"175.08,-218.79 165.22,-214.92 170.51,-224.09 175.08,-218.79\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"214.5\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;authority&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>1&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M214.5,-251.46C214.5,-245.24 214.5,-235.15 214.5,-226.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"218,-226.6 214.5,-216.6 211,-226.6 218,-226.6\"/>\n",
       "</g>\n",
       "<!-- 18 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>18</title>\n",
       "<text text-anchor=\"middle\" x=\"345.5\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;path&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;18 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>1&#45;&gt;18</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M230.04,-251.65C251.87,-243.61 291.91,-228.87 318.62,-219.03\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"319.56,-222.41 327.74,-215.67 317.14,-215.84 319.56,-222.41\"/>\n",
       "</g>\n",
       "<!-- 22 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>22</title>\n",
       "<text text-anchor=\"middle\" x=\"440.5\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;query&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;22 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>1&#45;&gt;22</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M230.03,-254.06C266.04,-246.37 355.93,-227.18 405.8,-216.53\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"406.44,-219.98 415.49,-214.46 404.98,-213.13 406.44,-219.98\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"7.5\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">ftp</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>2&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M61.65,-201.01C52.08,-193.57 35.69,-180.81 23.44,-171.28\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"25.95,-168.8 15.91,-165.42 21.65,-174.32 25.95,-168.8\"/>\n",
       "</g>\n",
       "<!-- 6 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>6</title>\n",
       "<text text-anchor=\"middle\" x=\"63.5\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;userinfo&gt;</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;6 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>5&#45;&gt;6</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M195.38,-201.01C169.62,-192.78 123.53,-178.06 93.23,-168.38\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"94.5,-165.11 83.91,-165.4 92.37,-171.77 94.5,-165.11\"/>\n",
       "</g>\n",
       "<!-- 8 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>8</title>\n",
       "<text text-anchor=\"middle\" x=\"131.5\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">@ (64)</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;8 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>5&#45;&gt;8</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M203.99,-201.01C190.89,-193.4 168.23,-180.23 151.72,-170.63\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"153.64,-167.7 143.24,-165.7 150.12,-173.75 153.64,-167.7\"/>\n",
       "</g>\n",
       "<!-- 9 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>9</title>\n",
       "<text text-anchor=\"middle\" x=\"188.5\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;host&gt;</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;9 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>5&#45;&gt;9</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M211.31,-201.21C207.8,-194.7 202.01,-183.96 197.15,-174.92\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"200.26,-173.32 192.43,-166.17 194.09,-176.64 200.26,-173.32\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>11</title>\n",
       "<text text-anchor=\"middle\" x=\"241.5\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">: (58)</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;11 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>5&#45;&gt;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M217.81,-201.21C221.49,-194.62 227.6,-183.72 232.69,-174.62\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"235.59,-176.6 237.42,-166.16 229.49,-173.18 235.59,-176.6\"/>\n",
       "</g>\n",
       "<!-- 12 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>12</title>\n",
       "<text text-anchor=\"middle\" x=\"293.5\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;port&gt;</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;12 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>5&#45;&gt;12</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M224.51,-201.01C236.85,-193.47 258.13,-180.48 273.8,-170.91\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"275.57,-173.92 282.28,-165.73 271.93,-167.95 275.57,-173.92\"/>\n",
       "</g>\n",
       "<!-- 7 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>7</title>\n",
       "<text text-anchor=\"middle\" x=\"63.5\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">user:password</text>\n",
       "</g>\n",
       "<!-- 6&#45;&gt;7 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>6&#45;&gt;7</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M63.5,-150.96C63.5,-144.74 63.5,-134.65 63.5,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"67,-126.1 63.5,-116.1 60,-126.1 67,-126.1\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>10</title>\n",
       "<text text-anchor=\"middle\" x=\"188.5\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">www.google.com</text>\n",
       "</g>\n",
       "<!-- 9&#45;&gt;10 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>9&#45;&gt;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M188.5,-150.96C188.5,-144.74 188.5,-134.65 188.5,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"192,-126.1 188.5,-116.1 185,-126.1 192,-126.1\"/>\n",
       "</g>\n",
       "<!-- 13 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>13</title>\n",
       "<text text-anchor=\"middle\" x=\"293.5\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;nat&gt;</text>\n",
       "</g>\n",
       "<!-- 12&#45;&gt;13 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>12&#45;&gt;13</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M293.5,-150.96C293.5,-144.74 293.5,-134.65 293.5,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"297,-126.1 293.5,-116.1 290,-126.1 297,-126.1\"/>\n",
       "</g>\n",
       "<!-- 14 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>14</title>\n",
       "<text text-anchor=\"middle\" x=\"263.5\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 13&#45;&gt;14 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>13&#45;&gt;14</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M289.82,-100.71C285.68,-94.05 278.8,-82.98 273.1,-73.82\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"276.26,-72.27 268.01,-65.63 270.32,-75.97 276.26,-72.27\"/>\n",
       "</g>\n",
       "<!-- 16 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>16</title>\n",
       "<text text-anchor=\"middle\" x=\"322.5\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 13&#45;&gt;16 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>13&#45;&gt;16</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M297.06,-100.71C301.01,-94.12 307.57,-83.22 313.04,-74.12\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"315.98,-76.01 318.14,-65.64 309.98,-72.41 315.98,-76.01\"/>\n",
       "</g>\n",
       "<!-- 15 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>15</title>\n",
       "<text text-anchor=\"middle\" x=\"263.5\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">5 (53)</text>\n",
       "</g>\n",
       "<!-- 14&#45;&gt;15 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>14&#45;&gt;15</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M263.5,-50.46C263.5,-44.24 263.5,-34.15 263.5,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"267,-25.6 263.5,-15.6 260,-25.6 267,-25.6\"/>\n",
       "</g>\n",
       "<!-- 17 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>17</title>\n",
       "<text text-anchor=\"middle\" x=\"322.5\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">3 (51)</text>\n",
       "</g>\n",
       "<!-- 16&#45;&gt;17 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>16&#45;&gt;17</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M322.5,-50.46C322.5,-44.24 322.5,-34.15 322.5,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"326,-25.6 322.5,-15.6 319,-25.6 326,-25.6\"/>\n",
       "</g>\n",
       "<!-- 19 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>19</title>\n",
       "<text text-anchor=\"middle\" x=\"345.5\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">/ (47)</text>\n",
       "</g>\n",
       "<!-- 18&#45;&gt;19 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>18&#45;&gt;19</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M345.5,-201.21C345.5,-194.99 345.5,-184.9 345.5,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"349,-176.35 345.5,-166.35 342,-176.35 349,-176.35\"/>\n",
       "</g>\n",
       "<!-- 20 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>20</title>\n",
       "<text text-anchor=\"middle\" x=\"392.5\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;id&gt;</text>\n",
       "</g>\n",
       "<!-- 18&#45;&gt;20 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>18&#45;&gt;20</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M351.26,-201.21C358.1,-194.19 369.72,-182.26 378.89,-172.85\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"381.16,-175.53 385.63,-165.92 376.15,-170.65 381.16,-175.53\"/>\n",
       "</g>\n",
       "<!-- 21 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>21</title>\n",
       "<text text-anchor=\"middle\" x=\"392.5\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">def</text>\n",
       "</g>\n",
       "<!-- 20&#45;&gt;21 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>20&#45;&gt;21</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M392.5,-150.96C392.5,-144.74 392.5,-134.65 392.5,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"396,-126.1 392.5,-116.1 389,-126.1 396,-126.1\"/>\n",
       "</g>\n",
       "<!-- 23 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>23</title>\n",
       "<text text-anchor=\"middle\" x=\"440.5\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">? (63)</text>\n",
       "</g>\n",
       "<!-- 22&#45;&gt;23 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>22&#45;&gt;23</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M440.5,-201.21C440.5,-194.99 440.5,-184.9 440.5,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"444,-176.35 440.5,-166.35 437,-176.35 444,-176.35\"/>\n",
       "</g>\n",
       "<!-- 24 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>24</title>\n",
       "<text text-anchor=\"middle\" x=\"502.5\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;params&gt;</text>\n",
       "</g>\n",
       "<!-- 22&#45;&gt;24 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>22&#45;&gt;24</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M448.35,-201.01C457.81,-193.65 473.96,-181.09 486.17,-171.59\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"487.97,-174.62 493.71,-165.71 483.67,-169.09 487.97,-174.62\"/>\n",
       "</g>\n",
       "<!-- 25 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>25</title>\n",
       "<text text-anchor=\"middle\" x=\"502.5\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;param&gt;</text>\n",
       "</g>\n",
       "<!-- 24&#45;&gt;25 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>24&#45;&gt;25</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M502.5,-150.96C502.5,-144.74 502.5,-134.65 502.5,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"506,-126.1 502.5,-116.1 499,-126.1 506,-126.1\"/>\n",
       "</g>\n",
       "<!-- 26 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>26</title>\n",
       "<text text-anchor=\"middle\" x=\"453.5\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;id&gt;</text>\n",
       "</g>\n",
       "<!-- 25&#45;&gt;26 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>25&#45;&gt;26</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M496.49,-100.71C489.36,-93.69 477.25,-81.76 467.69,-72.35\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"470.22,-69.92 460.64,-65.4 465.31,-74.91 470.22,-69.92\"/>\n",
       "</g>\n",
       "<!-- 28 -->\n",
       "<g id=\"node29\" class=\"node\">\n",
       "<title>28</title>\n",
       "<text text-anchor=\"middle\" x=\"502.5\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">= (61)</text>\n",
       "</g>\n",
       "<!-- 25&#45;&gt;28 -->\n",
       "<g id=\"edge28\" class=\"edge\">\n",
       "<title>25&#45;&gt;28</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M502.5,-100.71C502.5,-94.49 502.5,-84.4 502.5,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"506,-75.85 502.5,-65.85 499,-75.85 506,-75.85\"/>\n",
       "</g>\n",
       "<!-- 29 -->\n",
       "<g id=\"node30\" class=\"node\">\n",
       "<title>29</title>\n",
       "<text text-anchor=\"middle\" x=\"551.5\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;id&gt;</text>\n",
       "</g>\n",
       "<!-- 25&#45;&gt;29 -->\n",
       "<g id=\"edge29\" class=\"edge\">\n",
       "<title>25&#45;&gt;29</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M508.51,-100.71C515.64,-93.69 527.75,-81.76 537.31,-72.35\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"539.69,-74.91 544.36,-65.4 534.78,-69.92 539.69,-74.91\"/>\n",
       "</g>\n",
       "<!-- 27 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>27</title>\n",
       "<text text-anchor=\"middle\" x=\"453.5\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">abc</text>\n",
       "</g>\n",
       "<!-- 26&#45;&gt;27 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>26&#45;&gt;27</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M453.5,-50.46C453.5,-44.24 453.5,-34.15 453.5,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"457,-25.6 453.5,-15.6 450,-25.6 457,-25.6\"/>\n",
       "</g>\n",
       "<!-- 30 -->\n",
       "<g id=\"node31\" class=\"node\">\n",
       "<title>30</title>\n",
       "<text text-anchor=\"middle\" x=\"551.5\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">def</text>\n",
       "</g>\n",
       "<!-- 29&#45;&gt;30 -->\n",
       "<g id=\"edge30\" class=\"edge\">\n",
       "<title>29&#45;&gt;30</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M551.5,-50.46C551.5,-44.24 551.5,-34.15 551.5,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"555,-25.6 551.5,-15.6 548,-25.6 555,-25.6\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089a4200>"
      ]
     },
     "execution_count": 122,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display_tree(f.derivation_tree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:38.182157Z",
     "iopub.status.busy": "2025-01-16T09:40:38.181982Z",
     "iopub.status.idle": "2025-01-16T09:40:38.185187Z",
     "shell.execute_reply": "2025-01-16T09:40:38.184830Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'e+d+'"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = GrammarFuzzer(CGI_GRAMMAR, min_nonterminals=3, max_nonterminals=5)\n",
    "f.fuzz()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:38.187036Z",
     "iopub.status.busy": "2025-01-16T09:40:38.186942Z",
     "iopub.status.idle": "2025-01-16T09:40:38.636446Z",
     "shell.execute_reply": "2025-01-16T09:40:38.636058Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"243pt\" height=\"374pt\"\n",
       " viewBox=\"0.00 0.00 242.75 374.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 370)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-370 238.75,-370 238.75,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"54.12\" y=\"-352.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"54.12\" y=\"-302.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;string&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M54.12,-351.96C54.12,-345.74 54.12,-335.65 54.12,-326.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"57.63,-327.1 54.13,-317.1 50.63,-327.1 57.63,-327.1\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"22.12\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;letter&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M50.2,-301.71C45.79,-295.05 38.44,-283.98 32.37,-274.82\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"35.36,-273 26.92,-266.6 29.53,-276.87 35.36,-273\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"86.12\" y=\"-252.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;string&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>1&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M58.05,-301.71C62.46,-295.05 69.81,-283.98 75.88,-274.82\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"78.72,-276.87 81.33,-266.6 72.89,-273 78.72,-276.87\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"22.12\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;other&gt;</text>\n",
       "</g>\n",
       "<!-- 2&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>2&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M22.12,-251.46C22.12,-245.24 22.12,-235.15 22.12,-226.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"25.63,-226.6 22.13,-216.6 18.63,-226.6 25.63,-226.6\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"22.12\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">e (101)</text>\n",
       "</g>\n",
       "<!-- 3&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>3&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M22.12,-201.21C22.12,-194.99 22.12,-184.9 22.12,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"25.63,-176.35 22.13,-166.35 18.63,-176.35 25.63,-176.35\"/>\n",
       "</g>\n",
       "<!-- 6 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>6</title>\n",
       "<text text-anchor=\"middle\" x=\"85.12\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;letter&gt;</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;6 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>5&#45;&gt;6</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M86,-251.46C85.87,-245.24 85.66,-235.15 85.48,-226.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"88.99,-226.53 85.28,-216.6 81.99,-226.67 88.99,-226.53\"/>\n",
       "</g>\n",
       "<!-- 9 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>9</title>\n",
       "<text text-anchor=\"middle\" x=\"149.12\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;string&gt;</text>\n",
       "</g>\n",
       "<!-- 5&#45;&gt;9 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>5&#45;&gt;9</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M94.1,-251.26C103.72,-243.9 120.12,-231.34 132.53,-221.84\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"134.4,-224.81 140.21,-215.96 130.14,-219.26 134.4,-224.81\"/>\n",
       "</g>\n",
       "<!-- 7 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>7</title>\n",
       "<text text-anchor=\"middle\" x=\"85.12\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;plus&gt;</text>\n",
       "</g>\n",
       "<!-- 6&#45;&gt;7 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>6&#45;&gt;7</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M85.12,-201.21C85.12,-194.99 85.12,-184.9 85.12,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"88.63,-176.35 85.13,-166.35 81.63,-176.35 88.63,-176.35\"/>\n",
       "</g>\n",
       "<!-- 8 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>8</title>\n",
       "<text text-anchor=\"middle\" x=\"85.12\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">+ (43)</text>\n",
       "</g>\n",
       "<!-- 7&#45;&gt;8 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>7&#45;&gt;8</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M85.12,-150.96C85.12,-144.74 85.12,-134.65 85.12,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"88.63,-126.1 85.13,-116.1 81.63,-126.1 88.63,-126.1\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>10</title>\n",
       "<text text-anchor=\"middle\" x=\"147.12\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;letter&gt;</text>\n",
       "</g>\n",
       "<!-- 9&#45;&gt;10 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>9&#45;&gt;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M148.88,-201.21C148.62,-194.99 148.2,-184.9 147.84,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"151.35,-176.2 147.43,-166.35 144.35,-176.49 151.35,-176.2\"/>\n",
       "</g>\n",
       "<!-- 13 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>13</title>\n",
       "<text text-anchor=\"middle\" x=\"211.12\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;string&gt;</text>\n",
       "</g>\n",
       "<!-- 9&#45;&gt;13 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>9&#45;&gt;13</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M156.98,-201.01C166.44,-193.65 182.58,-181.09 194.79,-171.59\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"196.59,-174.62 202.34,-165.71 192.29,-169.09 196.59,-174.62\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>11</title>\n",
       "<text text-anchor=\"middle\" x=\"147.12\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;other&gt;</text>\n",
       "</g>\n",
       "<!-- 10&#45;&gt;11 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>10&#45;&gt;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M147.12,-150.96C147.12,-144.74 147.12,-134.65 147.12,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"150.63,-126.1 147.13,-116.1 143.63,-126.1 150.63,-126.1\"/>\n",
       "</g>\n",
       "<!-- 12 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>12</title>\n",
       "<text text-anchor=\"middle\" x=\"147.12\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">d (100)</text>\n",
       "</g>\n",
       "<!-- 11&#45;&gt;12 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>11&#45;&gt;12</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M147.12,-100.71C147.12,-94.49 147.12,-84.4 147.12,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"150.63,-75.85 147.13,-65.85 143.63,-75.85 150.63,-75.85\"/>\n",
       "</g>\n",
       "<!-- 14 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>14</title>\n",
       "<text text-anchor=\"middle\" x=\"211.12\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;letter&gt;</text>\n",
       "</g>\n",
       "<!-- 13&#45;&gt;14 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>13&#45;&gt;14</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M211.12,-150.96C211.12,-144.74 211.12,-134.65 211.12,-125.89\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"214.63,-126.1 211.13,-116.1 207.63,-126.1 214.63,-126.1\"/>\n",
       "</g>\n",
       "<!-- 15 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>15</title>\n",
       "<text text-anchor=\"middle\" x=\"211.12\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;plus&gt;</text>\n",
       "</g>\n",
       "<!-- 14&#45;&gt;15 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>14&#45;&gt;15</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M211.12,-100.71C211.12,-94.49 211.12,-84.4 211.12,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"214.63,-75.85 211.13,-65.85 207.63,-75.85 214.63,-75.85\"/>\n",
       "</g>\n",
       "<!-- 16 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>16</title>\n",
       "<text text-anchor=\"middle\" x=\"211.12\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">+ (43)</text>\n",
       "</g>\n",
       "<!-- 15&#45;&gt;16 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>15&#45;&gt;16</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M211.12,-50.46C211.12,-44.24 211.12,-34.15 211.12,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"214.63,-25.6 211.13,-15.6 207.63,-25.6 214.63,-25.6\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x1089a76b0>"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display_tree(f.derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "How do we stack up against `simple_grammar_fuzzer()`?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:38.638421Z",
     "iopub.status.busy": "2025-01-16T09:40:38.638298Z",
     "iopub.status.idle": "2025-01-16T09:40:39.959085Z",
     "shell.execute_reply": "2025-01-16T09:40:39.958771Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 \n"
     ]
    }
   ],
   "source": [
    "trials = 50\n",
    "xs = []\n",
    "ys = []\n",
    "f = GrammarFuzzer(EXPR_GRAMMAR, max_nonterminals=20)\n",
    "for i in range(trials):\n",
    "    with Timer() as t:\n",
    "        s = f.fuzz()\n",
    "    xs.append(len(s))\n",
    "    ys.append(t.elapsed_time())\n",
    "    print(i, end=\" \")\n",
    "print()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:39.960802Z",
     "iopub.status.busy": "2025-01-16T09:40:39.960678Z",
     "iopub.status.idle": "2025-01-16T09:40:39.962541Z",
     "shell.execute_reply": "2025-01-16T09:40:39.962301Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Average time: 0.0263434316823259\n"
     ]
    }
   ],
   "source": [
    "average_time = sum(ys) / trials\n",
    "print(\"Average time:\", average_time)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {
    "button": false,
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:39.964143Z",
     "iopub.status.busy": "2025-01-16T09:40:39.964045Z",
     "iopub.status.idle": "2025-01-16T09:40:40.026229Z",
     "shell.execute_reply": "2025-01-16T09:40:40.025931Z"
    },
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "plt.scatter(xs, ys)\n",
    "plt.title('Time required for generating an output');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Our test generation is much faster, but also our inputs are much smaller.  We see that with derivation trees, we can get much better control over grammar production."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Finally, how does `GrammarFuzzer` work with `expr_grammar`, where `simple_grammar_fuzzer()` failed?  It works without any issue:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:40.027959Z",
     "iopub.status.busy": "2025-01-16T09:40:40.027838Z",
     "iopub.status.idle": "2025-01-16T09:40:40.033241Z",
     "shell.execute_reply": "2025-01-16T09:40:40.033015Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'(9 + 7) * 7 * 2 * 5 + 7 * 2 / 6'"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = GrammarFuzzer(expr_grammar, max_nonterminals=10)\n",
    "f.fuzz()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "With `GrammarFuzzer`, we now have a solid foundation on which to build further fuzzers and illustrate more exciting concepts from the world of generating software tests.  Many of these do not even require writing a grammar – instead, they _infer_ a grammar from the domain at hand, and thus allow using grammar-based fuzzing even without writing a grammar.  Stay tuned!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    },
    "tags": []
   },
   "source": [
    "## Synopsis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Efficient Grammar Fuzzing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "jp-MarkdownHeadingCollapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    },
    "tags": []
   },
   "source": [
    "This chapter introduces `GrammarFuzzer`, an efficient grammar fuzzer that takes a grammar to produce syntactically valid input strings.  Here's a typical usage:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:40.034861Z",
     "iopub.status.busy": "2025-01-16T09:40:40.034745Z",
     "iopub.status.idle": "2025-01-16T09:40:40.036867Z",
     "shell.execute_reply": "2025-01-16T09:40:40.036524Z"
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "from Grammars import US_PHONE_GRAMMAR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:40.038651Z",
     "iopub.status.busy": "2025-01-16T09:40:40.038522Z",
     "iopub.status.idle": "2025-01-16T09:40:40.041474Z",
     "shell.execute_reply": "2025-01-16T09:40:40.041206Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'(771)306-0659'"
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phone_fuzzer = GrammarFuzzer(US_PHONE_GRAMMAR)\n",
    "phone_fuzzer.fuzz()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The `GrammarFuzzer` constructor takes a number of keyword arguments to control its behavior.  `start_symbol`, for instance, allows setting the symbol that expansion starts with (instead of `<start>`):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:40.043299Z",
     "iopub.status.busy": "2025-01-16T09:40:40.043135Z",
     "iopub.status.idle": "2025-01-16T09:40:40.045820Z",
     "shell.execute_reply": "2025-01-16T09:40:40.045518Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'409'"
      ]
     },
     "execution_count": 131,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "area_fuzzer = GrammarFuzzer(US_PHONE_GRAMMAR, start_symbol='<area>')\n",
    "area_fuzzer.fuzz()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Here's how to parameterize the `GrammarFuzzer` constructor:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:40.047526Z",
     "iopub.status.busy": "2025-01-16T09:40:40.047416Z",
     "iopub.status.idle": "2025-01-16T09:40:40.049267Z",
     "shell.execute_reply": "2025-01-16T09:40:40.048888Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# ignore\n",
    "import inspect"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:40.051202Z",
     "iopub.status.busy": "2025-01-16T09:40:40.051069Z",
     "iopub.status.idle": "2025-01-16T09:40:40.053004Z",
     "shell.execute_reply": "2025-01-16T09:40:40.052746Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Produce strings from `grammar`, starting with `start_symbol`.\n",
      "If `min_nonterminals` or `max_nonterminals` is given, use them as limits \n",
      "for the number of nonterminals produced.  \n",
      "If `disp` is set, display the intermediate derivation trees.\n",
      "If `log` is set, show intermediate steps as text on standard output.\n"
     ]
    }
   ],
   "source": [
    "# ignore\n",
    "print(inspect.getdoc(GrammarFuzzer.__init__))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:40.054671Z",
     "iopub.status.busy": "2025-01-16T09:40:40.054552Z",
     "iopub.status.idle": "2025-01-16T09:40:40.056329Z",
     "shell.execute_reply": "2025-01-16T09:40:40.056022Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "# ignore\n",
    "from ClassDiagram import display_class_hierarchy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:40.058005Z",
     "iopub.status.busy": "2025-01-16T09:40:40.057890Z",
     "iopub.status.idle": "2025-01-16T09:40:40.937315Z",
     "shell.execute_reply": "2025-01-16T09:40:40.936885Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"324pt\" height=\"458pt\"\n",
       " viewBox=\"0.00 0.00 323.62 458.25\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 454.25)\">\n",
       "<g id=\"a_graph0\"><a xlink:title=\"GrammarFuzzer class hierarchy\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-454.25 319.62,-454.25 319.62,4 -4,4\"/>\n",
       "</a>\n",
       "</g>\n",
       "<!-- GrammarFuzzer -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>GrammarFuzzer</title>\n",
       "<g id=\"a_node1\"><a xlink:href=\"#\" xlink:title=\"class GrammarFuzzer:&#10;Produce strings from grammars efficiently, using derivation trees.\">\n",
       "<polygon fill=\"none\" stroke=\"black\" points=\"0,-0.5 0,-327.75 178,-327.75 178,-0.5 0,-0.5\"/>\n",
       "<text text-anchor=\"start\" x=\"39.12\" y=\"-311.45\" font-family=\"Patua One, Helvetica, sans-serif\" font-weight=\"bold\" font-size=\"14.00\" fill=\"#b03a2e\">GrammarFuzzer</text>\n",
       "<polyline fill=\"none\" stroke=\"black\" points=\"0,-301.75 178,-301.75\"/>\n",
       "<g id=\"a_node1_0\"><a xlink:href=\"#\" xlink:title=\"GrammarFuzzer\">\n",
       "<g id=\"a_node1_1\"><a xlink:href=\"#\" xlink:title=\"__init__(self, grammar: Dict[str, List[Expansion]], start_symbol: str = &#39;&lt;start&gt;&#39;, min_nonterminals: int = 0, max_nonterminals: int = 10, disp: bool = False, log: Union[bool, int] = False) &#45;&gt; None:&#10;Produce strings from `grammar`, starting with `start_symbol`.&#10;If `min_nonterminals` or `max_nonterminals` is given, use them as limits&#10;for the number of nonterminals produced.&#10;If `disp` is set, display the intermediate derivation trees.&#10;If `log` is set, show intermediate steps as text on standard output.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-289.25\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-style=\"italic\" font-size=\"10.00\">__init__()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_2\"><a xlink:href=\"#\" xlink:title=\"fuzz(self) &#45;&gt; str:&#10;Produce a string from the grammar.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-276.5\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-style=\"italic\" font-size=\"10.00\">fuzz()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_3\"><a xlink:href=\"#\" xlink:title=\"fuzz_tree(self) &#45;&gt; DerivationTree:&#10;Produce a derivation tree from the grammar.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-263.75\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-size=\"10.00\">fuzz_tree()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_4\"><a xlink:href=\"#\" xlink:title=\"any_possible_expansions(self, node: DerivationTree) &#45;&gt; bool\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-250\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">any_possible_expansions()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_5\"><a xlink:href=\"#\" xlink:title=\"check_grammar(self) &#45;&gt; None:&#10;Check the grammar passed\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-237.25\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">check_grammar()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_6\"><a xlink:href=\"#\" xlink:title=\"choose_node_expansion(self, node: DerivationTree, children_alternatives: List[List[DerivationTree]]) &#45;&gt; int:&#10;Return index of expansion in `children_alternatives` to be selected.&#10;&#39;children_alternatives`: a list of possible children for `node`.&#10;Defaults to random. To be overloaded in subclasses.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-225.5\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-style=\"italic\" font-size=\"10.00\">choose_node_expansion()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_7\"><a xlink:href=\"#\" xlink:title=\"choose_tree_expansion(self, tree: DerivationTree, children: List[DerivationTree]) &#45;&gt; int:&#10;Return index of subtree in `children` to be selected for expansion.&#10;Defaults to random.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-211.75\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">choose_tree_expansion()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_8\"><a xlink:href=\"#\" xlink:title=\"expand_node(self, node: DerivationTree) &#45;&gt; DerivationTree\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-199\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expand_node()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_9\"><a xlink:href=\"#\" xlink:title=\"expand_node_by_cost(self, node: DerivationTree, choose: Callable = &lt;built&#45;in function min&gt;) &#45;&gt; DerivationTree\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-186.25\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expand_node_by_cost()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_10\"><a xlink:href=\"#\" xlink:title=\"expand_node_max_cost(self, node: DerivationTree) &#45;&gt; DerivationTree\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-173.5\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expand_node_max_cost()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_11\"><a xlink:href=\"#\" xlink:title=\"expand_node_min_cost(self, node: DerivationTree) &#45;&gt; DerivationTree\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-160.75\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expand_node_min_cost()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_12\"><a xlink:href=\"#\" xlink:title=\"expand_node_randomly(self, node: DerivationTree) &#45;&gt; DerivationTree:&#10;Choose a random expansion for `node` and return it\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-148\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expand_node_randomly()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_13\"><a xlink:href=\"#\" xlink:title=\"expand_tree(self, tree: DerivationTree) &#45;&gt; DerivationTree:&#10;Expand `tree` in a three&#45;phase strategy until all expansions are complete.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-135.25\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expand_tree()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_14\"><a xlink:href=\"#\" xlink:title=\"expand_tree_once(self, tree: DerivationTree) &#45;&gt; DerivationTree:&#10;Choose an unexpanded symbol in tree; expand it.&#10;Can be overloaded in subclasses.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-123.5\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-style=\"italic\" font-size=\"10.00\">expand_tree_once()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_15\"><a xlink:href=\"#\" xlink:title=\"expand_tree_with_strategy(self, tree: DerivationTree, expand_node_method: Callable, limit: Optional[int] = None):&#10;Expand tree using `expand_node_method` as node expansion function&#10;until the number of possible expansions reaches `limit`.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-109.75\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expand_tree_with_strategy()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_16\"><a xlink:href=\"#\" xlink:title=\"expansion_cost(self, expansion: Expansion, seen: Set[str] = set()) &#45;&gt; Union[int, float]\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-97\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expansion_cost()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_17\"><a xlink:href=\"#\" xlink:title=\"expansion_to_children(self, expansion: Expansion) &#45;&gt; List[DerivationTree]\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-84.25\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expansion_to_children()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_18\"><a xlink:href=\"#\" xlink:title=\"init_tree(self) &#45;&gt; DerivationTree\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-71.5\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">init_tree()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_19\"><a xlink:href=\"#\" xlink:title=\"log_tree(self, tree: DerivationTree) &#45;&gt; None:&#10;Output a tree if self.log is set; if self.display is also set, show the tree structure\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-58.75\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">log_tree()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_20\"><a xlink:href=\"#\" xlink:title=\"possible_expansions(self, node: DerivationTree) &#45;&gt; int\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-46\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">possible_expansions()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_21\"><a xlink:href=\"#\" xlink:title=\"process_chosen_children(self, chosen_children: List[DerivationTree], expansion: Expansion) &#45;&gt; List[DerivationTree]:&#10;Process children after selection. &#160;By default, does nothing.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-33.25\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">process_chosen_children()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_22\"><a xlink:href=\"#\" xlink:title=\"supported_opts(self) &#45;&gt; Set[str]:&#10;Set of supported options. To be overloaded in subclasses.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-21.5\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-style=\"italic\" font-size=\"10.00\">supported_opts()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_23\"><a xlink:href=\"#\" xlink:title=\"symbol_cost(self, symbol: str, seen: Set[str] = set()) &#45;&gt; Union[int, float]\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-7.75\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">symbol_cost()</text>\n",
       "</a>\n",
       "</g>\n",
       "</a>\n",
       "</g>\n",
       "</a>\n",
       "</g>\n",
       "</g>\n",
       "<!-- Fuzzer -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>Fuzzer</title>\n",
       "<g id=\"a_node2\"><a xlink:href=\"Fuzzer.ipynb\" xlink:title=\"class Fuzzer:&#10;Base class for fuzzers.\">\n",
       "<polygon fill=\"none\" stroke=\"black\" points=\"51,-364.75 51,-449.75 127,-449.75 127,-364.75 51,-364.75\"/>\n",
       "<text text-anchor=\"start\" x=\"68.38\" y=\"-433.45\" font-family=\"Patua One, Helvetica, sans-serif\" font-weight=\"bold\" font-size=\"14.00\" fill=\"#b03a2e\">Fuzzer</text>\n",
       "<polyline fill=\"none\" stroke=\"black\" points=\"51,-423.75 127,-423.75\"/>\n",
       "<g id=\"a_node2_24\"><a xlink:href=\"#\" xlink:title=\"Fuzzer\">\n",
       "<g id=\"a_node2_25\"><a xlink:href=\"Fuzzer.ipynb\" xlink:title=\"__init__(self) &#45;&gt; None:&#10;Constructor\">\n",
       "<text text-anchor=\"start\" x=\"59\" y=\"-411.25\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-style=\"italic\" font-size=\"10.00\">__init__()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node2_26\"><a xlink:href=\"Fuzzer.ipynb\" xlink:title=\"fuzz(self) &#45;&gt; str:&#10;Return fuzz input\">\n",
       "<text text-anchor=\"start\" x=\"59\" y=\"-398.5\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-style=\"italic\" font-size=\"10.00\">fuzz()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node2_27\"><a xlink:href=\"Fuzzer.ipynb\" xlink:title=\"run(self, runner: Fuzzer.Runner = &lt;Fuzzer.Runner object at 0x10653dfd0&gt;) &#45;&gt; Tuple[subprocess.CompletedProcess, str]:&#10;Run `runner` with fuzz input\">\n",
       "<text text-anchor=\"start\" x=\"59\" y=\"-385.75\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-size=\"10.00\">run()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node2_28\"><a xlink:href=\"Fuzzer.ipynb\" xlink:title=\"runs(self, runner: Fuzzer.Runner = &lt;Fuzzer.PrintRunner object at 0x10653db80&gt;, trials: int = 10) &#45;&gt; List[Tuple[subprocess.CompletedProcess, str]]:&#10;Run `runner` with fuzz input, `trials` times\">\n",
       "<text text-anchor=\"start\" x=\"59\" y=\"-373\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-size=\"10.00\">runs()</text>\n",
       "</a>\n",
       "</g>\n",
       "</a>\n",
       "</g>\n",
       "</a>\n",
       "</g>\n",
       "</g>\n",
       "<!-- GrammarFuzzer&#45;&gt;Fuzzer -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>GrammarFuzzer&#45;&gt;Fuzzer</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M89,-327.96C89,-336.73 89,-345.15 89,-353.01\"/>\n",
       "<polygon fill=\"none\" stroke=\"black\" points=\"85.5,-352.96 89,-362.96 92.5,-352.96 85.5,-352.96\"/>\n",
       "</g>\n",
       "<!-- Legend -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>Legend</title>\n",
       "<text text-anchor=\"start\" x=\"196.38\" y=\"-180.12\" font-family=\"Patua One, Helvetica, sans-serif\" font-weight=\"bold\" font-size=\"10.00\" fill=\"#b03a2e\">Legend</text>\n",
       "<text text-anchor=\"start\" x=\"196.38\" y=\"-170.12\" font-family=\"Patua One, Helvetica, sans-serif\" font-size=\"10.00\">• </text>\n",
       "<text text-anchor=\"start\" x=\"202.38\" y=\"-170.12\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-size=\"8.00\">public_method()</text>\n",
       "<text text-anchor=\"start\" x=\"196.38\" y=\"-160.12\" font-family=\"Patua One, Helvetica, sans-serif\" font-size=\"10.00\">• </text>\n",
       "<text text-anchor=\"start\" x=\"202.38\" y=\"-160.12\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"8.00\">private_method()</text>\n",
       "<text text-anchor=\"start\" x=\"196.38\" y=\"-150.12\" font-family=\"Patua One, Helvetica, sans-serif\" font-size=\"10.00\">• </text>\n",
       "<text text-anchor=\"start\" x=\"202.38\" y=\"-150.12\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-style=\"italic\" font-size=\"8.00\">overloaded_method()</text>\n",
       "<text text-anchor=\"start\" x=\"196.38\" y=\"-141.07\" font-family=\"Helvetica,sans-Serif\" font-size=\"9.00\">Hover over names to see doc</text>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/html": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"324pt\" height=\"458pt\"\n",
       " viewBox=\"0.00 0.00 323.62 458.25\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 454.25)\">\n",
       "<g id=\"a_graph0\"><a xlink:title=\"GrammarFuzzer class hierarchy\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-454.25 319.62,-454.25 319.62,4 -4,4\"/>\n",
       "</a>\n",
       "</g>\n",
       "<!-- GrammarFuzzer -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>GrammarFuzzer</title>\n",
       "<g id=\"a_node1\"><a xlink:href=\"#\" xlink:title=\"class GrammarFuzzer:&#10;Produce strings from grammars efficiently, using derivation trees.\">\n",
       "<polygon fill=\"none\" stroke=\"black\" points=\"0,-0.5 0,-327.75 178,-327.75 178,-0.5 0,-0.5\"/>\n",
       "<text text-anchor=\"start\" x=\"39.12\" y=\"-311.45\" font-family=\"Patua One, Helvetica, sans-serif\" font-weight=\"bold\" font-size=\"14.00\" fill=\"#b03a2e\">GrammarFuzzer</text>\n",
       "<polyline fill=\"none\" stroke=\"black\" points=\"0,-301.75 178,-301.75\"/>\n",
       "<g id=\"a_node1_0\"><a xlink:href=\"#\" xlink:title=\"GrammarFuzzer\">\n",
       "<g id=\"a_node1_1\"><a xlink:href=\"#\" xlink:title=\"__init__(self, grammar: Dict[str, List[Expansion]], start_symbol: str = &#39;&lt;start&gt;&#39;, min_nonterminals: int = 0, max_nonterminals: int = 10, disp: bool = False, log: Union[bool, int] = False) &#45;&gt; None:&#10;Produce strings from `grammar`, starting with `start_symbol`.&#10;If `min_nonterminals` or `max_nonterminals` is given, use them as limits&#10;for the number of nonterminals produced.&#10;If `disp` is set, display the intermediate derivation trees.&#10;If `log` is set, show intermediate steps as text on standard output.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-289.25\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-style=\"italic\" font-size=\"10.00\">__init__()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_2\"><a xlink:href=\"#\" xlink:title=\"fuzz(self) &#45;&gt; str:&#10;Produce a string from the grammar.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-276.5\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-style=\"italic\" font-size=\"10.00\">fuzz()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_3\"><a xlink:href=\"#\" xlink:title=\"fuzz_tree(self) &#45;&gt; DerivationTree:&#10;Produce a derivation tree from the grammar.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-263.75\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-size=\"10.00\">fuzz_tree()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_4\"><a xlink:href=\"#\" xlink:title=\"any_possible_expansions(self, node: DerivationTree) &#45;&gt; bool\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-250\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">any_possible_expansions()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_5\"><a xlink:href=\"#\" xlink:title=\"check_grammar(self) &#45;&gt; None:&#10;Check the grammar passed\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-237.25\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">check_grammar()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_6\"><a xlink:href=\"#\" xlink:title=\"choose_node_expansion(self, node: DerivationTree, children_alternatives: List[List[DerivationTree]]) &#45;&gt; int:&#10;Return index of expansion in `children_alternatives` to be selected.&#10;&#39;children_alternatives`: a list of possible children for `node`.&#10;Defaults to random. To be overloaded in subclasses.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-225.5\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-style=\"italic\" font-size=\"10.00\">choose_node_expansion()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_7\"><a xlink:href=\"#\" xlink:title=\"choose_tree_expansion(self, tree: DerivationTree, children: List[DerivationTree]) &#45;&gt; int:&#10;Return index of subtree in `children` to be selected for expansion.&#10;Defaults to random.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-211.75\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">choose_tree_expansion()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_8\"><a xlink:href=\"#\" xlink:title=\"expand_node(self, node: DerivationTree) &#45;&gt; DerivationTree\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-199\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expand_node()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_9\"><a xlink:href=\"#\" xlink:title=\"expand_node_by_cost(self, node: DerivationTree, choose: Callable = &lt;built&#45;in function min&gt;) &#45;&gt; DerivationTree\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-186.25\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expand_node_by_cost()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_10\"><a xlink:href=\"#\" xlink:title=\"expand_node_max_cost(self, node: DerivationTree) &#45;&gt; DerivationTree\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-173.5\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expand_node_max_cost()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_11\"><a xlink:href=\"#\" xlink:title=\"expand_node_min_cost(self, node: DerivationTree) &#45;&gt; DerivationTree\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-160.75\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expand_node_min_cost()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_12\"><a xlink:href=\"#\" xlink:title=\"expand_node_randomly(self, node: DerivationTree) &#45;&gt; DerivationTree:&#10;Choose a random expansion for `node` and return it\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-148\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expand_node_randomly()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_13\"><a xlink:href=\"#\" xlink:title=\"expand_tree(self, tree: DerivationTree) &#45;&gt; DerivationTree:&#10;Expand `tree` in a three&#45;phase strategy until all expansions are complete.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-135.25\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expand_tree()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_14\"><a xlink:href=\"#\" xlink:title=\"expand_tree_once(self, tree: DerivationTree) &#45;&gt; DerivationTree:&#10;Choose an unexpanded symbol in tree; expand it.&#10;Can be overloaded in subclasses.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-123.5\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-style=\"italic\" font-size=\"10.00\">expand_tree_once()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_15\"><a xlink:href=\"#\" xlink:title=\"expand_tree_with_strategy(self, tree: DerivationTree, expand_node_method: Callable, limit: Optional[int] = None):&#10;Expand tree using `expand_node_method` as node expansion function&#10;until the number of possible expansions reaches `limit`.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-109.75\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expand_tree_with_strategy()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_16\"><a xlink:href=\"#\" xlink:title=\"expansion_cost(self, expansion: Expansion, seen: Set[str] = set()) &#45;&gt; Union[int, float]\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-97\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expansion_cost()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_17\"><a xlink:href=\"#\" xlink:title=\"expansion_to_children(self, expansion: Expansion) &#45;&gt; List[DerivationTree]\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-84.25\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">expansion_to_children()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_18\"><a xlink:href=\"#\" xlink:title=\"init_tree(self) &#45;&gt; DerivationTree\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-71.5\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">init_tree()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_19\"><a xlink:href=\"#\" xlink:title=\"log_tree(self, tree: DerivationTree) &#45;&gt; None:&#10;Output a tree if self.log is set; if self.display is also set, show the tree structure\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-58.75\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">log_tree()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_20\"><a xlink:href=\"#\" xlink:title=\"possible_expansions(self, node: DerivationTree) &#45;&gt; int\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-46\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">possible_expansions()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_21\"><a xlink:href=\"#\" xlink:title=\"process_chosen_children(self, chosen_children: List[DerivationTree], expansion: Expansion) &#45;&gt; List[DerivationTree]:&#10;Process children after selection. &#160;By default, does nothing.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-33.25\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">process_chosen_children()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_22\"><a xlink:href=\"#\" xlink:title=\"supported_opts(self) &#45;&gt; Set[str]:&#10;Set of supported options. To be overloaded in subclasses.\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-21.5\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-style=\"italic\" font-size=\"10.00\">supported_opts()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node1_23\"><a xlink:href=\"#\" xlink:title=\"symbol_cost(self, symbol: str, seen: Set[str] = set()) &#45;&gt; Union[int, float]\">\n",
       "<text text-anchor=\"start\" x=\"8\" y=\"-7.75\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"10.00\">symbol_cost()</text>\n",
       "</a>\n",
       "</g>\n",
       "</a>\n",
       "</g>\n",
       "</a>\n",
       "</g>\n",
       "</g>\n",
       "<!-- Fuzzer -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>Fuzzer</title>\n",
       "<g id=\"a_node2\"><a xlink:href=\"Fuzzer.ipynb\" xlink:title=\"class Fuzzer:&#10;Base class for fuzzers.\">\n",
       "<polygon fill=\"none\" stroke=\"black\" points=\"51,-364.75 51,-449.75 127,-449.75 127,-364.75 51,-364.75\"/>\n",
       "<text text-anchor=\"start\" x=\"68.38\" y=\"-433.45\" font-family=\"Patua One, Helvetica, sans-serif\" font-weight=\"bold\" font-size=\"14.00\" fill=\"#b03a2e\">Fuzzer</text>\n",
       "<polyline fill=\"none\" stroke=\"black\" points=\"51,-423.75 127,-423.75\"/>\n",
       "<g id=\"a_node2_24\"><a xlink:href=\"#\" xlink:title=\"Fuzzer\">\n",
       "<g id=\"a_node2_25\"><a xlink:href=\"Fuzzer.ipynb\" xlink:title=\"__init__(self) &#45;&gt; None:&#10;Constructor\">\n",
       "<text text-anchor=\"start\" x=\"59\" y=\"-411.25\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-style=\"italic\" font-size=\"10.00\">__init__()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node2_26\"><a xlink:href=\"Fuzzer.ipynb\" xlink:title=\"fuzz(self) &#45;&gt; str:&#10;Return fuzz input\">\n",
       "<text text-anchor=\"start\" x=\"59\" y=\"-398.5\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-style=\"italic\" font-size=\"10.00\">fuzz()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node2_27\"><a xlink:href=\"Fuzzer.ipynb\" xlink:title=\"run(self, runner: Fuzzer.Runner = &lt;Fuzzer.Runner object at 0x10653dfd0&gt;) &#45;&gt; Tuple[subprocess.CompletedProcess, str]:&#10;Run `runner` with fuzz input\">\n",
       "<text text-anchor=\"start\" x=\"59\" y=\"-385.75\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-size=\"10.00\">run()</text>\n",
       "</a>\n",
       "</g>\n",
       "<g id=\"a_node2_28\"><a xlink:href=\"Fuzzer.ipynb\" xlink:title=\"runs(self, runner: Fuzzer.Runner = &lt;Fuzzer.PrintRunner object at 0x10653db80&gt;, trials: int = 10) &#45;&gt; List[Tuple[subprocess.CompletedProcess, str]]:&#10;Run `runner` with fuzz input, `trials` times\">\n",
       "<text text-anchor=\"start\" x=\"59\" y=\"-373\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-size=\"10.00\">runs()</text>\n",
       "</a>\n",
       "</g>\n",
       "</a>\n",
       "</g>\n",
       "</a>\n",
       "</g>\n",
       "</g>\n",
       "<!-- GrammarFuzzer&#45;&gt;Fuzzer -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>GrammarFuzzer&#45;&gt;Fuzzer</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M89,-327.96C89,-336.73 89,-345.15 89,-353.01\"/>\n",
       "<polygon fill=\"none\" stroke=\"black\" points=\"85.5,-352.96 89,-362.96 92.5,-352.96 85.5,-352.96\"/>\n",
       "</g>\n",
       "<!-- Legend -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>Legend</title>\n",
       "<text text-anchor=\"start\" x=\"196.38\" y=\"-180.12\" font-family=\"Patua One, Helvetica, sans-serif\" font-weight=\"bold\" font-size=\"10.00\" fill=\"#b03a2e\">Legend</text>\n",
       "<text text-anchor=\"start\" x=\"196.38\" y=\"-170.12\" font-family=\"Patua One, Helvetica, sans-serif\" font-size=\"10.00\">• </text>\n",
       "<text text-anchor=\"start\" x=\"202.38\" y=\"-170.12\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-weight=\"bold\" font-size=\"8.00\">public_method()</text>\n",
       "<text text-anchor=\"start\" x=\"196.38\" y=\"-160.12\" font-family=\"Patua One, Helvetica, sans-serif\" font-size=\"10.00\">• </text>\n",
       "<text text-anchor=\"start\" x=\"202.38\" y=\"-160.12\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-size=\"8.00\">private_method()</text>\n",
       "<text text-anchor=\"start\" x=\"196.38\" y=\"-150.12\" font-family=\"Patua One, Helvetica, sans-serif\" font-size=\"10.00\">• </text>\n",
       "<text text-anchor=\"start\" x=\"202.38\" y=\"-150.12\" font-family=\"'Fira Mono', 'Source Code Pro', 'Courier', monospace\" font-style=\"italic\" font-size=\"8.00\">overloaded_method()</text>\n",
       "<text text-anchor=\"start\" x=\"196.38\" y=\"-141.07\" font-family=\"Helvetica,sans-Serif\" font-size=\"9.00\">Hover over names to see doc</text>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x108b30fb0>"
      ]
     },
     "execution_count": 135,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ignore\n",
    "display_class_hierarchy([GrammarFuzzer],\n",
    "                        public_methods=[\n",
    "                            Fuzzer.__init__,\n",
    "                            Fuzzer.fuzz,\n",
    "                            Fuzzer.run,\n",
    "                            Fuzzer.runs,\n",
    "                            GrammarFuzzer.__init__,\n",
    "                            GrammarFuzzer.fuzz,\n",
    "                            GrammarFuzzer.fuzz_tree,\n",
    "                        ],\n",
    "                        types={\n",
    "                            'DerivationTree': DerivationTree,\n",
    "                            'Expansion': Expansion,\n",
    "                            'Grammar': Grammar\n",
    "                        },\n",
    "                        project='fuzzingbook')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Derivation Trees"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Internally, `GrammarFuzzer` makes use of [derivation trees](#Derivation-Trees), which it expands step by step.  After producing a string, the tree produced can be accessed in the `derivation_tree` attribute."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:40.939758Z",
     "iopub.status.busy": "2025-01-16T09:40:40.939593Z",
     "iopub.status.idle": "2025-01-16T09:40:41.334656Z",
     "shell.execute_reply": "2025-01-16T09:40:41.334303Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Generated by graphviz version 12.2.1 (20241206.2353)\n",
       " -->\n",
       "<!-- Pages: 1 -->\n",
       "<svg width=\"636pt\" height=\"223pt\"\n",
       " viewBox=\"0.00 0.00 635.75 223.25\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 219.25)\">\n",
       "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-219.25 631.75,-219.25 631.75,4 -4,4\"/>\n",
       "<!-- 0 -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>0</title>\n",
       "<text text-anchor=\"middle\" x=\"265.12\" y=\"-201.95\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;start&gt;</text>\n",
       "</g>\n",
       "<!-- 1 -->\n",
       "<g id=\"node2\" class=\"node\">\n",
       "<title>1</title>\n",
       "<text text-anchor=\"middle\" x=\"265.12\" y=\"-151.7\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;phone&#45;number&gt;</text>\n",
       "</g>\n",
       "<!-- 0&#45;&gt;1 -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>0&#45;&gt;1</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M265.12,-201.21C265.12,-194.99 265.12,-184.9 265.12,-176.14\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"268.63,-176.35 265.13,-166.35 261.63,-176.35 268.63,-176.35\"/>\n",
       "</g>\n",
       "<!-- 2 -->\n",
       "<g id=\"node3\" class=\"node\">\n",
       "<title>2</title>\n",
       "<text text-anchor=\"middle\" x=\"113.12\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">( (40)</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;2 -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>1&#45;&gt;2</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M245.87,-150.76C218.79,-142.17 169.38,-126.48 139.08,-116.86\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"140.15,-113.53 129.56,-113.84 138.03,-120.2 140.15,-113.53\"/>\n",
       "</g>\n",
       "<!-- 3 -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>3</title>\n",
       "<text text-anchor=\"middle\" x=\"166.12\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;area&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;3 -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>1&#45;&gt;3</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M252.59,-150.76C236.59,-142.97 208.64,-129.35 188.85,-119.7\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"190.44,-116.58 179.92,-115.35 187.38,-122.88 190.44,-116.58\"/>\n",
       "</g>\n",
       "<!-- 10 -->\n",
       "<g id=\"node11\" class=\"node\">\n",
       "<title>10</title>\n",
       "<text text-anchor=\"middle\" x=\"232.12\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">) (41)</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;10 -->\n",
       "<g id=\"edge10\" class=\"edge\">\n",
       "<title>1&#45;&gt;10</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M261.08,-150.96C256.53,-144.3 248.95,-133.23 242.69,-124.07\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"245.59,-122.12 237.06,-115.84 239.82,-126.07 245.59,-122.12\"/>\n",
       "</g>\n",
       "<!-- 11 -->\n",
       "<g id=\"node12\" class=\"node\">\n",
       "<title>11</title>\n",
       "<text text-anchor=\"middle\" x=\"299.12\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;exchange&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;11 -->\n",
       "<g id=\"edge11\" class=\"edge\">\n",
       "<title>1&#45;&gt;11</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M269.29,-150.96C274.04,-144.23 281.96,-132.98 288.45,-123.77\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"291.15,-126.02 294.05,-115.83 285.43,-121.99 291.15,-126.02\"/>\n",
       "</g>\n",
       "<!-- 18 -->\n",
       "<g id=\"node19\" class=\"node\">\n",
       "<title>18</title>\n",
       "<text text-anchor=\"middle\" x=\"366.12\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&#45; (45)</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;18 -->\n",
       "<g id=\"edge18\" class=\"edge\">\n",
       "<title>1&#45;&gt;18</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M277.92,-150.76C294.31,-142.93 323.02,-129.22 343.22,-119.57\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"344.56,-122.81 352.07,-115.34 341.54,-116.49 344.56,-122.81\"/>\n",
       "</g>\n",
       "<!-- 19 -->\n",
       "<g id=\"node20\" class=\"node\">\n",
       "<title>19</title>\n",
       "<text text-anchor=\"middle\" x=\"489.12\" y=\"-101.45\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;line&gt;</text>\n",
       "</g>\n",
       "<!-- 1&#45;&gt;19 -->\n",
       "<g id=\"edge19\" class=\"edge\">\n",
       "<title>1&#45;&gt;19</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M293.49,-150.76C336.22,-141.56 416.65,-124.24 459.88,-114.92\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"460.51,-118.37 469.54,-112.84 459.03,-111.53 460.51,-118.37\"/>\n",
       "</g>\n",
       "<!-- 4 -->\n",
       "<g id=\"node5\" class=\"node\">\n",
       "<title>4</title>\n",
       "<text text-anchor=\"middle\" x=\"34.12\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;lead&#45;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 3&#45;&gt;4 -->\n",
       "<g id=\"edge4\" class=\"edge\">\n",
       "<title>3&#45;&gt;4</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M149.41,-100.51C127.19,-92.39 87.66,-77.94 61.17,-68.26\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"62.72,-65.1 52.13,-64.96 60.32,-71.68 62.72,-65.1\"/>\n",
       "</g>\n",
       "<!-- 6 -->\n",
       "<g id=\"node7\" class=\"node\">\n",
       "<title>6</title>\n",
       "<text text-anchor=\"middle\" x=\"107.12\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 3&#45;&gt;6 -->\n",
       "<g id=\"edge6\" class=\"edge\">\n",
       "<title>3&#45;&gt;6</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M158.89,-100.71C150.04,-93.47 134.81,-81.02 123.15,-71.48\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"125.55,-68.92 115.59,-65.3 121.12,-74.34 125.55,-68.92\"/>\n",
       "</g>\n",
       "<!-- 8 -->\n",
       "<g id=\"node9\" class=\"node\">\n",
       "<title>8</title>\n",
       "<text text-anchor=\"middle\" x=\"166.12\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 3&#45;&gt;8 -->\n",
       "<g id=\"edge8\" class=\"edge\">\n",
       "<title>3&#45;&gt;8</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M166.12,-100.71C166.12,-94.49 166.12,-84.4 166.12,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"169.63,-75.85 166.13,-65.85 162.63,-75.85 169.63,-75.85\"/>\n",
       "</g>\n",
       "<!-- 5 -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>5</title>\n",
       "<text text-anchor=\"middle\" x=\"34.12\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">7 (55)</text>\n",
       "</g>\n",
       "<!-- 4&#45;&gt;5 -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>4&#45;&gt;5</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M34.12,-50.46C34.12,-44.24 34.12,-34.15 34.12,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"37.63,-25.6 34.13,-15.6 30.63,-25.6 37.63,-25.6\"/>\n",
       "</g>\n",
       "<!-- 7 -->\n",
       "<g id=\"node8\" class=\"node\">\n",
       "<title>7</title>\n",
       "<text text-anchor=\"middle\" x=\"107.12\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">7 (55)</text>\n",
       "</g>\n",
       "<!-- 6&#45;&gt;7 -->\n",
       "<g id=\"edge7\" class=\"edge\">\n",
       "<title>6&#45;&gt;7</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M107.12,-50.46C107.12,-44.24 107.12,-34.15 107.12,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"110.63,-25.6 107.13,-15.6 103.63,-25.6 110.63,-25.6\"/>\n",
       "</g>\n",
       "<!-- 9 -->\n",
       "<g id=\"node10\" class=\"node\">\n",
       "<title>9</title>\n",
       "<text text-anchor=\"middle\" x=\"166.12\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">1 (49)</text>\n",
       "</g>\n",
       "<!-- 8&#45;&gt;9 -->\n",
       "<g id=\"edge9\" class=\"edge\">\n",
       "<title>8&#45;&gt;9</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M166.12,-50.46C166.12,-44.24 166.12,-34.15 166.12,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"169.63,-25.6 166.13,-15.6 162.63,-25.6 169.63,-25.6\"/>\n",
       "</g>\n",
       "<!-- 12 -->\n",
       "<g id=\"node13\" class=\"node\">\n",
       "<title>12</title>\n",
       "<text text-anchor=\"middle\" x=\"239.12\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;lead&#45;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 11&#45;&gt;12 -->\n",
       "<g id=\"edge12\" class=\"edge\">\n",
       "<title>11&#45;&gt;12</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M291.77,-100.71C282.77,-93.47 267.28,-81.02 255.42,-71.48\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"257.71,-68.83 247.72,-65.29 253.32,-74.28 257.71,-68.83\"/>\n",
       "</g>\n",
       "<!-- 14 -->\n",
       "<g id=\"node15\" class=\"node\">\n",
       "<title>14</title>\n",
       "<text text-anchor=\"middle\" x=\"312.12\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 11&#45;&gt;14 -->\n",
       "<g id=\"edge14\" class=\"edge\">\n",
       "<title>11&#45;&gt;14</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M300.72,-100.71C302.41,-94.41 305.18,-84.17 307.56,-75.33\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"310.9,-76.37 310.12,-65.8 304.14,-74.55 310.9,-76.37\"/>\n",
       "</g>\n",
       "<!-- 16 -->\n",
       "<g id=\"node17\" class=\"node\">\n",
       "<title>16</title>\n",
       "<text text-anchor=\"middle\" x=\"371.12\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 11&#45;&gt;16 -->\n",
       "<g id=\"edge16\" class=\"edge\">\n",
       "<title>11&#45;&gt;16</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M308.24,-100.51C319.45,-93 338.73,-80.09 352.99,-70.53\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"354.69,-73.6 361.05,-65.13 350.79,-67.79 354.69,-73.6\"/>\n",
       "</g>\n",
       "<!-- 13 -->\n",
       "<g id=\"node14\" class=\"node\">\n",
       "<title>13</title>\n",
       "<text text-anchor=\"middle\" x=\"239.12\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">3 (51)</text>\n",
       "</g>\n",
       "<!-- 12&#45;&gt;13 -->\n",
       "<g id=\"edge13\" class=\"edge\">\n",
       "<title>12&#45;&gt;13</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M239.12,-50.46C239.12,-44.24 239.12,-34.15 239.12,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"242.63,-25.6 239.13,-15.6 235.63,-25.6 242.63,-25.6\"/>\n",
       "</g>\n",
       "<!-- 15 -->\n",
       "<g id=\"node16\" class=\"node\">\n",
       "<title>15</title>\n",
       "<text text-anchor=\"middle\" x=\"312.12\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">0 (48)</text>\n",
       "</g>\n",
       "<!-- 14&#45;&gt;15 -->\n",
       "<g id=\"edge15\" class=\"edge\">\n",
       "<title>14&#45;&gt;15</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M312.12,-50.46C312.12,-44.24 312.12,-34.15 312.12,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"315.63,-25.6 312.13,-15.6 308.63,-25.6 315.63,-25.6\"/>\n",
       "</g>\n",
       "<!-- 17 -->\n",
       "<g id=\"node18\" class=\"node\">\n",
       "<title>17</title>\n",
       "<text text-anchor=\"middle\" x=\"371.12\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">6 (54)</text>\n",
       "</g>\n",
       "<!-- 16&#45;&gt;17 -->\n",
       "<g id=\"edge17\" class=\"edge\">\n",
       "<title>16&#45;&gt;17</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M371.12,-50.46C371.12,-44.24 371.12,-34.15 371.12,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"374.63,-25.6 371.13,-15.6 367.63,-25.6 374.63,-25.6\"/>\n",
       "</g>\n",
       "<!-- 20 -->\n",
       "<g id=\"node21\" class=\"node\">\n",
       "<title>20</title>\n",
       "<text text-anchor=\"middle\" x=\"430.12\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 19&#45;&gt;20 -->\n",
       "<g id=\"edge20\" class=\"edge\">\n",
       "<title>19&#45;&gt;20</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M481.89,-100.71C473.04,-93.47 457.81,-81.02 446.15,-71.48\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"448.55,-68.92 438.59,-65.3 444.12,-74.34 448.55,-68.92\"/>\n",
       "</g>\n",
       "<!-- 22 -->\n",
       "<g id=\"node23\" class=\"node\">\n",
       "<title>22</title>\n",
       "<text text-anchor=\"middle\" x=\"489.12\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 19&#45;&gt;22 -->\n",
       "<g id=\"edge22\" class=\"edge\">\n",
       "<title>19&#45;&gt;22</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M489.12,-100.71C489.12,-94.49 489.12,-84.4 489.12,-75.64\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"492.63,-75.85 489.13,-65.85 485.63,-75.85 492.63,-75.85\"/>\n",
       "</g>\n",
       "<!-- 24 -->\n",
       "<g id=\"node25\" class=\"node\">\n",
       "<title>24</title>\n",
       "<text text-anchor=\"middle\" x=\"548.12\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 19&#45;&gt;24 -->\n",
       "<g id=\"edge24\" class=\"edge\">\n",
       "<title>19&#45;&gt;24</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M496.36,-100.71C505.21,-93.47 520.44,-81.02 532.1,-71.48\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"534.13,-74.34 539.66,-65.3 529.7,-68.92 534.13,-74.34\"/>\n",
       "</g>\n",
       "<!-- 26 -->\n",
       "<g id=\"node27\" class=\"node\">\n",
       "<title>26</title>\n",
       "<text text-anchor=\"middle\" x=\"607.12\" y=\"-51.2\" font-family=\"Times,serif\" font-size=\"14.00\">&lt;digit&gt;</text>\n",
       "</g>\n",
       "<!-- 19&#45;&gt;26 -->\n",
       "<g id=\"edge26\" class=\"edge\">\n",
       "<title>19&#45;&gt;26</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M504.07,-100.51C523.58,-92.54 558.01,-78.46 581.67,-68.79\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"582.97,-72.03 590.9,-65.01 580.32,-65.55 582.97,-72.03\"/>\n",
       "</g>\n",
       "<!-- 21 -->\n",
       "<g id=\"node22\" class=\"node\">\n",
       "<title>21</title>\n",
       "<text text-anchor=\"middle\" x=\"430.12\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">0 (48)</text>\n",
       "</g>\n",
       "<!-- 20&#45;&gt;21 -->\n",
       "<g id=\"edge21\" class=\"edge\">\n",
       "<title>20&#45;&gt;21</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M430.12,-50.46C430.12,-44.24 430.12,-34.15 430.12,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"433.63,-25.6 430.13,-15.6 426.63,-25.6 433.63,-25.6\"/>\n",
       "</g>\n",
       "<!-- 23 -->\n",
       "<g id=\"node24\" class=\"node\">\n",
       "<title>23</title>\n",
       "<text text-anchor=\"middle\" x=\"489.12\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">6 (54)</text>\n",
       "</g>\n",
       "<!-- 22&#45;&gt;23 -->\n",
       "<g id=\"edge23\" class=\"edge\">\n",
       "<title>22&#45;&gt;23</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M489.12,-50.46C489.12,-44.24 489.12,-34.15 489.12,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"492.63,-25.6 489.13,-15.6 485.63,-25.6 492.63,-25.6\"/>\n",
       "</g>\n",
       "<!-- 25 -->\n",
       "<g id=\"node26\" class=\"node\">\n",
       "<title>25</title>\n",
       "<text text-anchor=\"middle\" x=\"548.12\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">5 (53)</text>\n",
       "</g>\n",
       "<!-- 24&#45;&gt;25 -->\n",
       "<g id=\"edge25\" class=\"edge\">\n",
       "<title>24&#45;&gt;25</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M548.12,-50.46C548.12,-44.24 548.12,-34.15 548.12,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"551.63,-25.6 548.13,-15.6 544.63,-25.6 551.63,-25.6\"/>\n",
       "</g>\n",
       "<!-- 27 -->\n",
       "<g id=\"node28\" class=\"node\">\n",
       "<title>27</title>\n",
       "<text text-anchor=\"middle\" x=\"607.12\" y=\"-0.95\" font-family=\"Times,serif\" font-size=\"14.00\">9 (57)</text>\n",
       "</g>\n",
       "<!-- 26&#45;&gt;27 -->\n",
       "<g id=\"edge27\" class=\"edge\">\n",
       "<title>26&#45;&gt;27</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M607.12,-50.46C607.12,-44.24 607.12,-34.15 607.12,-25.39\"/>\n",
       "<polygon fill=\"black\" stroke=\"black\" points=\"610.63,-25.6 607.13,-15.6 603.63,-25.6 610.63,-25.6\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<graphviz.graphs.Digraph at 0x108b33290>"
      ]
     },
     "execution_count": 136,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display_tree(phone_fuzzer.derivation_tree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "In the internal representation of a derivation tree, a _node_ is a pair (`symbol`, `children`).  For nonterminals, `symbol` is the symbol that is being expanded, and `children` is a list of further nodes.  For terminals, `symbol` is the terminal string, and `children` is empty."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:41.336398Z",
     "iopub.status.busy": "2025-01-16T09:40:41.336270Z",
     "iopub.status.idle": "2025-01-16T09:40:41.338767Z",
     "shell.execute_reply": "2025-01-16T09:40:41.338510Z"
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "('<start>',\n",
       " [('<phone-number>',\n",
       "   [('(', []),\n",
       "    ('<area>',\n",
       "     [('<lead-digit>', [('7', [])]),\n",
       "      ('<digit>', [('7', [])]),\n",
       "      ('<digit>', [('1', [])])]),\n",
       "    (')', []),\n",
       "    ('<exchange>',\n",
       "     [('<lead-digit>', [('3', [])]),\n",
       "      ('<digit>', [('0', [])]),\n",
       "      ('<digit>', [('6', [])])]),\n",
       "    ('-', []),\n",
       "    ('<line>',\n",
       "     [('<digit>', [('0', [])]),\n",
       "      ('<digit>', [('6', [])]),\n",
       "      ('<digit>', [('5', [])]),\n",
       "      ('<digit>', [('9', [])])])])])"
      ]
     },
     "execution_count": 137,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phone_fuzzer.derivation_tree"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The chapter contains various helpers to work with derivation trees, including visualization tools – notably, `display_tree()`, above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": true,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Lessons Learned\n",
    "\n",
    "* _Derivation trees_ are important for expressing input structure\n",
    "* _Grammar fuzzing based on derivation trees_ \n",
    "    1. is much more efficient than string-based grammar fuzzing,\n",
    "    2. gives much better control over input generation, and\n",
    "    3. effectively avoids running into infinite expansions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "## Next Steps\n",
    "\n",
    "Congratulations!  You have reached one of the central \"hubs\" of the book.  From here, there is a wide range of techniques that build on grammar fuzzing."
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### Extending Grammars\n",
    "\n",
    "First, we have a number of techniques that all _extend_ grammars in some form:\n",
    "\n",
    "* [Parsing and recombining inputs](Parser.ipynb) allows making use of existing inputs, again using derivation trees\n",
    "* [Covering grammar expansions](GrammarCoverageFuzzer.ipynb) allows for _combinatorial_ coverage\n",
    "* [Assigning _probabilities_ to individual expansions](ProbabilisticGrammarFuzzer.ipynb) gives additional control over expansions\n",
    "* [Assigning _constraints_ to individual expansions](GeneratorGrammarFuzzer.ipynb) allows expressing _semantic constraints_ on individual rules."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### Applying Grammars\n",
    "\n",
    "Second, we can _apply_ grammars in a variety of contexts that all involve some form of learning it automatically:\n",
    "\n",
    "* [Fuzzing APIs](APIFuzzer.ipynb), learning a grammar from APIs\n",
    "* [Fuzzing graphical user interfaces](WebFuzzer.ipynb), learning a grammar from user interfaces for subsequent fuzzing\n",
    "* [Mining grammars](GrammarMiner.ipynb), learning a grammar for arbitrary input formats\n",
    "\n",
    "Keep on expanding!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Background\n",
    "\n",
    "Derivation trees (then frequently called _parse trees_) are a standard data structure into which *parsers* decompose inputs.  The *Dragon Book* (also known as *Compilers: Principles, Techniques, and Tools*) \\cite{Aho2006} discusses parsing into derivation trees as part of compiling programs.  We also use derivation trees [when parsing and recombining inputs](Parser.ipynb).\n",
    "\n",
    "The key idea in this chapter, namely expanding until a limit of symbols is reached, and then always choosing the shortest path, stems from Luke \\cite{Luke2000}."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Exercises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise 1: Caching Method Results\n",
    "\n",
    "Tracking `GrammarFuzzer` reveals that some methods are called again and again, always with the same values.  \n",
    "\n",
    "Set up a class `FasterGrammarFuzzer` with a _cache_ that checks whether the method has been called before, and if so, return the previously computed \"memoized\" value.  Do this for `expansion_to_children()`.  Compare the number of invocations before and after the optimization."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "fragment"
    },
    "solution": "hidden",
    "solution2": "hidden",
    "solution2_first": true,
    "solution_first": true
   },
   "source": [
    "**Important**: For `expansion_to_children()`, make sure that each list returned is an individual copy.  If you return the same (cached) list, this will interfere with the in-place modification of `GrammarFuzzer`.  Use the Python `copy.deepcopy()` function for this purpose."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    },
    "solution2": "hidden"
   },
   "source": [
    "**Solution.** Let us demonstrate this for `expansion_to_children()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:41.340718Z",
     "iopub.status.busy": "2025-01-16T09:40:41.340611Z",
     "iopub.status.idle": "2025-01-16T09:40:41.342157Z",
     "shell.execute_reply": "2025-01-16T09:40:41.341933Z"
    },
    "slideshow": {
     "slide_type": "skip"
    },
    "solution2": "hidden"
   },
   "outputs": [],
   "source": [
    "import copy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:41.343681Z",
     "iopub.status.busy": "2025-01-16T09:40:41.343584Z",
     "iopub.status.idle": "2025-01-16T09:40:41.346571Z",
     "shell.execute_reply": "2025-01-16T09:40:41.346341Z"
    },
    "slideshow": {
     "slide_type": "skip"
    },
    "solution2": "hidden"
   },
   "outputs": [],
   "source": [
    "class FasterGrammarFuzzer(GrammarFuzzer):\n",
    "    \"\"\"Variant of `GrammarFuzzer` with memoized values\"\"\"\n",
    "\n",
    "    def __init__(self, *args, **kwargs) -> None:\n",
    "        super().__init__(*args, **kwargs)\n",
    "        self._expansion_cache: Dict[Expansion, List[DerivationTree]] = {}\n",
    "        self._expansion_invocations = 0\n",
    "        self._expansion_invocations_cached = 0\n",
    "\n",
    "    def expansion_to_children(self, expansion: Expansion) \\\n",
    "            -> List[DerivationTree]:\n",
    "        self._expansion_invocations += 1\n",
    "        if expansion in self._expansion_cache:\n",
    "            self._expansion_invocations_cached += 1\n",
    "            cached_result = copy.deepcopy(self._expansion_cache[expansion])\n",
    "            return cached_result\n",
    "\n",
    "        result = super().expansion_to_children(expansion)\n",
    "        self._expansion_cache[expansion] = copy.deepcopy(result)\n",
    "        return result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:41.347999Z",
     "iopub.status.busy": "2025-01-16T09:40:41.347901Z",
     "iopub.status.idle": "2025-01-16T09:40:41.351487Z",
     "shell.execute_reply": "2025-01-16T09:40:41.351222Z"
    },
    "slideshow": {
     "slide_type": "skip"
    },
    "solution2": "hidden"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'+-++9.2 * 8.9 / 7 + 3'"
      ]
     },
     "execution_count": 140,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = FasterGrammarFuzzer(EXPR_GRAMMAR, min_nonterminals=3, max_nonterminals=5)\n",
    "f.fuzz()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:41.352877Z",
     "iopub.status.busy": "2025-01-16T09:40:41.352792Z",
     "iopub.status.idle": "2025-01-16T09:40:41.354678Z",
     "shell.execute_reply": "2025-01-16T09:40:41.354446Z"
    },
    "slideshow": {
     "slide_type": "skip"
    },
    "solution2": "hidden"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "131"
      ]
     },
     "execution_count": 141,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f._expansion_invocations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 142,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:41.356035Z",
     "iopub.status.busy": "2025-01-16T09:40:41.355951Z",
     "iopub.status.idle": "2025-01-16T09:40:41.357900Z",
     "shell.execute_reply": "2025-01-16T09:40:41.357667Z"
    },
    "slideshow": {
     "slide_type": "skip"
    },
    "solution2": "hidden"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "107"
      ]
     },
     "execution_count": 142,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f._expansion_invocations_cached"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:41.359280Z",
     "iopub.status.busy": "2025-01-16T09:40:41.359196Z",
     "iopub.status.idle": "2025-01-16T09:40:41.360915Z",
     "shell.execute_reply": "2025-01-16T09:40:41.360677Z"
    },
    "slideshow": {
     "slide_type": "skip"
    },
    "solution2": "hidden"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "81.68% of invocations can be cached\n"
     ]
    }
   ],
   "source": [
    "print(\"%.2f%% of invocations can be cached\" %\n",
    "      (f._expansion_invocations_cached * 100 / f._expansion_invocations))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    },
    "solution2": "hidden",
    "solution2_first": true
   },
   "source": [
    "### Exercise 2: Grammar Pre-Compilation\n",
    "\n",
    "Some methods such as `symbol_cost()` or `expansion_cost()` return a value that is dependent on the grammar only.  Set up a class `EvenFasterGrammarFuzzer()` that pre-computes these values once upon initialization, such that later invocations of `symbol_cost()` or `expansion_cost()` need only look up these values."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    },
    "solution2": "hidden"
   },
   "source": [
    "**Solution.** Here's a possible solution, using a hack to substitute the `symbol_cost()` and `expansion_cost()` functions once the pre-computed values are set up."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 144,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:41.362357Z",
     "iopub.status.busy": "2025-01-16T09:40:41.362282Z",
     "iopub.status.idle": "2025-01-16T09:40:41.365131Z",
     "shell.execute_reply": "2025-01-16T09:40:41.364918Z"
    },
    "slideshow": {
     "slide_type": "skip"
    },
    "solution2": "hidden"
   },
   "outputs": [],
   "source": [
    "class EvenFasterGrammarFuzzer(GrammarFuzzer):\n",
    "    \"\"\"Variant of `GrammarFuzzer` with precomputed costs\"\"\"\n",
    "\n",
    "    def __init__(self, *args, **kwargs) -> None:\n",
    "        super().__init__(*args, **kwargs)\n",
    "        self._symbol_costs: Dict[str, Union[int, float]] = {}\n",
    "        self._expansion_costs: Dict[Expansion, Union[int, float]] = {}\n",
    "        self.precompute_costs()\n",
    "\n",
    "    def new_symbol_cost(self, symbol: str,\n",
    "                        seen: Set[str] = set()) -> Union[int, float]:\n",
    "        return self._symbol_costs[symbol]\n",
    "\n",
    "    def new_expansion_cost(self, expansion: Expansion,\n",
    "                           seen: Set[str] = set()) -> Union[int, float]:\n",
    "        return self._expansion_costs[expansion]\n",
    "\n",
    "    def precompute_costs(self) -> None:\n",
    "        for symbol in self.grammar:\n",
    "            self._symbol_costs[symbol] = super().symbol_cost(symbol)\n",
    "            for expansion in self.grammar[symbol]:\n",
    "                self._expansion_costs[expansion] = \\\n",
    "                    super().expansion_cost(expansion)\n",
    "\n",
    "        # Make sure we now call the caching methods\n",
    "        self.symbol_cost = self.new_symbol_cost  # type: ignore\n",
    "        self.expansion_cost = self.new_expansion_cost  # type: ignore"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:41.366481Z",
     "iopub.status.busy": "2025-01-16T09:40:41.366402Z",
     "iopub.status.idle": "2025-01-16T09:40:41.368647Z",
     "shell.execute_reply": "2025-01-16T09:40:41.368437Z"
    },
    "slideshow": {
     "slide_type": "skip"
    },
    "solution2": "hidden"
   },
   "outputs": [],
   "source": [
    "f = EvenFasterGrammarFuzzer(EXPR_GRAMMAR)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    },
    "solution2": "hidden"
   },
   "source": [
    "Here are the individual costs:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:41.370086Z",
     "iopub.status.busy": "2025-01-16T09:40:41.370005Z",
     "iopub.status.idle": "2025-01-16T09:40:41.371982Z",
     "shell.execute_reply": "2025-01-16T09:40:41.371771Z"
    },
    "slideshow": {
     "slide_type": "skip"
    },
    "solution2": "hidden"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'<start>': 6,\n",
       " '<expr>': 5,\n",
       " '<term>': 4,\n",
       " '<factor>': 3,\n",
       " '<integer>': 2,\n",
       " '<digit>': 1}"
      ]
     },
     "execution_count": 146,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f._symbol_costs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:41.373417Z",
     "iopub.status.busy": "2025-01-16T09:40:41.373337Z",
     "iopub.status.idle": "2025-01-16T09:40:41.375438Z",
     "shell.execute_reply": "2025-01-16T09:40:41.375229Z"
    },
    "slideshow": {
     "slide_type": "skip"
    },
    "solution2": "hidden"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'<expr>': 6,\n",
       " '<term> + <expr>': 10,\n",
       " '<term> - <expr>': 10,\n",
       " '<term>': 5,\n",
       " '<factor> * <term>': 8,\n",
       " '<factor> / <term>': 8,\n",
       " '<factor>': 4,\n",
       " '+<factor>': 4,\n",
       " '-<factor>': 4,\n",
       " '(<expr>)': 6,\n",
       " '<integer>.<integer>': 5,\n",
       " '<integer>': 3,\n",
       " '<digit><integer>': 4,\n",
       " '<digit>': 2,\n",
       " '0': 1,\n",
       " '1': 1,\n",
       " '2': 1,\n",
       " '3': 1,\n",
       " '4': 1,\n",
       " '5': 1,\n",
       " '6': 1,\n",
       " '7': 1,\n",
       " '8': 1,\n",
       " '9': 1}"
      ]
     },
     "execution_count": 147,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f._expansion_costs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:41.376817Z",
     "iopub.status.busy": "2025-01-16T09:40:41.376737Z",
     "iopub.status.idle": "2025-01-16T09:40:41.382290Z",
     "shell.execute_reply": "2025-01-16T09:40:41.382009Z"
    },
    "slideshow": {
     "slide_type": "skip"
    },
    "solution2": "hidden"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'(8 * 0 + 8 + 4 * 3) * +(6 - 7 / 5) / 3 * 6'"
      ]
     },
     "execution_count": 148,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = EvenFasterGrammarFuzzer(EXPR_GRAMMAR)\n",
    "f.fuzz()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "subslide"
    },
    "solution": "hidden",
    "solution2": "hidden",
    "solution2_first": true,
    "solution_first": true
   },
   "source": [
    "### Exercise 3: Maintaining Trees to be Expanded\n",
    "\n",
    "In `expand_tree_once()`, the algorithm traverses the tree again and again to find nonterminals that still can be extended.  Speed up the process by keeping a list of nonterminal symbols in the tree that still can be expanded."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    },
    "solution": "hidden",
    "solution2": "hidden"
   },
   "source": [
    "**Solution.** Left as exercise for the reader."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise 4: Alternate Random Expansions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We could define `expand_node_randomly()` such that it simply invokes `expand_node_by_cost(node, random.choice)`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2025-01-16T09:40:41.385321Z",
     "iopub.status.busy": "2025-01-16T09:40:41.385198Z",
     "iopub.status.idle": "2025-01-16T09:40:41.387354Z",
     "shell.execute_reply": "2025-01-16T09:40:41.387024Z"
    },
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "class ExerciseGrammarFuzzer(GrammarFuzzer):\n",
    "    def expand_node_randomly(self, node: DerivationTree) -> DerivationTree:\n",
    "        if self.log:\n",
    "            print(\"Expanding\", all_terminals(node), \"randomly by cost\")\n",
    "\n",
    "        return self.expand_node_by_cost(node, random.choice)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    },
    "solution2": "hidden",
    "solution2_first": true
   },
   "source": [
    "What is the difference between the original implementation and this alternative?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "button": false,
    "new_sheet": false,
    "run_control": {
     "read_only": false
    },
    "slideshow": {
     "slide_type": "skip"
    },
    "solution": "hidden",
    "solution2": "hidden"
   },
   "source": [
    "**Solution.** The alternative in `ExerciseGrammarFuzzer` has another probability distribution.  In the original `GrammarFuzzer`, all expansions have the same likelihood of being expanded.  In `ExerciseGrammarFuzzer`, first, a cost is chosen (randomly); then, one of the expansions with this cost is chosen (again randomly).  This means that expansions whose cost is unique have a higher chance of being selected."
   ]
  }
 ],
 "metadata": {
  "ipub": {
   "bibliography": "fuzzingbook.bib",
   "toc": true
  },
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.12.8"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": true,
   "title_cell": "",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": true
  },
  "toc-autonumbering": false,
  "toc-showcode": false,
  "toc-showmarkdowntxt": false,
  "varInspector": {
   "cols": {
    "lenName": 16,
    "lenType": 16,
    "lenVar": 40
   },
   "kernels_config": {
    "python": {
     "delete_cmd_postfix": "",
     "delete_cmd_prefix": "del ",
     "library": "var_list.py",
     "varRefreshCmd": "print(var_dic_list())"
    },
    "r": {
     "delete_cmd_postfix": ") ",
     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
    }
   },
   "types_to_exclude": [
    "module",
    "function",
    "builtin_function_or_method",
    "instance",
    "_Feature"
   ],
   "window_display": false
  },
  "vscode": {
   "interpreter": {
    "hash": "4185989cf89c47c310c2629adcadd634093b57a2c49dffb5ae8d0d14fa302f2b"
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}