701 lines
16 KiB
Plaintext
701 lines
16 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"outputs": [],
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"source": [
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"import automata.regex.regex as re\n",
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"from automata.fa.nfa import NFA\n",
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"from IPython.display import Math\n",
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"\n",
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"debug = True\n",
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"\n",
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"def regex_test(regex: str, inputs: list[str]):\n",
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" re.validate(regex)\n",
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" nfa = NFA.from_regex(regex)\n",
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" for i in inputs:\n",
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" print(f\"{i}: {nfa.accepts_input(i)}\")\n",
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" \n",
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"def latexify(regex: str):\n",
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" steps = regex\n",
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" steps = steps.replace('()', '\\\\epsilon')\n",
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" steps = steps.replace('(', '\\\\left(')\n",
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" steps = steps.replace(')', '\\\\right)')\n",
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" steps = steps.replace('|', '\\\\cup')\n",
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" steps = steps.replace('&', '\\\\land')\n",
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" steps = steps.replace('*', '^*')\n",
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" steps = '$' + steps + '$'\n",
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" return steps"
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],
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"metadata": {
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"collapsed": false,
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"ExecuteTime": {
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"end_time": "2024-03-22T22:22:26.498255Z",
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"start_time": "2024-03-22T22:22:26.217210Z"
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}
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},
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"id": "aa9fd8c82c66ac9a",
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"execution_count": 1
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},
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{
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"cell_type": "code",
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"{'aaaab', 'a', 'b', 'aaab', 'ab', 'aaaa', 'aab'}\n",
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"['baaaa', 'ba', 'baab', 'baaaaa', 'baa', 'baaab']\n",
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"['', 'b', 'bb', 'bbb', 'bba', 'ba', 'bab']\n"
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]
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}
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],
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"source": [
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"l1 = ['b', 'ba']\n",
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"l2 = ['a', 'aab', 'aaaa']\n",
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"l3 = []\n",
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"for i in range(0, 5):\n",
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" l3.append('a' * i + 'b')\n",
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"# turn all of these into sets\n",
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"l1 = set(l1)\n",
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"l2 = set(l2)\n",
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"l3 = set(l3)\n",
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"\n",
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"# l2 union l3\n",
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"l2.union(l3)\n",
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"if debug:\n",
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" print(l2.union(l3))\n",
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"# l1 times l2\n",
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"a = []\n",
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"for i in l1:\n",
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" for j in l2:\n",
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" a.append(i + j)\n",
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"if debug:\n",
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" print(a)\n",
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"\n",
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"# kleen star of l1 up to length 3\n",
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"a = ['']\n",
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"for i in l1:\n",
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" a.append(i)\n",
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" for j in l1:\n",
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" a.append(i + j)\n",
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" for k in l1:\n",
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" a.append(i + j + k)\n",
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"# remove all strings of length greater than 3\n",
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"a = [x for x in a if len(x) <= 3]\n",
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"if debug:\n",
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" print(a)"
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],
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"metadata": {
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"collapsed": true,
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"ExecuteTime": {
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"end_time": "2024-03-22T22:22:26.505139Z",
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"start_time": "2024-03-22T22:22:26.499628Z"
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}
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},
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"id": "initial_id",
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"execution_count": 2
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},
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{
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"cell_type": "markdown",
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"source": [
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"# Question 28\n",
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"## b.\n",
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"aaab, aaaa, a, aaaab, b, ab, aab\n",
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"\n",
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"## c.\n",
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"baaaaa, baaab, baa, baaaa, baab, ba\n",
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"\n",
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"## e. \n",
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"$\\epsilon$, ba, bab, b, bba, bb, bbb\n",
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"\n",
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"## h.\n",
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"Yes, because the Kleen star of $L_1$ includes all original strings in the language, and the string 'b', where n is zero is included in $(L_1)^*$, so it is possible for $(L_1)^*$ to have a string where there is more bs than as. "
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],
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"metadata": {
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"collapsed": false
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},
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"id": "823eb4c275b95911"
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},
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{
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"cell_type": "markdown",
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"source": [
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"$\\pagebreak$"
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],
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"metadata": {
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"collapsed": false
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},
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"id": "ae5527f9dab42f59"
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},
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{
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"cell_type": "markdown",
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"source": [
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"# Question 30\n",
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"## c."
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],
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"metadata": {
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"collapsed": false
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},
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"id": "5f0f62a85e3cddd0"
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},
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{
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"cell_type": "code",
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"aabb: False\n",
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"aabbbb: False\n",
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"aabbbbb: True\n",
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"aaabbbbbb: False\n",
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"aaaabbbbbbbbb: True\n"
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]
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},
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{
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"data": {
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"text/plain": "<IPython.core.display.Math object>",
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"text/latex": "$\\displaystyle \\left(aa\\right)^*\\left(bbbb\\right)^*b$"
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# Define a regex where 'a' is a multiple of 2 and 'b' is a multiple of 4 + 1\n",
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"q31c = '(aa)*(bbbb)*b'\n",
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"inputs = [\n",
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"\t'aabb',\n",
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"\t'aabbbb',\n",
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"\t'aabbbbb',\n",
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"\t'aaabbbbbb',\n",
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"\t'aaaabbbbbbbbb',\n",
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"]\n",
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"if debug:\n",
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" regex_test(q31c, inputs)\n",
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"Math(latexify(q31c))"
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],
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"metadata": {
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"collapsed": false,
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"ExecuteTime": {
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"end_time": "2024-03-22T22:22:26.515494Z",
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"start_time": "2024-03-22T22:22:26.506412Z"
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}
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},
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"id": "c0d81b8fb61431de",
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"execution_count": 3
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},
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{
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"cell_type": "markdown",
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"source": [
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"## d."
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],
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"metadata": {
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"collapsed": false
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},
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"id": "53e09c458eafb680"
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},
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{
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"cell_type": "code",
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"ab: True\n",
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"bab: False\n",
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"bbab: True\n",
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"bbbabb: True\n",
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"bbbabbb: False\n",
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"abb: False\n",
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"abbb: True\n",
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"ba: True\n",
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"bba: False\n",
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"bbba: True\n"
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]
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},
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{
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"data": {
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"text/plain": "<IPython.core.display.Math object>",
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"text/latex": "$\\displaystyle \\left(\\left(bb\\right)^*ba\\left(bb\\right)^*\\right)\\cup\\left(\\left(bb\\right)^*ab\\left(bb\\right)^*\\right)$"
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# Define a regex where the number of as is 1 and the number of bs is odd\n",
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"q31e = \"((bb)*ba(bb)*)|((bb)*ab(bb)*)\"\n",
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"inputs = [\n",
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" 'ab',\n",
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" 'bab',\n",
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" 'bbab',\n",
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" 'bbbabb',\n",
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" 'bbbabbb',\n",
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" 'abb',\n",
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" 'abbb',\n",
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" 'ba',\n",
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" 'bba',\n",
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" 'bbba',\n",
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"]\n",
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"if debug:\n",
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" regex_test(q31e, inputs)\n",
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"Math(latexify(q31e))"
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],
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"metadata": {
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"collapsed": false,
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"ExecuteTime": {
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"end_time": "2024-03-22T22:22:26.522964Z",
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"start_time": "2024-03-22T22:22:26.517188Z"
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}
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},
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"id": "75f2af38daefc05b",
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"execution_count": 4
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},
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{
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"cell_type": "markdown",
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"source": [
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"$\\pagebreak$"
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],
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"metadata": {
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"collapsed": false
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},
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"id": "9271ff95c0925816"
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},
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{
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"cell_type": "markdown",
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"source": [
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"# Question 31\n",
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"## c."
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],
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"metadata": {
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"collapsed": false
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},
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"id": "c4c41d43969e22bb"
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},
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{
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"cell_type": "code",
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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": True\n",
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"0: True\n",
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"1: True\n",
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"00: True\n",
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"01: True\n",
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"10: True\n",
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"11: True\n",
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"000: True\n",
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"001: True\n",
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"010: True\n",
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"011: True\n",
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"100: True\n",
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"101: True\n",
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"0000: True\n",
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"0001: True\n",
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"0010: True\n",
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"0011: True\n",
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"0100: True\n",
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"0101: True\n",
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"0110: True\n",
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"0111: True\n",
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"1000: True\n",
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"00000: False\n",
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"11111: False\n"
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]
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},
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{
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"data": {
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"text/plain": "<IPython.core.display.Math object>",
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"text/latex": "$\\displaystyle \\epsilon\\cup\\left(0\\cup1\\right)\\cup\\left(0\\cup1\\right)\\left(0\\cup1\\right)\\cup\\left(0\\cup1\\right)\\left(0\\cup1\\right)\\left(0\\cup1\\right)\\cup\\left(0\\cup1\\right)\\left(0\\cup1\\right)\\left(0\\cup1\\right)\\left(0\\cup1\\right)$"
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},
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"execution_count": 5,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# Define a regex where the number of 1s or 0s is less than or equal to 4\n",
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"q31c = '()|(0|1)|(0|1)(0|1)|(0|1)(0|1)(0|1)|(0|1)(0|1)(0|1)(0|1)'\n",
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"inputs = [\n",
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" '',\n",
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" '0',\n",
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" '1',\n",
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" '00',\n",
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" '01',\n",
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" '10',\n",
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" '11',\n",
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" '000',\n",
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" '001',\n",
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" '010',\n",
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" '011',\n",
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" '100',\n",
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" '101',\n",
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" '0000',\n",
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" '0001',\n",
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" '0010',\n",
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" '0011',\n",
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" '0100',\n",
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" '0101',\n",
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" '0110',\n",
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" '0111',\n",
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" '1000',\n",
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" '00000',\n",
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" '11111',\n",
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"]\n",
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"if debug:\n",
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" regex_test(q31c, inputs)\n",
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"Math(latexify(q31c))"
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],
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"metadata": {
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"collapsed": false,
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"ExecuteTime": {
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||
"end_time": "2024-03-22T22:22:26.531449Z",
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"start_time": "2024-03-22T22:22:26.524511Z"
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}
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},
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"id": "404b569fb1f1b3dd",
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"execution_count": 5
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},
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{
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"cell_type": "markdown",
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"source": [
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"## e."
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],
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"metadata": {
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"collapsed": false
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},
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"id": "64279b55169daadf"
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},
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{
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"cell_type": "code",
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"0100001: True\n",
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"0100000100001: True\n",
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"1000001: True\n",
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"00000010000100000: True\n",
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"1000100001: False\n"
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]
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},
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{
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"data": {
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"text/plain": "<IPython.core.display.Math object>",
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"text/latex": "$\\displaystyle 0^*1\\left(00000^*1\\right)^*\\left(00000^*1\\right)0^*$"
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# Define a regex where the number of 1s is greater or equal to 2 and there are at least four zeros between each 1 and the next 1\n",
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"q31e = '0*1(00000*1)*(00000*1)0*'\n",
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"# 0*1 = 0 or more zeros followed by a 1\n",
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"# (00000*1) = 4 or more zeros followed by a 1\n",
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"inputs = [\n",
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" '0100001',\n",
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" '0100000100001',\n",
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" '1000001',\n",
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" '00000010000100000',\n",
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" '1000100001',\n",
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"]\n",
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"if debug:\n",
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" regex_test(q31e, inputs)\n",
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"Math(latexify(q31e))"
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],
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"metadata": {
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||
"collapsed": false,
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"ExecuteTime": {
|
||
"end_time": "2024-03-22T22:22:26.538114Z",
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||
"start_time": "2024-03-22T22:22:26.532887Z"
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}
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},
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||
"id": "91ecde46ccfd6037",
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"execution_count": 6
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},
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{
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"cell_type": "markdown",
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"source": [
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"$\\pagebreak$"
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],
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"metadata": {
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||
"collapsed": false
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||
},
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||
"id": "a5ed03840b4cc1a0"
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},
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{
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||
"cell_type": "markdown",
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"source": [
|
||
"# Question 32"
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],
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||
"metadata": {
|
||
"collapsed": false
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||
},
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||
"id": "bf475fac64302b91"
|
||
},
|
||
{
|
||
"cell_type": "code",
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||
"outputs": [
|
||
{
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||
"name": "stdout",
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"output_type": "stream",
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"text": [
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"00100: True\n",
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"00100: True\n",
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"00100: False\n"
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]
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},
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||
{
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"data": {
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||
"text/plain": "<IPython.core.display.Math object>",
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"text/latex": "$\\displaystyle 00100$"
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||
},
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||
"execution_count": 7,
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||
"metadata": {},
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||
"output_type": "execute_result"
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||
}
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],
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"source": [
|
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"# Let R1 be the regular expression 1* 0 1* 0 1* 0 1* 0 1*\n",
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"# Let R2 be the regular expression 0* (1 ∪ ε) 0* (1 ∪ ε) 0* (1 ∪ ε) 0* (1 ∪ ε) 0*\n",
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"# Let R3 be the regular expression (0 ∪ 1)* 00 (0 ∪ 1)\n",
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"# Was not supposed to be accepted by all, only bt 2 and 3\n",
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"r1 = '1*01*01*01*01*'\n",
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"r2 = '0*(1|())0*(1|())0*(1|())0*(1|())0*'\n",
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"r3 = '(0|1)*00(0|1)'\n",
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"inputs = ['00100']\n",
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"if debug:\n",
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" regex_test(r1, inputs)\n",
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||
" regex_test(r2, inputs)\n",
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||
" regex_test(r3, inputs)\n",
|
||
"Math(latexify(inputs[0]))"
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||
],
|
||
"metadata": {
|
||
"collapsed": false,
|
||
"ExecuteTime": {
|
||
"end_time": "2024-03-22T22:22:26.547221Z",
|
||
"start_time": "2024-03-22T22:22:26.539392Z"
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||
}
|
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},
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||
"id": "e3c432e6556167c7",
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||
"execution_count": 7
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||
},
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||
{
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||
"cell_type": "markdown",
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"source": [
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"$\\pagebreak$"
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||
],
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||
"metadata": {
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||
"collapsed": false
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||
},
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||
"id": "77d9bf7018723f74"
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},
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||
{
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||
"cell_type": "markdown",
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"source": [
|
||
"# Question 33"
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||
],
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||
"metadata": {
|
||
"collapsed": false
|
||
},
|
||
"id": "afd92f4bb203df3a"
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"p: True\n",
|
||
"cp: True\n",
|
||
"pc: True\n",
|
||
"cpc: True\n",
|
||
"ccpcc: True\n",
|
||
"ccpccpcc: True\n",
|
||
"ccpccpccpcc: True\n",
|
||
"ccpccpccpccpcc: True\n",
|
||
"cpcpcpcpcpc: True\n",
|
||
"pp: True\n"
|
||
]
|
||
},
|
||
{
|
||
"data": {
|
||
"text/plain": "<IPython.core.display.Math object>",
|
||
"text/latex": "$\\displaystyle \\left(c^*pc^*\\right)^*$"
|
||
},
|
||
"execution_count": 8,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"# Define a regex where no p is touching another p, and any number of c's can be between any two p's\n",
|
||
"q33 = '(c*pc*)*'\n",
|
||
"inputs = [\n",
|
||
" 'p',\n",
|
||
" 'cp',\n",
|
||
" 'pc',\n",
|
||
" 'cpc',\n",
|
||
" 'ccpcc',\n",
|
||
" 'ccpccpcc',\n",
|
||
" 'ccpccpccpcc',\n",
|
||
" 'ccpccpccpccpcc',\n",
|
||
" 'cpcpcpcpcpc',\n",
|
||
" 'pp', # Not supposed to be here\n",
|
||
"]\n",
|
||
"if debug:\n",
|
||
" regex_test(q33, inputs)\n",
|
||
"Math(latexify(q33))"
|
||
],
|
||
"metadata": {
|
||
"collapsed": false,
|
||
"ExecuteTime": {
|
||
"end_time": "2024-03-22T22:22:26.555155Z",
|
||
"start_time": "2024-03-22T22:22:26.549229Z"
|
||
}
|
||
},
|
||
"id": "5b8c9ad34523ea81",
|
||
"execution_count": 8
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"source": [
|
||
"$\\pagebreak$"
|
||
],
|
||
"metadata": {
|
||
"collapsed": false
|
||
},
|
||
"id": "567a0f904e3a8b66"
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"source": [
|
||
"# Question 34"
|
||
],
|
||
"metadata": {
|
||
"collapsed": false
|
||
},
|
||
"id": "8580d9b65031d9f7"
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"0: True\n",
|
||
"1: True\n",
|
||
"00: True\n",
|
||
"01: True\n",
|
||
"10: True\n",
|
||
"11: True\n",
|
||
"000: False\n",
|
||
"001: False\n",
|
||
"010: False\n",
|
||
"0000: True\n",
|
||
"00001: True\n",
|
||
"001100: False\n",
|
||
"011001110: False\n"
|
||
]
|
||
},
|
||
{
|
||
"data": {
|
||
"text/plain": "<IPython.core.display.Math object>",
|
||
"text/latex": "$\\displaystyle \\left(\\left(0\\cup1\\right)\\left(0\\cup1\\right)\\left(0\\cup1\\right)\\right)^*\\left(\\left(0\\cup1\\right)\\cup\\left(00\\cup01\\cup10\\cup11\\right)\\right)$"
|
||
},
|
||
"execution_count": 9,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"# Define a regex where the length of the string is not a multiple of 3, on input {0, 1}\n",
|
||
"q34 = '((0|1)(0|1)(0|1))*((0|1)|(00|01|10|11))'\n",
|
||
"inputs = [\n",
|
||
" '0', # 1\n",
|
||
" '1', # 1\n",
|
||
" '00', # 2\n",
|
||
" '01', # 2\n",
|
||
" '10', # 2 \n",
|
||
" '11', # 2\n",
|
||
" '000', # 3\n",
|
||
" '001', # 3\n",
|
||
" '010', # 3\n",
|
||
" '0000', # 4\n",
|
||
" '00001', # 5\n",
|
||
" '001100', # 6\n",
|
||
" '011001110', # 9\n",
|
||
"]\n",
|
||
"if debug:\n",
|
||
" regex_test(q34, inputs)\n",
|
||
"Math(latexify(q34))"
|
||
],
|
||
"metadata": {
|
||
"collapsed": false,
|
||
"ExecuteTime": {
|
||
"end_time": "2024-03-22T22:22:26.563180Z",
|
||
"start_time": "2024-03-22T22:22:26.556508Z"
|
||
}
|
||
},
|
||
"id": "6aa37e012a2b072d",
|
||
"execution_count": 9
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"source": [
|
||
"$\\pagebreak$"
|
||
],
|
||
"metadata": {
|
||
"collapsed": false
|
||
},
|
||
"id": "28524e4a3d3c6e8e"
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"source": [
|
||
"# Question 35\n",
|
||
"## a.\n",
|
||
"$$L = \\{ w \\in \\{ 0, 1 \\}^* \\ | n_0(w) \\text{ is a multiple of } 4 \\}$$\n",
|
||
"\n",
|
||
"## b.\n",
|
||
"$$L = \\{ a^x b^y c^z \\ | x \\geq 1, y \\geq 2, z \\geq 3 \\}$$"
|
||
],
|
||
"metadata": {
|
||
"collapsed": false
|
||
},
|
||
"id": "1f0065878ddf68f8"
|
||
}
|
||
],
|
||
"metadata": {
|
||
"kernelspec": {
|
||
"display_name": "Python 3",
|
||
"language": "python",
|
||
"name": "python3"
|
||
},
|
||
"language_info": {
|
||
"codemirror_mode": {
|
||
"name": "ipython",
|
||
"version": 2
|
||
},
|
||
"file_extension": ".py",
|
||
"mimetype": "text/x-python",
|
||
"name": "python",
|
||
"nbconvert_exporter": "python",
|
||
"pygments_lexer": "ipython2",
|
||
"version": "2.7.6"
|
||
}
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 5
|
||
}
|