2024-01-18 17:27:31 -04:00
{
"cells": [
{
"cell_type": "markdown",
"source": [
"### <center> Assignment 1 \n",
"#### <center> January 18, 2024\n",
"##### <center> Isaac Shoebottom (3429069)"
],
"metadata": {
"collapsed": false
},
"id": "9ec6bd31bcc8447b"
},
{
"cell_type": "code",
"execution_count": 1,
"id": "7709a8186ebbb39a",
"metadata": {
"collapsed": false,
"ExecuteTime": {
2024-01-19 14:00:50 -04:00
"end_time": "2024-01-19T18:00:01.812390600Z",
"start_time": "2024-01-19T18:00:01.808045100Z"
2024-01-18 17:27:31 -04:00
}
},
"outputs": [],
"source": [
2024-01-18 17:48:05 -04:00
"# Helper functions\n",
"def get_all_strings_with_a_given_alphabet_and_length(_alphabet, _length):\n",
"\t_strings = ['']\n",
"\tfor itr in range(_length):\n",
"\t\t_string = [s + c for s in _strings for c in _alphabet]\n",
"\t\t_strings += _string\n",
"\n",
"\t# Remove duplicates (set() isn't as nice as it doesn't preserve order, at least for verification purposes)\n",
"\t_strings = list(dict.fromkeys(_strings))\n",
"\treturn _strings\n",
"\n",
"\n",
"def replace_empty_string_with_symbol(_strings):\n",
"\tsymbol = 'ε'\n",
"\tfor itr in range(len(_strings)):\n",
"\t\tif _strings[itr] == '':\n",
"\t\t\t_strings[itr] = symbol\n",
"\treturn _strings\n",
"\n",
"\n",
"def print_strings(_strings):\n",
"\t_strings = replace_empty_string_with_symbol(_strings)\n",
"\t# print(len(strings))\n",
"\tprint(_strings)"
2024-01-18 17:27:31 -04:00
]
},
{
"cell_type": "markdown",
"source": [
"<div style=\"page-break-after: always;\"></div>\n",
"Question 1"
],
"metadata": {
"collapsed": false
},
"id": "cf6146860173f94f"
},
{
"cell_type": "markdown",
"source": [
"(a) $L_1 = \\{0^m 1^n | m,n \\in \\mathbb{Z}^*, n \\leq 2, m \\leq n\\}$"
],
"metadata": {
"collapsed": false
},
"id": "b12165c136baa772"
},
{
"cell_type": "code",
"execution_count": 2,
"id": "a3ebc6e89e38cc93",
"metadata": {
"collapsed": false,
"ExecuteTime": {
2024-01-19 14:00:50 -04:00
"end_time": "2024-01-19T18:00:01.833000200Z",
"start_time": "2024-01-19T18:00:01.813388900Z"
2024-01-18 17:27:31 -04:00
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"['ε', '1', '01', '11', '011', '0011']\n"
]
}
],
"source": [
"# Language rule 1, n is less than or equal to 2\n",
"n = 2\n",
"# Language rule 2, m is less than or equal to n\n",
"m = n\n",
"strings = []\n",
"# Language rule 3, n and m are both non-negative integers\n",
"for i in range(0, n + 1):\n",
"\tfor j in range(0, m + 1):\n",
"\t\tif j > i:\n",
"\t\t\tcontinue\n",
"\t\tstrings.append('0' * j + '1' * i)\n",
"print_strings(strings)"
]
},
{
"cell_type": "markdown",
"id": "b821cc2e0085663a",
"metadata": {
"collapsed": false
},
"source": [
"(b) $L_2 = \\{a^m b^n | m,n \\in \\mathbb{Z}^*, 2 \\leq n \\leq 6, n \\text{ is a multiple of } m\\}$"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "1d32fddeec3be96f",
"metadata": {
"collapsed": false,
"ExecuteTime": {
2024-01-19 14:00:50 -04:00
"end_time": "2024-01-19T18:00:01.833998900Z",
"start_time": "2024-01-19T18:00:01.819489500Z"
2024-01-18 17:27:31 -04:00
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"['abb', 'aabb', 'abbb', 'aaabbb', 'abbbb', 'aabbbb', 'aaaabbbb', 'abbbbb', 'aaaaabbbbb', 'abbbbbb', 'aabbbbbb', 'aaabbbbbb', 'aaaaaabbbbbb']\n"
]
}
],
"source": [
"possible_n = [2, 3, 4, 5, 6]\n",
"\n",
"strings = []\n",
"for i in possible_n:\n",
"\tinteger_divisors = []\n",
"\tfor j in range(1, i + 1):\n",
"\t\tif i % j == 0:\n",
"\t\t\tinteger_divisors.append(j)\n",
"\tfor j in integer_divisors:\n",
"\t\tstrings.append('a' * j + 'b' * i)\n",
"print_strings(strings)"
]
},
{
"cell_type": "markdown",
"id": "2056f304860447ea",
"metadata": {
"collapsed": false
},
"source": [
"(c) $L_3 = \\{a^n b^{2n-1} c^{3n-2} | n \\in \\mathbb{Z}^+, n \\leq 3\\}$"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "a5ba1ad20a2b94e0",
"metadata": {
"collapsed": false,
"ExecuteTime": {
2024-01-19 14:00:50 -04:00
"end_time": "2024-01-19T18:00:01.833998900Z",
"start_time": "2024-01-19T18:00:01.824503300Z"
2024-01-18 17:27:31 -04:00
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"['abc', 'aabbbcccc', 'aaabbbbbccccccc']\n"
]
}
],
"source": [
"possible_n = [1, 2, 3]\n",
"\n",
"strings = []\n",
"for i in possible_n:\n",
"\tan = i\n",
"\tbn = 2 * i - 1\n",
"\tcn = 3 * i - 2\n",
"\tstrings.append('a' * an + 'b' * bn + 'c' * cn)\n",
"print_strings(strings)"
]
},
{
"cell_type": "markdown",
"id": "520023ded45b27da",
"metadata": {
"collapsed": false
},
"source": [
"(d) $L_4 = \\{w \\in \\{a,b\\}^* | |w| \\leq 3 \\text{ and } w \\text{ is a palindrome} \\}$"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "452a62fd365b4f58",
"metadata": {
"collapsed": false,
"ExecuteTime": {
2024-01-19 14:00:50 -04:00
"end_time": "2024-01-19T18:00:01.836315700Z",
"start_time": "2024-01-19T18:00:01.830000300Z"
2024-01-18 17:27:31 -04:00
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"['ε', 'a', 'b', 'aa', 'bb', 'aaa', 'aba', 'bab', 'bbb']\n"
]
}
],
"source": [
"alphabet = ['a', 'b']\n",
"# include empty string\n",
"strings = get_all_strings_with_a_given_alphabet_and_length(alphabet, 3)\n",
"def is_palindrome(s):\n",
"\treturn s == s[::-1]\n",
"strings = list(filter(is_palindrome, strings))\n",
"print_strings(strings)"
]
},
{
"cell_type": "markdown",
"id": "9dff8b1af013d957",
"metadata": {
"collapsed": false
},
"source": [
"<div style=\"page-break-after: always;\"></div>\n",
"Question 2"
]
},
{
"cell_type": "markdown",
"id": "ad64b6ae580ba4dd",
"metadata": {
"collapsed": false
},
"source": [
"(a) $L_5 = \\{w \\in \\{a,b\\}^* | |w| \\leq 3 \\text{ and } n_a(w) \\text{ is even} \\}$"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "d5167f62fda41135",
"metadata": {
"collapsed": false,
"ExecuteTime": {
2024-01-19 14:00:50 -04:00
"end_time": "2024-01-19T18:00:01.843841300Z",
"start_time": "2024-01-19T18:00:01.834999100Z"
2024-01-18 17:27:31 -04:00
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"['ε', 'b', 'aa', 'bb', 'aab', 'aba', 'baa', 'bbb']\n"
]
}
],
"source": [
"alphabet = ['a', 'b']\n",
"strings = get_all_strings_with_a_given_alphabet_and_length(alphabet, 3)\n",
"\n",
"def number_of_a_is_even(s):\n",
"\treturn s.count('a') % 2 == 0\n",
"\n",
"strings = list(filter(number_of_a_is_even, strings))\n",
"print_strings(strings)"
]
},
{
"cell_type": "markdown",
"id": "9f8b05d94c453058",
"metadata": {
"collapsed": false
},
"source": [
"(b) $L_6 = \\{a^m b^n c^p | m,n,p \\in \\mathbb{Z}^*, m + n + p \\leq 4, m > n, n = p\\}$"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "9b3b29ce2400f4ab",
"metadata": {
"collapsed": false,
"ExecuteTime": {
2024-01-19 14:00:50 -04:00
"end_time": "2024-01-19T18:00:01.924237300Z",
"start_time": "2024-01-19T18:00:01.838835700Z"
2024-01-18 17:27:31 -04:00
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"['a', 'aa', 'aabc', 'aaa', 'aaaa']\n"
]
}
],
"source": [
"possible_m = [0, 1, 2, 3, 4]\n",
"possible_n = [0, 1, 2, 3, 4]\n",
"possible_p = [0, 1, 2, 3, 4]\n",
"\n",
"strings = []\n",
"for m in possible_m:\n",
"\tfor n in possible_n:\n",
"\t\tfor p in possible_p:\n",
"\t\t\t# First rule\n",
"\t\t\tif m + n + p <= 4:\n",
"\t\t\t\t# Second rule\n",
"\t\t\t\tif m > n:\n",
"\t\t\t\t\t# Third rule\n",
"\t\t\t\t\tif n == p:\n",
"\t\t\t\t\t\tstrings.append('a' * m + 'b' * n + 'c' * p)\n",
"print_strings(strings)"
]
},
{
"cell_type": "markdown",
"id": "7626dea00c2aeda4",
"metadata": {
"collapsed": false
},
"source": [
"(c) $L_7 = \\{w \\in \\{a,b,c\\}^* | |w| \\leq 4, n_a(w) > n_b(w), n_b(w) = n_c(w)\\}$"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "8cbf5de6f2655673",
"metadata": {
"collapsed": false,
"ExecuteTime": {
2024-01-19 14:00:50 -04:00
"end_time": "2024-01-19T18:00:01.925239100Z",
"start_time": "2024-01-19T18:00:01.844838400Z"
2024-01-18 17:27:31 -04:00
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"['a', 'aa', 'aaa', 'aaaa', 'aabc', 'aacb', 'abac', 'abca', 'acab', 'acba', 'baac', 'baca', 'bcaa', 'caab', 'caba', 'cbaa']\n"
]
}
],
"source": [
"alphabet = ['a', 'b', 'c']\n",
"strings = get_all_strings_with_a_given_alphabet_and_length(alphabet, 4)\n",
"\n",
"def number_of_a_is_greater_than_number_of_b(s):\n",
"\treturn s.count('a') > s.count('b')\n",
"\n",
"def number_of_b_is_equal_to_number_of_c(s):\n",
"\treturn s.count('b') == s.count('c')\n",
"\n",
"strings = list(filter(number_of_a_is_greater_than_number_of_b, strings))\n",
"strings = list(filter(number_of_b_is_equal_to_number_of_c, strings))\n",
"\n",
"print_strings(strings)"
]
},
{
"cell_type": "markdown",
"id": "88faa9e0347d6fd8",
"metadata": {
"collapsed": false
},
"source": [
"<div style=\"page-break-after: always;\"></div>\n",
"Question 3"
]
},
{
"cell_type": "markdown",
"id": "efdcdcb4f9a0298",
"metadata": {
"collapsed": false
},
"source": [
2024-01-19 14:00:50 -04:00
"First, it could be helpful to find the set of all valid games to compare against. This is a little tricky as first you need to know the rules of the game and encode them correctly as a language. The rules that I have discerned are as follows:\n",
2024-01-18 17:27:31 -04:00
"1. Each valid game has 3 a's or 3 b's, denoting a victory for one side\n",
2024-01-19 14:00:50 -04:00
"2. Once either side has won, the game does not continue, and the string is complete\n",
2024-01-18 17:27:31 -04:00
"\n",
"$$L = \\{w \\in \\{a,b\\}^* | n_a(w) = 3 \\text{ or } n_b(w) = 3 \\text{ and } w \\text{ does not end with the opposite letter}\\}$$"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "660a123373b6ff39",
"metadata": {
"collapsed": false,
"ExecuteTime": {
2024-01-19 14:00:50 -04:00
"end_time": "2024-01-19T18:00:01.926239100Z",
"start_time": "2024-01-19T18:00:01.849939800Z"
2024-01-18 17:27:31 -04:00
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"['aaa', 'bbb', 'aaba', 'abaa', 'abbb', 'baaa', 'babb', 'bbab', 'aabba', 'aabbb', 'ababa', 'ababb', 'abbaa', 'abbab', 'baaba', 'baabb', 'babaa', 'babab', 'bbaaa', 'bbaab']\n"
]
}
],
"source": [
"alphabet = ['a', 'b']\n",
"strings = get_all_strings_with_a_given_alphabet_and_length(alphabet, 5)\n",
"\n",
"def number_of_a_is_equal_to_3(s):\n",
"\treturn s.count('a') == 3\n",
"\n",
"def number_of_b_is_equal_to_3(s):\n",
"\treturn s.count('b') == 3\n",
"\n",
"# A game stops when a player has 3 a's or 3 b's, so there should be no trailing wins from the losing player\n",
"def valid_game(s):\n",
"\tif number_of_a_is_equal_to_3(s):\n",
"\t\treturn not s.endswith('b')\n",
"\telif number_of_b_is_equal_to_3(s):\n",
"\t\treturn not s.endswith('a')\n",
"\n",
"valid_strings = list(filter(valid_game, strings))\n",
"print_strings(valid_strings)"
]
},
{
"cell_type": "markdown",
"id": "bcbd29f7048b2af2",
"metadata": {
"collapsed": false
},
"source": [
"(a) $L_1 = \\{w \\in \\{a,b\\}^* | n_a(w) = 3 \\text{ or } n_b(w) = 3 \\}$\n",
"\n",
2024-01-19 14:00:50 -04:00
"This language is not representable with code, as the set is produces is infinite. This language places no restrictions on the length of the string, so \"aaabbbbbbbbbbbbbbbbbbbbbbbbbbb\" is a valid string in this language. So here is a few examples of valid strings in this language that are not valid in the original:\n",
2024-01-18 17:27:31 -04:00
"\n",
"1. \"aaabbbbbbbbbbbbbbbbbbbbbbbbbbb\"\n",
"2. \"bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbaaa\"\n",
"3. \"bababaaaaaaaaaaaaaaaaaaaaaaaaaa\""
]
},
{
"cell_type": "markdown",
"id": "7afad368b9ad3a09",
"metadata": {
"collapsed": false
},
"source": [
"(b) $L_2 = \\{w \\in \\{a,b\\}^* | |w| \\leq 5, \\text{ and } n_a(w) = 3 \\text{ or } n_b(w) = 3 \\}$\n",
"\n",
2024-01-19 14:00:50 -04:00
"This language differs as it is not encoding the rule that the game stops once a player has 3 a's or 3 b's. This can be seen in these strings that are valid in this language but not in the original language L:"
2024-01-18 17:27:31 -04:00
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "53dc46669ffac28c",
"metadata": {
"collapsed": false,
"ExecuteTime": {
2024-01-19 14:00:50 -04:00
"end_time": "2024-01-19T18:00:01.926239100Z",
"start_time": "2024-01-19T18:00:01.854940200Z"
2024-01-18 17:27:31 -04:00
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"['aaab', 'bbba', 'aaabb', 'aabab', 'abaab', 'abbba', 'baaab', 'babba', 'bbaba', 'bbbaa']\n"
]
}
],
"source": [
"strings = get_all_strings_with_a_given_alphabet_and_length(alphabet, 5)\n",
"\n",
"def number_of_a_or_b_is_equal_to_3(s):\n",
"\treturn s.count('a') == 3 or s.count('b') == 3\n",
"\n",
"strings = list(filter(number_of_a_or_b_is_equal_to_3, strings))\n",
"strings = [i for i in strings if i not in valid_strings] # strings - valid_strings\n",
"print_strings(strings)"
]
},
{
"cell_type": "markdown",
"id": "990482f3b9b9e5b6",
"metadata": {
"collapsed": false
},
"source": [
"(c) $L_3 = \\{w \\in \\{a,b\\}^* | (n_a(w) = 3 \\text{ and } n_b(w) < 3) \\text{ or } (n_b(w) = 3 \\text{ and } n_a(w) < 3) \\}$\n",
"\n",
2024-01-19 14:00:50 -04:00
"This language is similar to the previous one, in that it does not encode the rule that the game stops once a player has 3 a's or 3 b's. The difference in this language is that the limit of string length 5 is not explicitly stated, however it is encoded to be that way by saying that if there are 3 a's then there must be less than 3 b's, and vice versa. This means that the effective maximum length of the string is 5, and thus the language is equivalent to the language from the previous question.\n",
2024-01-18 17:27:31 -04:00
"$$\n",
"x, y \\in \\mathbb{Z}^* \\\\\n",
"x = 3 \\\\\n",
"y = < 3 \\\\\n",
"< 3 = \\{0, 1, 2\\} \\\\\n",
"x + y = \\{3, 4, 5\\}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "397defd138f4789a",
"metadata": {
"collapsed": false,
"ExecuteTime": {
2024-01-19 14:00:50 -04:00
"end_time": "2024-01-19T18:00:01.927237Z",
"start_time": "2024-01-19T18:00:01.860724500Z"
2024-01-18 17:27:31 -04:00
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"['aaab', 'bbba', 'aaabb', 'aabab', 'abaab', 'abbba', 'baaab', 'babba', 'bbaba', 'bbbaa']\n"
]
}
],
"source": [
"strings = get_all_strings_with_a_given_alphabet_and_length(alphabet, 5)\n",
"\n",
"def opposite_number_of_winner_is_less_than_3(s):\n",
"\tif number_of_a_is_equal_to_3(s):\n",
"\t\treturn s.count('b') < 3\n",
"\telif number_of_b_is_equal_to_3(s):\n",
"\t\treturn s.count('a') < 3\n",
"\n",
"strings = list(filter(opposite_number_of_winner_is_less_than_3, strings))\n",
"strings = [i for i in strings if i not in valid_strings] # strings - valid_strings\n",
"print_strings(strings)"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"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": "3.11.5"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
