Compare commits
No commits in common. "65f5adc2aca1ced0a35f3b2b82ad9296db7d9f4e" and "91d8538cc995f21183389db47910320e53b2f137" have entirely different histories.
65f5adc2ac
...
91d8538cc9
2
.gitignore
vendored
2
.gitignore
vendored
@ -1,2 +0,0 @@
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.ipynb_checkpoints
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*.pdf
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@ -2,10 +2,9 @@
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<excludeFolder url="file://$MODULE_DIR$/.idea" />
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<excludeFolder url="file://$MODULE_DIR$/.venv" />
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<excludeFolder url="file://$MODULE_DIR$/.ipynb_checkpoints" />
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||||||
</content>
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@ -7,5 +7,5 @@
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@ -1,25 +0,0 @@
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@ -1,25 +0,0 @@
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@ -1,17 +0,0 @@
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<configuration default="false" name="Export A1" type="ShConfigurationType">
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<option name="SCRIPT_TEXT" value="jupyter-nbconvert.exe "Assignment 1.ipynb" --to webpdf --no-input" />
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</component>
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60
2024-1-15.py
60
2024-1-15.py
@ -48,70 +48,10 @@ def p3():
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return strings
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return strings
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def valid():
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alphabet = ['a', 'b']
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strings = get_all_strings_with_a_given_alphabet_and_length(alphabet, 5)
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def number_of_a_is_equal_to_3(s):
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return s.count('a') == 3
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def number_of_b_is_equal_to_3(s):
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return s.count('b') == 3
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# 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
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def valid_game(s):
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if number_of_a_is_equal_to_3(s):
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return not s.endswith('b')
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elif number_of_b_is_equal_to_3(s):
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return not s.endswith('a')
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strings = list(filter(valid_game, strings))
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return strings
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def p5():
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alphabet = ['a', 'b']
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valid_strings = valid()
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strings = get_all_strings_with_a_given_alphabet_and_length(alphabet, 5)
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||||||
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||||||
def number_of_a_or_b_is_equal_to_3(s):
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return s.count('a') == 3 or s.count('b') == 3
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strings = list(filter(number_of_a_or_b_is_equal_to_3, strings))
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strings = [i for i in strings if i not in valid_strings] # strings - valid_strings
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return strings
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||||||
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||||||
def p6():
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||||||
alphabet = ['a', 'b']
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valid_strings = valid()
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strings = get_all_strings_with_a_given_alphabet_and_length(alphabet, 5)
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||||||
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||||||
def number_of_a_is_equal_to_3(s):
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||||||
return s.count('a') == 3
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||||||
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||||||
def number_of_b_is_equal_to_3(s):
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||||||
return s.count('b') == 3
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||||||
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||||||
def opposite_number_of_winner_is_less_than_3(s):
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if number_of_a_is_equal_to_3(s):
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return s.count('b') < 3
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||||||
elif number_of_b_is_equal_to_3(s):
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||||||
return s.count('a') < 3
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||||||
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||||||
strings = list(filter(opposite_number_of_winner_is_less_than_3, strings))
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||||||
strings = [i for i in strings if i not in valid_strings] # strings - valid_strings
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||||||
return strings
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||||||
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||||||
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||||||
def main():
|
def main():
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||||||
print_strings(p1())
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print_strings(p1())
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||||||
print_strings(p2())
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print_strings(p2())
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||||||
print_strings(p3())
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print_strings(p3())
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print_strings(valid())
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print_strings(p5())
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||||||
print_strings(p6())
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||||||
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||||||
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||||||
if __name__ == '__main__':
|
if __name__ == '__main__':
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||||||
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|||||||
@ -1,563 +0,0 @@
|
|||||||
{
|
|
||||||
"cells": [
|
|
||||||
{
|
|
||||||
"cell_type": "markdown",
|
|
||||||
"source": [
|
|
||||||
"### <center> Assignment 1 \n",
|
|
||||||
"#### <center> January 18, 2024\n",
|
|
||||||
"##### <center> Isaac Shoebottom (3429069)"
|
|
||||||
],
|
|
||||||
"metadata": {
|
|
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"collapsed": false
|
|
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},
|
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"id": "9ec6bd31bcc8447b"
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|
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},
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|
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{
|
|
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"cell_type": "code",
|
|
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"execution_count": 1,
|
|
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"id": "7709a8186ebbb39a",
|
|
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"metadata": {
|
|
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"collapsed": false,
|
|
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"ExecuteTime": {
|
|
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"end_time": "2024-01-18T20:59:35.482159200Z",
|
|
||||||
"start_time": "2024-01-18T20:59:35.470889900Z"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"outputs": [],
|
|
||||||
"source": [
|
|
||||||
"# util contains functions that make it easier to form strings \n",
|
|
||||||
"# from languages, and give output that matches with expected \n",
|
|
||||||
"# output for the homework, (replace empty string with epsilon)\n",
|
|
||||||
"from util import *"
|
|
||||||
]
|
|
||||||
},
|
|
||||||
{
|
|
||||||
"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
|
|
||||||
},
|
|
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"id": "b12165c136baa772"
|
|
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},
|
|
||||||
{
|
|
||||||
"cell_type": "code",
|
|
||||||
"execution_count": 2,
|
|
||||||
"id": "a3ebc6e89e38cc93",
|
|
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"metadata": {
|
|
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"collapsed": false,
|
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"ExecuteTime": {
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|
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"end_time": "2024-01-18T20:59:35.494611400Z",
|
|
||||||
"start_time": "2024-01-18T20:59:35.477374Z"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"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": {
|
|
||||||
"end_time": "2024-01-18T20:59:35.536014100Z",
|
|
||||||
"start_time": "2024-01-18T20:59:35.486263100Z"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"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": {
|
|
||||||
"end_time": "2024-01-18T20:59:35.536014100Z",
|
|
||||||
"start_time": "2024-01-18T20:59:35.491094800Z"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"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": {
|
|
||||||
"end_time": "2024-01-18T20:59:35.539017300Z",
|
|
||||||
"start_time": "2024-01-18T20:59:35.497609900Z"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"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": {
|
|
||||||
"end_time": "2024-01-18T20:59:35.581083800Z",
|
|
||||||
"start_time": "2024-01-18T20:59:35.542564Z"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"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": {
|
|
||||||
"end_time": "2024-01-18T20:59:35.594528300Z",
|
|
||||||
"start_time": "2024-01-18T20:59:35.585083400Z"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"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": {
|
|
||||||
"end_time": "2024-01-18T20:59:35.649706300Z",
|
|
||||||
"start_time": "2024-01-18T20:59:35.597530600Z"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"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": [
|
|
||||||
"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. The rules that I have discerned are as follows:\n",
|
|
||||||
"1. Each valid game has 3 a's or 3 b's, denoting a victory for one side\n",
|
|
||||||
"2. Once it has either victories on either side, the game does not continue, and the string is complete\n",
|
|
||||||
"\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": {
|
|
||||||
"end_time": "2024-01-18T20:59:35.650706300Z",
|
|
||||||
"start_time": "2024-01-18T20:59:35.603524100Z"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"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",
|
|
||||||
"This one 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 language L but are in this language:\n",
|
|
||||||
"\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",
|
|
||||||
"This language has the problem 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:"
|
|
||||||
]
|
|
||||||
},
|
|
||||||
{
|
|
||||||
"cell_type": "code",
|
|
||||||
"execution_count": 10,
|
|
||||||
"id": "53dc46669ffac28c",
|
|
||||||
"metadata": {
|
|
||||||
"collapsed": false,
|
|
||||||
"ExecuteTime": {
|
|
||||||
"end_time": "2024-01-18T20:59:35.652706Z",
|
|
||||||
"start_time": "2024-01-18T20:59:35.612375700Z"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"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",
|
|
||||||
"This language has a similar problem 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",
|
|
||||||
"$$\n",
|
|
||||||
"x, y \\in \\mathbb{Z}^* \\\\\n",
|
|
||||||
"x = 3 \\\\\n",
|
|
||||||
"y = < 3 \\\\\n",
|
|
||||||
"< 3 = \\{0, 1, 2\\} \\\\\n",
|
|
||||||
"x + y = \\{3, 4, 5\\}\n",
|
|
||||||
"$$"
|
|
||||||
]
|
|
||||||
},
|
|
||||||
{
|
|
||||||
"cell_type": "code",
|
|
||||||
"execution_count": 11,
|
|
||||||
"id": "397defd138f4789a",
|
|
||||||
"metadata": {
|
|
||||||
"collapsed": false,
|
|
||||||
"ExecuteTime": {
|
|
||||||
"end_time": "2024-01-18T20:59:35.691666Z",
|
|
||||||
"start_time": "2024-01-18T20:59:35.656707400Z"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"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
|
|
||||||
}
|
|
||||||
12
util.py
12
util.py
@ -9,15 +9,5 @@ def get_all_strings_with_a_given_alphabet_and_length(alphabet, length):
|
|||||||
return strings
|
return strings
|
||||||
|
|
||||||
|
|
||||||
def replace_empty_string_with_symbol(strings):
|
|
||||||
symbol = 'ε'
|
|
||||||
for i in range(len(strings)):
|
|
||||||
if strings[i] == '':
|
|
||||||
strings[i] = symbol
|
|
||||||
return strings
|
|
||||||
|
|
||||||
|
|
||||||
def print_strings(strings):
|
def print_strings(strings):
|
||||||
strings = replace_empty_string_with_symbol(strings)
|
print(strings, len(strings))
|
||||||
# print(len(strings))
|
|
||||||
print(strings)
|
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user