CS2333/Assignment 1.ipynb

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   "source": [
    "### <center> Assignment 1 \n",
    "#### <center> January 18, 2024\n",
    "##### <center> Isaac Shoebottom (3429069)"
   ],
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   "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"
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   "source": [
    "(a) $L_1 = \\{0^m 1^n | m,n \\in \\mathbb{Z}^*, n \\leq 2, m \\leq n\\}$"
   ],
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   "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)"
   ]
  },
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   "source": [
    "(b) $L_2 = \\{a^m b^n | m,n \\in \\mathbb{Z}^*, 2 \\leq n \\leq 6, n \\text{ is a multiple of } m\\}$"
   ]
  },
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   "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)"
   ]
  },
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   "source": [
    "(c) $L_3 = \\{a^n b^{2n-1} c^{3n-2} | n \\in \\mathbb{Z}^+, n \\leq 3\\}$"
   ]
  },
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   "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)"
   ]
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   "source": [
    "(d) $L_4 = \\{w \\in \\{a,b\\}^* | |w| \\leq 3 \\text{ and } w \\text{ is a palindrome} \\}$"
   ]
  },
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   },
   "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)"
   ]
  },
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   "source": [
    "<div style=\"page-break-after: always;\"></div>\n",
    "Question 2"
   ]
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   "source": [
    "(a) $L_5 = \\{w \\in \\{a,b\\}^* | |w| \\leq 3 \\text{ and } n_a(w) \\text{ is even} \\}$"
   ]
  },
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   "outputs": [
    {
     "name": "stdout",
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     "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)"
   ]
  },
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   "source": [
    "(b) $L_6 = \\{a^m b^n c^p | m,n,p \\in \\mathbb{Z}^*, m + n + p \\leq 4, m > n, n = p\\}$"
   ]
  },
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   },
   "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)"
   ]
  },
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   "source": [
    "(c) $L_7 = \\{w \\in \\{a,b,c\\}^* | |w| \\leq 4, n_a(w) > n_b(w), n_b(w) = n_c(w)\\}$"
   ]
  },
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   },
   "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)"
   ]
  },
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   "source": [
    "<div style=\"page-break-after: always;\"></div>\n",
    "Question 3"
   ]
  },
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   "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",
    "$$L = \\{w \\in \\{a,b\\}^* | n_a(w) = 3 \\text{ or } n_b(w) = 3 \\text{ and } w \\text{ does not end with the opposite letter}\\}$$"
   ]
  },
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   },
   "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)"
   ]
  },
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   "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\""
   ]
  },
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   "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:"
   ]
  },
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   },
   "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",
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   "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",
    "$$\n",
    "x, y \\in \\mathbb{Z}^* \\\\\n",
    "x = 3 \\\\\n",
    "y = < 3 \\\\\n",
    "< 3 = \\{0, 1, 2\\} \\\\\n",
    "x + y = \\{3, 4, 5\\}\n",
    "$$"
   ]
  },
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   },
   "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)"
   ]
  }
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