JEKULcalculator.ipynb

{
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      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
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      "source": [
        "<a href=\"https://colab.research.google.com/github/GammaBLo/TeamLocustResearch/blob/main/JEKULcalculator.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "HugejeoN8qGo"
      },
      "source": [
        "Conversion module: This module should contain functions that can convert decimal numbers to binary and vice versa. This module is important because the arithmetic operations module requires numbers to be represented in binary.\n",
        "\n",
        "Logical operations module: This module should contain functions that perform logical operations such as AND, OR, XOR, and NOT. These operations are often used in computer systems to manipulate data.\n",
        "\n",
        "Bitwise shift module: This module should contain functions that perform bitwise shift operations such as left shift and right shift. These operations are useful when working with binary numbers and can be used to multiply or divide a number by a power of 2.\n",
        "\n",
        "Memory module: This module should contain functions that allow you to read and write to memory. In computer systems, memory is used to store data and programs, so this module is essential when working with computer systems.\n",
        "\n",
        "Control module: This module should contain functions that control the flow of a program. This includes conditional statements, loops, and function calls. This module is important because it allows you to write programs that can perform complex tasks."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "b8jmSIxB8yq7"
      },
      "source": [
        "Conversion module:     \n",
        "Decimal (base 10)\n",
        "Binary (base 2)\n",
        "Octal (base 8)\n",
        "Hexadecimal (base 16)\n",
        "Decimal to Binary: divide the decimal number by 2 repeatedly, taking note of the remainder at each step. The binary number is the sequence of remainders, read from bottom to top.\n",
        "\n",
        "Binary to Decimal: multiply each digit of the binary number with its corresponding power of 2, and sum the results.\n",
        "\n",
        "Decimal to Octal: divide the decimal number by 8 repeatedly, taking note of the remainder at each step. The octal number is the sequence of remainders, read from bottom to top.\n",
        "\n",
        "Octal to Decimal: multiply each digit of the octal number with its corresponding power of 8, and sum the results.\n",
        "\n",
        "Decimal to Hexadecimal: divide the decimal number by 16 repeatedly, taking note of the remainder at each step. If the remainder is greater than 9, use the corresponding letter (A-F) instead. The hexadecimal number is the sequence of remainders, read from bottom to top.\n",
        "\n",
        "Hexadecimal to Decimal: multiply each digit of the hexadecimal number with its corresponding power of 16, and sum the results.\n",
        "\n",
        "Binary to Octal: group the binary digits into groups of three, starting from the rightmost digit, and convert each group to its corresponding octal digit.\n",
        "\n",
        "Octal to Binary: convert each octal digit to its corresponding binary representation, and concatenate the results.\n",
        "\n",
        "Binary to Hexadecimal: group the binary digits into groups of four, starting from the rightmost digit, and convert each group to its corresponding hexadecimal digit.\n",
        "\n",
        "Hexadecimal to Binary: convert each hexadecimal digit to its corresponding binary representation, and concatenate the results.\n",
        "\n",
        "Octal to Hexadecimal: convert the octal number to binary, then convert the binary number to hexadecimal.\n",
        "\n",
        "Hexadecimal to Octal: convert the hexadecimal number to binary, then convert the binary number to octal."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "WOmhuX788mfS"
      },
      "outputs": [],
      "source": [
        "!pip install pi_logic\n",
        "\n",
        "def to_binary(n):\n",
        "    \"\"\"\n",
        "    Converts an integer to its binary representation.\n",
        "    \"\"\"\n",
        "    return bin(n & int(\"1\" * 32, 2))[2:].zfill(32)\n",
        "\n",
        "def from_binary(s):\n",
        "    \"\"\"\n",
        "    Converts a binary string to an integer.\n",
        "    \"\"\"\n",
        "    return int(s, 2)\n",
        "\n",
        "def to_hex(n):\n",
        "    \"\"\"\n",
        "    Converts an integer to its hexadecimal representation.\n",
        "    \"\"\"\n",
        "    return hex(n & int(\"F\" * 8, 16))[2:].zfill(8)\n",
        "\n",
        "def from_hex(s):\n",
        "    \"\"\"\n",
        "    Converts a hexadecimal string to an integer.\n",
        "    \"\"\"\n",
        "    return int(s, 16)\n",
        "\n",
        "def to_base32(n):\n",
        "    \"\"\"\n",
        "    Converts an integer to its base 32 representation.\n",
        "    \"\"\"\n",
        "    alphabet = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ234567'\n",
        "    result = ''\n",
        "    while n > 0:\n",
        "        n, i = divmod(n, 32)\n",
        "        result = alphabet[i] + result\n",
        "    return result\n",
        "\n",
        "def from_base32(s):\n",
        "    \"\"\"\n",
        "    Converts a base 32 string to an integer.\n",
        "    \"\"\"\n",
        "    alphabet = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ234567'\n",
        "    result = 0\n",
        "    for c in s:\n",
        "        result = result * 32 + alphabet.index(c)\n",
        "    return result\n",
        "\n",
        "def to_base64(n):\n",
        "    \"\"\"\n",
        "    Converts an integer to its base 64 representation.\n",
        "    \"\"\"\n",
        "    alphabet = 'ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/'\n",
        "    result = ''\n",
        "    while n > 0:\n",
        "        n, i = divmod(n, 64)\n",
        "        result = alphabet[i] + result\n",
        "    return result\n",
        "\n",
        "def from_base64(s):\n",
        "    \"\"\"\n",
        "    Converts a base 64 string to an integer.\n",
        "    \"\"\"\n",
        "    alphabet = 'ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/'\n",
        "    result = 0\n",
        "    for c in s:\n",
        "        result = result * 64 + alphabet.index(c)\n",
        "    return result\n",
        "\n",
        "    # Decimal to Base-3\n",
        "def decimal_to_base3(decimal):\n",
        "    if decimal == 0:\n",
        "        return '0'\n",
        "    digits = []\n",
        "    while decimal > 0:\n",
        "        remainder = decimal % 3\n",
        "        digits.append(str(remainder))\n",
        "        decimal = decimal // 3\n",
        "    return ''.join(reversed(digits))\n",
        "\n",
        "# Base-3 to Decimal\n",
        "def base3_to_decimal(base3):\n",
        "    decimal = 0\n",
        "    for i in range(len(base3)):\n",
        "        digit = int(base3[i])\n",
        "        decimal += digit * (3 ** i)\n",
        "    return decimal\n",
        "\n",
        "# Decimal to Base-4\n",
        "def decimal_to_base4(decimal):\n",
        "    if decimal == 0:\n",
        "        return '0'\n",
        "    digits = []\n",
        "    while decimal > 0:\n",
        "        remainder = decimal % 4\n",
        "        digits.append(str(remainder))\n",
        "        decimal = decimal // 4\n",
        "    return ''.join(reversed(digits))\n",
        "\n",
        "# Base-4 to Decimal\n",
        "def base4_to_decimal(base4):\n",
        "    decimal = 0\n",
        "    for i in range(len(base4)):\n",
        "        digit = int(base4[i])\n",
        "        decimal += digit * (4 ** i)\n",
        "    return decimal\n",
        "\n",
        "def dec_to_ternary(decimal):\n",
        "    if decimal == 0:\n",
        "        return '0'\n",
        "    ternary = ''\n",
        "    while decimal > 0:\n",
        "        remainder = decimal % 3\n",
        "        ternary = str(remainder) + ternary\n",
        "        decimal = decimal // 3\n",
        "    return ternary\n",
        "\n",
        "def ternary_to_dec(ternary):\n",
        "    decimal = 0\n",
        "    power = 0\n",
        "    for digit in ternary[::-1]:\n",
        "        decimal += int(digit) * 3**power\n",
        "        power += 1\n",
        "    return decimal\n",
        "\n",
        "def dec_to_quaternary(decimal):\n",
        "    if decimal == 0:\n",
        "        return '0'\n",
        "    quaternary = ''\n",
        "    while decimal > 0:\n",
        "        remainder = decimal % 4\n",
        "        quaternary = str(remainder) + quaternary\n",
        "        decimal = decimal // 4\n",
        "    return quaternary\n",
        "\n",
        "def quaternary_to_dec(quaternary):\n",
        "    decimal = 0\n",
        "    power = 0\n",
        "    for digit in quaternary[::-1]:\n",
        "        decimal += int(digit) * 4**power\n",
        "        power += 1\n",
        "    return decimal\n",
        "\n",
        "def dec_to_quinary(decimal):\n",
        "    if decimal == 0:\n",
        "        return '0'\n",
        "    quinary = ''\n",
        "    while decimal > 0:\n",
        "        remainder = decimal % 5\n",
        "        quinary = str(remainder) + quinary\n",
        "        decimal = decimal // 5\n",
        "    return quinary\n",
        "\n",
        "def quinary_to_dec(quinary):\n",
        "    decimal = 0\n",
        "    power = 0\n",
        "    for digit in quinary[::-1]:\n",
        "        decimal += int(digit) * 5**power\n",
        "        power += 1\n",
        "    return decimal\n",
        "\n",
        "\n",
        "def base_n_to_decimal(number, base):\n",
        "    \"\"\"\n",
        "    Converts a number in base-n to decimal.\n",
        "    \"\"\"\n",
        "    decimal = 0\n",
        "    for index, digit in enumerate(reversed(str(number))):\n",
        "        decimal += int(digit) * (base ** index)\n",
        "    return decimal\n",
        "\n",
        "\n",
        "def decimal_to_base_n(number, base):\n",
        "    \"\"\"\n",
        "    Converts a decimal number to base-n.\n",
        "    \"\"\"\n",
        "    if number == 0:\n",
        "        return 0\n",
        "\n",
        "    result = \"\"\n",
        "    while number > 0:\n",
        "        remainder = number % base\n",
        "        result = str(remainder) + result\n",
        "        number //= base\n",
        "\n",
        "    return result\n",
        "\n",
        "def convert_dec_to_base(decimal, base):\n",
        "    if base == 2:\n",
        "        return dec_to_bin(decimal)\n",
        "    elif base == 3:\n",
        "        return dec_to_ternary(decimal)\n",
        "    elif base == 4:\n",
        "        return dec_to_quaternary(decimal)\n",
        "    elif base == 5:\n",
        "        return dec_to_quinary(decimal)\n",
        "    elif base == 8:\n",
        "        return dec_to_oct(decimal)\n",
        "    elif base == 10:\n",
        "        return str(decimal)\n",
        "    elif base == 16:\n",
        "        return dec_to_hex(decimal)\n",
        "    else:\n",
        "        return None\n",
        "\n",
        "def convert_base_to_dec(number, base):\n",
        "    if base == 2:\n",
        "        return bin_to_dec(number)\n",
        "    elif base == 3:\n",
        "        return ternary_to_dec(number)\n",
        "    elif base == 4:\n",
        "        return quaternary_to_dec(number)\n",
        "    elif base == 5:\n",
        "        return quinary_to_dec(number)\n",
        "    elif base == 8:\n",
        "        return oct_to_dec(number)\n",
        "    elif base == 10:\n",
        "        return int(number)\n",
        "    elif base == 16:\n",
        "        return hex_to_dec(number)\n",
        "    else:\n",
        "        return None\n",
        "\n",
        "def convert_base_to_base(number, base1, base2):\n",
        "    decimal = convert_base_to_dec(number, base1)\n",
        "    if decimal is not None:\n",
        "        return convert_dec_to_base(decimal, base2)\n",
        "    else:\n",
        "        return None\n",
        "\n",
        "def binary_to_ns_math(binary_str):\n",
        "\n",
        "    \"\"\"\n",
        "    Converts a binary string to our non-standard math system.\n",
        "\n",
        "    Args:\n",
        "        binary_str (str): A binary number as a string.\n",
        "\n",
        "    Returns:\n",
        "        str: The converted number in our non-standard math system.\n",
        "    \"\"\"\n",
        "    # Convert binary string to integer\n",
        "    binary_int = int(binary_str, 2)\n",
        "\n",
        "    # Reduce number modulo 17\n",
        "    ns_math_int = binary_int % 17\n",
        "\n",
        "    # Convert to non-standard math system\n",
        "    if ns_math_int == 0:\n",
        "        return \"O\"\n",
        "    elif ns_math_int == 1:\n",
        "        return \"!\"\n",
        "    elif ns_math_int == 2:\n",
        "        return \"\\\"\"\n",
        "    elif ns_math_int == 3:\n",
        "        return \"#\"\n",
        "    elif ns_math_int == 4:\n",
        "        return \"$\"\n",
        "    elif ns_math_int == 5:\n",
        "        return \"%\"\n",
        "    elif ns_math_int == 6:\n",
        "        return \"&\"\n",
        "    elif ns_math_int == 7:\n",
        "        return \"'\"\n",
        "    elif ns_math_int == 8:\n",
        "        return \"(\"\n",
        "    elif ns_math_int == 9:\n",
        "        return \")\"\n",
        "    elif ns_math_int == 10:\n",
        "        return \"*\"\n",
        "    elif ns_math_int == 11:\n",
        "        return \"+\"\n",
        "    elif ns_math_int == 12:\n",
        "        return \",\"\n",
        "    elif ns_math_int == 13:\n",
        "        return \"-\"\n",
        "    elif ns_math_int == 14:\n",
        "        return \".\"\n",
        "    elif ns_math_int == 15:\n",
        "        return \"/\"\n",
        "    elif ns_math_int == 16:\n",
        "        return \"1\"\n",
        "    else:\n",
        "        return \"\"\n",
        "\n",
        "def ns_math_to_binary(ns_math_str):\n",
        "    \"\"\"\n",
        "    Converts a number in our non-standard math system to binary.\n",
        "\n",
        "    Args:\n",
        "        ns_math_str (str): A number in our non-standard math system.\n",
        "\n",
        "    Returns:\n",
        "        str: The converted binary string.\n",
        "    \"\"\"\n",
        "    # Convert non-standard math system to integer\n",
        "    if ns_math_str == \"O\":\n",
        "        ns_math_int = 0\n",
        "    elif ns_math_str == \"!\":\n",
        "        ns_math_int = 1\n",
        "    elif ns_math_str == \"\\\"\":\n",
        "        ns_math_int = 2\n",
        "    elif ns_math_str == \"#\":\n",
        "        ns_math_int = 3\n",
        "    elif ns_math_str == \"$\":\n",
        "        ns_math_int = 4\n",
        "    elif ns_math_str == \"%\":\n",
        "        ns_math_int = 5\n",
        "    elif ns_math_str == \"&\":\n",
        "        ns_math_int = 6\n",
        "    elif ns_math_str == \"'\":\n",
        "        ns_math_int = 7\n",
        "    elif ns_math_str == \"(\":\n",
        "        ns_math_int = 8\n",
        "    elif ns_math_str == \")\":\n",
        "        ns_math_int = 9\n",
        "    elif ns_math_str == \"*\":\n",
        "        ns_math_int = 10\n",
        "    elif ns_math_str == \"+\":\n",
        "        ns_math_int = 11\n",
        "    elif ns_math_str == \",\":\n",
        "        ns_math_int = 12\n",
        "    elif ns_math_str == \"-\":\n",
        "        ns_math_int = 13\n",
        "    elif ns_math_str == \".\":\n",
        "        ns_math_int = 14\n",
        "    elif ns_math_str == \"/\":\n",
        "        ns_math_int = 15\n",
        "    elif ns_math_str == \"1\":\n",
        "        ns_math_int = 16\n",
        "    else:\n",
        "        ns_math_int = -1\n",
        "\n",
        "    # Convert integer to binary string\n",
        "    if ns_math_int >= 0:\n",
        "        return format(ns_math_int, '04b')\n",
        "    else:\n",
        "        return \"\"\n",
        "\n",
        "def decimal_to_nonstandard(decimal):\n",
        "    \"\"\"\n",
        "    Converts a decimal number to a non-standard math representation.\n",
        "\n",
        "    Args:\n",
        "        decimal (int): A decimal number.\n",
        "\n",
        "    Returns:\n",
        "        str: The non-standard math representation of the decimal number.\n",
        "    \"\"\"\n",
        "    binary = bin(decimal)[2:].zfill(8)  # Convert decimal to binary string\n",
        "    return nonstandard_math(binary)\n",
        "\n",
        "def byte_to_nonstandard(byte):\n",
        "    \"\"\"\n",
        "    Converts a byte to a non-standard math representation.\n",
        "\n",
        "    Args:\n",
        "        byte (str): A byte as a string of 0s and 1s.\n",
        "\n",
        "    Returns:\n",
        "        str: The non-standard math representation of the byte.\n",
        "    \"\"\"\n",
        "    decimal = byte_to_decimal(byte)  # Convert byte to decimal\n",
        "    return decimal_to_nonstandard(decimal)  # Convert decimal to non-standard math"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "!pip=get pi_logic"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ATyNA55Gs0ZG",
        "outputId": "7002f2e2-ffd1-4b08-81cf-7fc5a0e04ebc"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "/bin/bash: line 1: pi_logic: command not found\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "e36IJhA69H7V"
      },
      "source": [
        " Logical operations module\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "lAtuxyOB9LIx"
      },
      "outputs": [],
      "source": [
        "def logical_and(a, b):\n",
        "    \"\"\"\n",
        "    Performs a logical AND operation on two binary numbers.\n",
        "\n",
        "    Args:\n",
        "        a (str): A binary number as a string.\n",
        "        b (str): A binary number as a string.\n",
        "\n",
        "    Returns:\n",
        "        str: The result of the logical AND operation as a binary number.\n",
        "    \"\"\"\n",
        "    # Convert binary strings to integers\n",
        "    a_int = int(a, 2)\n",
        "    b_int = int(b, 2)\n",
        "\n",
        "    # Perform logical AND operation\n",
        "    result = a_int & b_int\n",
        "\n",
        "    # Convert result back to binary string and return\n",
        "    return format(result, 'b')\n",
        "\n",
        "\n",
        "def logical_or(a, b):\n",
        "    \"\"\"\n",
        "    Performs a logical OR operation on two binary numbers.\n",
        "\n",
        "    Args:\n",
        "        a (str): A binary number as a string.\n",
        "        b (str): A binary number as a string.\n",
        "\n",
        "    Returns:\n",
        "        str: The result of the logical OR operation as a binary number.\n",
        "    \"\"\"\n",
        "    # Convert binary strings to integers\n",
        "    a_int = int(a, 2)\n",
        "    b_int = int(b, 2)\n",
        "\n",
        "    # Perform logical OR operation\n",
        "    result = a_int | b_int\n",
        "\n",
        "    # Convert result back to binary string and return\n",
        "    return format(result, 'b')\n",
        "\n",
        "\n",
        "def logical_xor(a, b):\n",
        "    \"\"\"\n",
        "    Performs a logical XOR operation on two binary numbers.\n",
        "\n",
        "    Args:\n",
        "        a (str): A binary number as a string.\n",
        "        b (str): A binary number as a string.\n",
        "\n",
        "    Returns:\n",
        "        str: The result of the logical XOR operation as a binary number.\n",
        "    \"\"\"\n",
        "    # Convert binary strings to integers\n",
        "    a_int = int(a, 2)\n",
        "    b_int = int(b, 2)\n",
        "\n",
        "    # Perform logical XOR operation\n",
        "    result = a_int ^ b_int\n",
        "\n",
        "    # Convert result back to binary string and return\n",
        "    return format(result, 'b')\n",
        "\n",
        "\n",
        "def logical_not(a):\n",
        "    \"\"\"\n",
        "    Performs a logical NOT operation on a binary number.\n",
        "\n",
        "    Args:\n",
        "        a (str): A binary number as a string.\n",
        "\n",
        "    Returns:\n",
        "        str: The result of the logical NOT operation as a binary number.\n",
        "    \"\"\"\n",
        "    # Convert binary string to integer\n",
        "    a_int = int(a, 2)\n",
        "\n",
        "    # Perform logical NOT operation\n",
        "    result = ~a_int\n",
        "\n",
        "    # Convert result back to binary string and return\n",
        "    return format(result & 0xffffffff, 'b')\n",
        "\n",
        "def byte_to_ns_math(byte):\n",
        "    \"\"\"\n",
        "    Converts a byte to a non-standard math representation.\n",
        "\n",
        "    Args:\n",
        "        byte (int): A byte to be converted to non-standard math.\n",
        "\n",
        "    Returns:\n",
        "        str: The non-standard math representation of the byte.\n",
        "    \"\"\"\n",
        "    # Reduce byte modulo 17\n",
        "    mod = byte % 17\n",
        "\n",
        "    # Convert to non-standard math representation\n",
        "    if mod == 0:\n",
        "        return '000'\n",
        "    elif mod == 1:\n",
        "        return '001'\n",
        "    elif mod == 2:\n",
        "        return '010'\n",
        "    elif mod == 3:\n",
        "        return '011'\n",
        "    elif mod == 4:\n",
        "        return '100'\n",
        "    elif mod == 5:\n",
        "        return '101'\n",
        "    elif mod == 6:\n",
        "        return '110'\n",
        "    else:\n",
        "        return '111'\n",
        "\n",
        "def ns_math_to_byte(ns_math):\n",
        "    \"\"\"\n",
        "    Converts a non-standard math representation to a byte.\n",
        "\n",
        "    Args:\n",
        "        ns_math (str): A non-standard math representation.\n",
        "\n",
        "    Returns:\n",
        "        int: The byte value of the non-standard math representation.\n",
        "    \"\"\"\n",
        "    # Convert non-standard math to binary string\n",
        "    binary_str = ns_math.replace('0', '000').replace('1', '001').replace('2', '010').replace('3', '011').replace('4', '100').replace('5', '101').replace('6', '110').replace('7', '111')\n",
        "\n",
        "    # Convert binary string to integer\n",
        "    return int(binary_str, 2)\n",
        "\n",
        "def logical_and_ns_math(a, b):\n",
        "    \"\"\"\n",
        "    Performs a logical AND operation on two non-standard math numbers.\n",
        "\n",
        "    Args:\n",
        "        a (str): A non-standard math number as a string.\n",
        "        b (str): A non-standard math number as a string.\n",
        "\n",
        "    Returns:\n",
        "        str: The result of the logical AND operation as a non-standard math number.\n",
        "    \"\"\"\n",
        "    # Convert non-standard math strings to binary strings\n",
        "    binary_a = a.replace('0', '000').replace('1', '001').replace('2', '010').replace('3', '011').replace('4', '100').replace('5', '101').replace('6', '110').replace('7', '111')\n",
        "    binary_b = b.replace('0', '000').replace('1', '001').replace('2', '010').replace('3', '011').replace('4', '100').replace('5', '101').replace('6', '110').replace('7', '111')\n",
        "\n",
        "    # Convert binary strings to integers\n",
        "    a_int = int(binary_a, 2)\n",
        "    b_int = int(binary_b, 2)\n",
        "\n",
        "    # Perform logical AND operation\n",
        "    result = a_int & b_int\n",
        "\n",
        "    # Convert result back to non-standard math string and return\n",
        "    return byte_to_ns_math(result)\n",
        "\n",
        "def and_ns_math(a, b):\n",
        "    return ns_math_table[byte_to_ns_math[ord(a)] & byte_to_ns_math[ord(b)]]\n",
        "\n",
        "def or_ns_math(a, b):\n",
        "    return ns_math_table[byte_to_ns_math[ord(a)] | byte_to_ns_math[ord(b)]]\n",
        "\n",
        "def xor_ns_math(a, b):\n",
        "    return ns_math_table[byte_to_ns_math[ord(a)] ^ byte_to_ns_math[ord(b)]]\n",
        "\n",
        "def not_ns_math(a):\n",
        "    return ns_math_table[255 - byte_to_ns_math[ord(a)]]\n",
        "\n",
        "def nand_ns_math(a, b):\n",
        "    return ns_math_table[~(byte_to_ns_math[ord(a)] & byte_to_ns_math[ord(b)])]\n",
        "\n",
        "def nor_ns_math(a, b):\n",
        "    return ns_math_table[~(byte_to_ns_math[ord(a)] | byte_to_ns_math[ord(b)])]\n",
        "\n",
        "def xnor_ns_math(a, b):\n",
        "    return ns_math_table[~(byte_to_ns_math[ord(a)] ^ byte_to_ns_math[ord(b)])]\n",
        "\n",
        "def add_ns_math(a, b):\n",
        "    return ns_math_table[(byte_to_ns_math[ord(a)] + byte_to_ns_math[ord(b)]) % 17]\n",
        "\n",
        "def sub_ns_math(a, b):\n",
        "    return ns_math_table[(byte_to_ns_math[ord(a)] - byte_to_ns_math[ord(b)]) % 17]\n",
        "\n",
        "def mul_ns_math(a, b):\n",
        "    return ns_math_table[(byte_to_ns_math[ord(a)] * byte_to_ns_math[ord(b)]) % 17]\n",
        "\n",
        "def div_ns_math(a, b):\n",
        "    return ns_math_table[(byte_to_ns_math[ord(a)] / byte_to_ns_math[ord(b)]) % 17]\n",
        "\n",
        "def ns_math_to_bin(ns_math):\n",
        "    \"\"\"\n",
        "    Convert non-standard math to binary.\n",
        "\n",
        "    Args:\n",
        "        ns_math (str): Non-standard math string.\n",
        "\n",
        "    Returns:\n",
        "        str: Binary string.\n",
        "    \"\"\"\n",
        "    # Convert non-standard math to binary string\n",
        "    binary_str = ns_math.replace('0', '000').replace('1', '001').replace('2', '010').replace('3', '011').replace('4', '100').replace('5', '101').replace('6', '110').replace('7', '111')\n",
        "    return binary_str\n",
        "\n",
        "def ns_math_to_bin(ns_math):\n",
        "    \"\"\"\n",
        "    Converts non-standard math to binary string.\n",
        "\n",
        "    Args:\n",
        "    ns_math (str): The non-standard math value to be converted to binary.\n",
        "\n",
        "    Returns:\n",
        "    binary_str (str): The binary representation of the non-standard math value.\n",
        "    \"\"\"\n",
        "    # Convert non-standard math to binary string\n",
        "    binary_str = ns_math.replace('0', '000').replace('1', '001').replace('2', '010').replace('3', '011').replace('4', '100').replace('5', '101').replace('6', '110').replace('7', '111')\n",
        "    return binary_str\n",
        "\n",
        "\n",
        "\n",
        "def ns_math_table():\n",
        "    \"\"\"\n",
        "    Creates a lookup table for byte-to-non-standard math conversion.\n",
        "\n",
        "    Returns:\n",
        "        list: A list where each element is a tuple of (byte, ns_math) pairs.\n",
        "    \"\"\"\n",
        "    # Define the lookup table\n",
        "    lookup = []\n",
        "\n",
        "    # Iterate over all possible byte values\n",
        "    for byte in range(256):\n",
        "        # Convert byte to non-standard math representation\n",
        "        ns_math = byte_to_ns_math(byte)\n",
        "\n",
        "        # Add to lookup table\n",
        "        lookup.append((byte, ns_math))\n",
        "\n",
        "    return lookup\n",
        "\n",
        "\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "0POywnmu9Yli"
      },
      "source": [
        "Bitwise Shift Module"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "veE3YJX-9Zwn"
      },
      "outputs": [],
      "source": [
        "from typing import List, Tuple\n",
        "\n",
        "\n",
        "def left_shift(n: int, shift: int) -> int:\n",
        "    \"\"\"Shifts a number n to the left by shift bits.\"\"\"\n",
        "    return n << shift\n",
        "\n",
        "\n",
        "def right_shift(n: int, shift: int) -> int:\n",
        "    \"\"\"Shifts a number n to the right by shift bits.\"\"\"\n",
        "    return n >> shift\n",
        "\n",
        "\n",
        "def rotate_left(n: int, shift: int, bits: int) -> int:\n",
        "    \"\"\"Rotates a number n to the left by shift bits.\"\"\"\n",
        "    mask = (2 ** bits) - 1\n",
        "    return ((n << shift) | (n >> (bits - shift))) & mask\n",
        "\n",
        "\n",
        "def rotate_right(n: int, shift: int, bits: int) -> int:\n",
        "    \"\"\"Rotates a number n to the right by shift bits.\"\"\"\n",
        "    mask = (2 ** bits) - 1\n",
        "    return ((n >> shift) | (n << (bits - shift))) & mask\n",
        "\n",
        "\n",
        "def get_bits(n: int, bits: int) -> List[int]:\n",
        "    \"\"\"Returns a list of bits for the number n, up to the specified number of bits.\"\"\"\n",
        "    mask = (1 << bits) - 1\n",
        "    return [(n >> i) & 1 for i in range(bits)][::-1]\n",
        "\n",
        "\n",
        "def set_bits(n: int, bits: int, value: List[int]) -> int:\n",
        "    \"\"\"Sets the first n bits of the number to the specified value.\"\"\"\n",
        "    mask = (1 << bits) - 1\n",
        "    value = value[::-1]\n",
        "    return ((n >> bits) << bits) | (sum([value[i] << i for i in range(bits)]) & mask)\n",
        "\n",
        "\n",
        "def get_bit(n: int, bit: int) -> int:\n",
        "    \"\"\"Returns the bit value at the specified position.\"\"\"\n",
        "    return (n >> bit) & 1\n",
        "\n",
        "\n",
        "def set_bit(n: int, bit: int, value: int) -> int:\n",
        "    \"\"\"Sets the bit at the specified position to the specified value.\"\"\"\n",
        "    mask = ~(1 << bit)\n",
        "    return (n & mask) | (value << bit)\n",
        "\n",
        "def is_power_of_2(n: int) -> bool:\n",
        "    if n <= 0:\n",
        "        return False\n",
        "    else:\n",
        "        return (n & (n - 1)) == 0\n",
        "\n",
        "def bitwise_and(num1, num2):\n",
        "    \"\"\"Performs bitwise AND operation on two binary numbers.\"\"\"\n",
        "    result = ''\n",
        "    for i in range(len(num1)):\n",
        "        if num1[i] == '1' and num2[i] == '1':\n",
        "            result += '1'\n",
        "        else:\n",
        "            result += '0'\n",
        "    return result\n",
        "\n",
        "\n",
        "def bitwise_or(num1, num2):\n",
        "    \"\"\"Performs bitwise OR operation on two binary numbers.\"\"\"\n",
        "    result = ''\n",
        "    for i in range(len(num1)):\n",
        "        if num1[i] == '1' or num2[i] == '1':\n",
        "            result += '1'\n",
        "        else:\n",
        "            result += '0'\n",
        "    return result\n",
        "\n",
        "\n",
        "def bitwise_xor(num1, num2):\n",
        "    \"\"\"Performs bitwise XOR operation on two binary numbers.\"\"\"\n",
        "    result = ''\n",
        "    for i in range(len(num1)):\n",
        "        if num1[i] != num2[i]:\n",
        "            result += '1'\n",
        "        else:\n",
        "            result += '0'\n",
        "    return result\n",
        "\n",
        "\n",
        "def bitwise_not(num):\n",
        "    \"\"\"Performs bitwise NOT operation on a binary number.\"\"\"\n",
        "    result = ''\n",
        "    for i in range(len(num)):\n",
        "        if num[i] == '1':\n",
        "            result += '0'\n",
        "        else:\n",
        "            result += '1'\n",
        "    return result\n",
        "\n",
        "def bitwise_left_shift(num, n):\n",
        "    \"\"\"Performs bitwise left shift operation on a binary number.\"\"\"\n",
        "    shifted_num = num + '0' * n\n",
        "    return shifted_num[n:]\n",
        "\n",
        "\n",
        "def bitwise_right_shift(num, n):\n",
        "    \"\"\"Performs bitwise right shift operation on a binary number.\"\"\"\n",
        "    shifted_num = '0' * n + num\n",
        "    return shifted_num[:-n]\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "pslXBvuW9gme"
      },
      "source": [
        "Memory module:\n",
        "A function toA function to initialize memory, which takes the size of the memory and returns an initialized memory object.\n",
        " clear memory, which sets all memory locations to zero.\n",
        "A function to copy data from one memory location to another.\n",
        "A function to compare two memory locations and return a boolean indicating whether they are equal.\n",
        "Functions to perform arithmetic operations on memory locations, such as addition and subtraction.\n",
        "These functions should be designed to work with binary data, and take into account any byte ordering or endianness issues. Additionally, the module should have error checking and handling to prevent data corruption or\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "CfvOdJbm9tjp"
      },
      "outputs": [],
      "source": [
        "import struct\n",
        "\n",
        "# Memory map\n",
        "memory = {}\n",
        "\n",
        "# Supported data types\n",
        "data_types = {\n",
        "    'byte': 'B',\n",
        "    'word': 'H',\n",
        "    'dword': 'I',\n",
        "    'qword': 'Q',\n",
        "    'float': 'f',\n",
        "    'double': 'd',\n",
        "}\n",
        "\n",
        "# Error checking function to ensure memory addresses are within valid range\n",
        "def check_memory_address(address):\n",
        "    if address not in memory:\n",
        "        raise ValueError(\"Invalid memory address\")\n",
        "\n",
        "# Read a value from memory\n",
        "def read_memory(address, data_type='word'):\n",
        "    check_memory_address(address)\n",
        "    if data_type not in data_types:\n",
        "        raise ValueError(\"Invalid data type\")\n",
        "    return struct.unpack(data_types[data_type], memory[address])[0]\n",
        "\n",
        "# Write a value to memory\n",
        "def write_memory(address, value, data_type='word'):\n",
        "    if data_type not in data_types:\n",
        "        raise ValueError(\"Invalid data type\")\n",
        "    memory[address] = struct.pack(data_types[data_type], value)\n",
        "\n",
        "# Dynamically allocate memory\n",
        "def allocate_memory(size, data_type='word'):\n",
        "    if data_type not in data_types:\n",
        "        raise ValueError(\"Invalid data type\")\n",
        "    address = len(memory)\n",
        "    for i in range(size):\n",
        "        memory[address + i] = struct.pack(data_types[data_type], 0)\n",
        "    return address\n",
        "\n",
        "# Free dynamically allocated memory\n",
        "def free_memory(address, size):\n",
        "    for i in range(size):\n",
        "        del memory[address + i]\n",
        "\n",
        "# Search for a value in memory\n",
        "def search_memory(value, data_type='word'):\n",
        "    if data_type not in data_types:\n",
        "        raise ValueError(\"Invalid data type\")\n",
        "    for address, data in memory.items():\n",
        "        if struct.unpack(data_types[data_type], data)[0] == value:\n",
        "            return address\n",
        "    return None\n",
        "\n",
        "# Sort memory contents\n",
        "def sort_memory(reverse=False):\n",
        "    global memory\n",
        "    memory = dict(sorted(memory.items(), key=lambda x: x[1], reverse=reverse))\n",
        "\n",
        "# Copy a memory block\n",
        "def copy_memory(src_address, dest_address, size):\n",
        "    check_memory_address(src_address)\n",
        "    for i in range(size):\n",
        "        memory[dest_address + i] = memory[src_address + i]\n",
        "\n",
        "# Move a memory block\n",
        "def move_memory(src_address, dest_address, size):\n",
        "    check_memory_address(src_address)\n",
        "    for i in range(size):\n",
        "        memory[dest_address + i] = memory[src_address + i]\n",
        "        del memory[src_address + i]\n",
        "\n",
        "# Error correction and detection mechanisms\n",
        "def generate_checksum():\n",
        "    return sum(memory.values()) % 256\n",
        "\n",
        "def generate_parity_bit():\n",
        "    parity_bit = 0\n",
        "    for data in memory.values():\n",
        "        parity_bit ^= data\n",
        "    return parity_bit\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "9yEDPi3ZLa0s"
      },
      "outputs": [],
      "source": [
        "def ns_math_table():\n",
        "    \"\"\"\n",
        "    Creates a lookup table for byte-to-non-standard math conversion.\n",
        "\n",
        "    Returns:\n",
        "        list: A list where each element is a tuple of (byte, ns_math) pairs.\n",
        "    \"\"\"\n",
        "    # Define the lookup table\n",
        "    lookup = []\n",
        "\n",
        "    # Iterate over all possible byte values\n",
        "    for byte in range(256):\n",
        "        # Convert byte to non-standard math representation\n",
        "        ns_math = byte_to_ns_math(byte)\n",
        "\n",
        "        # Add to lookup table\n",
        "        lookup.append((byte, ns_math))\n",
        "\n",
        "    return lookup\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "122Tz7TXMw8i"
      },
      "outputs": [],
      "source": [
        "def create_subtraction_table():\n",
        "    \"\"\"\n",
        "    Creates a subtraction table for non-standard math.\n",
        "\n",
        "    Returns:\n",
        "        List[List[str]]: A 17x17 list representing the subtraction table.\n",
        "    \"\"\"\n",
        "    table = [['000' for _ in range(17)] for _ in range(17)]\n",
        "    for i in range(17):\n",
        "        for j in range(17):\n",
        "            # Convert to byte and subtract\n",
        "            result = (i - j) % 17\n",
        "            # Convert back to non-standard math\n",
        "            table[i][j] = byte_to_ns_math(result)\n",
        "    return table\n",
        "\n",
        "\n",
        "def create_multiplication_table():\n",
        "    \"\"\"\n",
        "    Creates a multiplication table for non-standard math.\n",
        "\n",
        "    Returns:\n",
        "        List[List[str]]: A 17x17 list representing the multiplication table.\n",
        "    \"\"\"\n",
        "    table = [['000' for _ in range(17)] for _ in range(17)]\n",
        "    for i in range(17):\n",
        "        for j in range(17):\n",
        "            # Convert to byte and multiply\n",
        "            result = (i * j) % 17\n",
        "            # Convert back to non-standard math\n",
        "            table[i][j] = byte_to_ns_math(result)\n",
        "    return table\n",
        "\n",
        "\n",
        "def create_division_table():\n",
        "    \"\"\"\n",
        "    Creates a division table for non-standard math.\n",
        "\n",
        "    Returns:\n",
        "        List[List[str]]: A 17x17 list representing the division table.\n",
        "    \"\"\"\n",
        "    table = [['000' for _ in range(17)] for _ in range(17)]\n",
        "    for i in range(17):\n",
        "        for j in range(17):\n",
        "            if j == 0:\n",
        "                # Division by zero is undefined\n",
        "                table[i][j] = '000'\n",
        "            else:\n",
        "                # Convert to byte and divide\n",
        "                result = (i * modinv(j, 17)) % 17\n",
        "                # Convert back to non-standard math\n",
        "                table[i][j] = byte_to_ns_math(result)\n",
        "    return table\n",
        "\n",
        "\n",
        "def create_modulo_table():\n",
        "    \"\"\"\n",
        "    Creates a modulo table for non-standard math.\n",
        "\n",
        "    Returns:\n",
        "        List[List[str]]: A 17x17 list representing the modulo table.\n",
        "    \"\"\"\n",
        "    table = [['000' for _ in range(17)] for _ in range(17)]\n",
        "    for i in range(17):\n",
        "        for j in range(17):\n",
        "            if j == 0:\n",
        "                # Modulo by zero is undefined\n",
        "                table[i][j] = '000'\n",
        "            else:\n",
        "                # Convert to byte and take modulo\n",
        "                result = i % j\n",
        "                # Convert back to non-standard math\n",
        "                table[i][j] = byte_to_ns_math(result)\n",
        "    return table\n",
        "\n",
        "\n",
        "def create_power_table():\n",
        "    \"\"\"\n",
        "    Creates a power table for non-standard math.\n",
        "\n",
        "    Returns:\n",
        "        List[List[str]]: A 17x17 list representing the power table.\n",
        "    \"\"\"\n",
        "    table = [['000' for _ in range(17)] for _ in range(17)]\n",
        "    for i in range(17):\n",
        "        for j in range(17):\n",
        "            # Convert to byte and raise to power\n",
        "            result = pow(i, j, 17)\n",
        "            # Convert back to non-standard math\n",
        "            table[i][j] = byte_to_ns_math(result)\n",
        "    return table\n",
        "\n",
        "def create_addition_table():\n",
        "    \"\"\"Creates an addition table for non-standard math.\"\"\"\n",
        "    ns_math_values = ['000', '001', '010', '011', '100', '101', '110', '111']\n",
        "    addition_table = []\n",
        "    for i in range(len(ns_math_values)):\n",
        "        row = []\n",
        "        for j in range(len(ns_math_values)):\n",
        "            byte1 = ns_math_to_byte(ns_math_values[i])\n",
        "            byte2 = ns_math_to_byte(ns_math_values[j])\n",
        "            sum_byte = byte1 + byte2\n",
        "            ns_math_sum = byte_to_ns_math(sum_byte % 256)\n",
        "            row.append(ns_math_sum)\n",
        "        addition_table.append(row)\n",
        "    return addition_table\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 416
        },
        "id": "ibPmMwlTOm9f",
        "outputId": "5e1a4541-6767-489e-e53c-f8eb190adfdb"
      },
      "outputs": [
        {
          "ename": "AttributeError",
          "evalue": "ignored",
          "output_type": "error",
          "traceback": [
            "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
            "\u001b[0;31mAttributeError\u001b[0m                            Traceback (most recent call last)",
            "\u001b[0;32m<ipython-input-7-0b74a30958b7>\u001b[0m in \u001b[0;36m<cell line: 125>\u001b[0;34m()\u001b[0m\n\u001b[1;32m    123\u001b[0m \u001b[0maddition_table\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcreate_addition_table\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    124\u001b[0m \u001b[0mdivision_table\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcreate_division_table\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 125\u001b[0;31m \u001b[0mmultiplication_table\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcreate_multiplication_table\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    126\u001b[0m \u001b[0mshift_table\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcreate_shift_table\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    127\u001b[0m \u001b[0mbitwise_and_table\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcreate_bitwise_and_table\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;32m<ipython-input-7-0b74a30958b7>\u001b[0m in \u001b[0;36mcreate_multiplication_table\u001b[0;34m()\u001b[0m\n\u001b[1;32m      5\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mj\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m16\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      6\u001b[0m             \u001b[0;31m# Perform multiplication in non-standard math\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 7\u001b[0;31m             \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mbyte_to_ns_math\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mns_math_to_byte\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mns_math_table\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mns_math_to_byte\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mns_math_table\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0;36m17\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      8\u001b[0m             \u001b[0mrow\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      9\u001b[0m         \u001b[0mtable\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrow\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;32m<ipython-input-2-57cabb42aa4e>\u001b[0m in \u001b[0;36mns_math_to_byte\u001b[0;34m(ns_math)\u001b[0m\n\u001b[1;32m    126\u001b[0m     \"\"\"\n\u001b[1;32m    127\u001b[0m     \u001b[0;31m# Convert non-standard math to binary string\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 128\u001b[0;31m     \u001b[0mbinary_str\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mns_math\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'0'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'000'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'1'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'001'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'2'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'010'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'3'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'011'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'4'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'100'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'5'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'101'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'6'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'110'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'7'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'111'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    129\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    130\u001b[0m     \u001b[0;31m# Convert binary string to integer\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;31mAttributeError\u001b[0m: 'tuple' object has no attribute 'replace'"
          ]
        }
      ],
      "source": [
        "def create_multiplication_table():\n",
        "    table = []\n",
        "    for i in range(16):\n",
        "        row = []\n",
        "        for j in range(16):\n",
        "            # Perform multiplication in non-standard math\n",
        "            result = byte_to_ns_math((ns_math_to_byte(ns_math_table[i])) * (ns_math_to_byte(ns_math_table[j])) % 17)\n",
        "            row.append(result)\n",
        "        table.append(row)\n",
        "    return table\n",
        "\n",
        "def create_shift_table():\n",
        "    table = []\n",
        "    for i in range(16):\n",
        "        row = []\n",
        "        for j in range(4):\n",
        "            # Perform shift in non-standard math\n",
        "            result = byte_to_ns_math((ns_math_to_byte(ns_math_table[i])) << j)\n",
        "            row.append(result)\n",
        "        table.append(row)\n",
        "    return table\n",
        "\n",
        "def create_bitwise_and_table():\n",
        "    table = []\n",
        "    for i in range(16):\n",
        "        row = []\n",
        "        for j in range(16):\n",
        "            # Perform bitwise AND in non-standard math\n",
        "            result = byte_to_ns_math(ns_math_to_byte(ns_math_table[i]) & ns_math_to_byte(ns_math_table[j]))\n",
        "            row.append(result)\n",
        "        table.append(row)\n",
        "    return table\n",
        "\n",
        "def create_bitwise_or_table():\n",
        "    table = []\n",
        "    for i in range(16):\n",
        "        row = []\n",
        "        for j in range(16):\n",
        "            # Perform bitwise OR in non-standard math\n",
        "            result = byte_to_ns_math(ns_math_to_byte(ns_math_table[i]) | ns_math_to_byte(ns_math_table[j]))\n",
        "            row.append(result)\n",
        "        table.append(row)\n",
        "    return table\n",
        "\n",
        "def create_bitwise_xor_table():\n",
        "    table = []\n",
        "    for i in range(16):\n",
        "        row = []\n",
        "        for j in range(16):\n",
        "            # Perform bitwise XOR in non-standard math\n",
        "            result = byte_to_ns_math(ns_math_to_byte(ns_math_table[i]) ^ ns_math_to_byte(ns_math_table[j]))\n",
        "            row.append(result)\n",
        "        table.append(row)\n",
        "    return table\n",
        "\n",
        " # Define the modinv function\n",
        "def modinv(a, m):\n",
        "    \"\"\"\n",
        "    Computes the modular inverse of a modulo m using the extended Euclidean algorithm.\n",
        "\n",
        "    Args:\n",
        "        a (int): The integer to compute the modular inverse of.\n",
        "        m (int): The modulus.\n",
        "\n",
        "    Returns:\n",
        "        int: The modular inverse of a modulo m.\n",
        "    \"\"\"\n",
        "    a = a % m\n",
        "    for x in range(1, m):\n",
        "        if (a * x) % m == 1:\n",
        "            return x\n",
        "    return 1\n",
        "\n",
        "# Define the create_division_table function\n",
        "def create_division_table():\n",
        "    \"\"\"\n",
        "    Creates a division table for non-standard math.\n",
        "\n",
        "    Returns:\n",
        "        list of list of str: A division table for non-standard math.\n",
        "    \"\"\"\n",
        "    table = [['-' for _ in range(17)] for _ in range(17)]\n",
        "\n",
        "    for i in range(17):\n",
        "        for j in range(17):\n",
        "            if j == 0:\n",
        "                table[i][j] = 'ERR'\n",
        "            else:\n",
        "                # Convert to byte and divide\n",
        "                result = (i * modinv(j, 17)) % 17\n",
        "                # Convert back to non-standard math\n",
        "                table[i][j] = byte_to_ns_math(result)\n",
        "\n",
        "    return table\n",
        "\n",
        "\n",
        "def create_ns_math_table():\n",
        "    \"\"\"\n",
        "    Creates a lookup table for byte-to-non-standard math conversion.\n",
        "\n",
        "    Returns:\n",
        "        list: A list where each element is a tuple of (byte, ns_math) pairs.\n",
        "    \"\"\"\n",
        "    # Define the lookup table\n",
        "    lookup = []\n",
        "\n",
        "    # Iterate over all possible byte values\n",
        "    for byte in range(256):\n",
        "        # Convert byte to non-standard math representation\n",
        "        ns_math = byte_to_ns_math(byte)\n",
        "\n",
        "        # Add to lookup table\n",
        "        lookup.append((byte, ns_math))\n",
        "\n",
        "    return lookup\n",
        "\n",
        "\n",
        "# Generate all tables and write them to files\n",
        "ns_math_table = create_ns_math_table()\n",
        "power_table = create_power_table()\n",
        "subtraction_table = create_subtraction_table()\n",
        "modulo_table = create_modulo_table()\n",
        "addition_table = create_addition_table()\n",
        "division_table = create_division_table()\n",
        "multiplication_table = create_multiplication_table()\n",
        "shift_table = create_shift_table()\n",
        "bitwise_and_table = create_bitwise_and_table()\n",
        "bitwise_or_table = create_bitwise_or_table()\n",
        "bitwise_xor_table = create_bitwise_xor_table()\n",
        "\n",
        "with open('ns_math_table.txt', 'w') as f:\n",
        "    for row in ns_math_table:\n",
        "        f.write(' '.join(row) + '\\n')\n",
        "\n",
        "with open('power_table.txt', 'w') as f:\n",
        "    for row in power_table:\n",
        "        f.write(' '.join(row) + '\\n')\n",
        "\n",
        "with open('subtraction_table.txt', 'w') as f:\n",
        "    for row in subtraction_table:\n",
        "        f.write(' '.join(row) + '\\n')\n",
        "\n",
        "with open('modulo_table.txt', 'w') as f:\n",
        "    for row in modulo_table:\n",
        "        f.write(' '.join(row) + '\\n')\n",
        "\n",
        "with open('addition_table.txt', 'w') as f:\n",
        "    for row in addition_table:\n",
        "        f.write(' '.join(row) + '\\n')\n",
        "\n",
        "with open('division_table.txt', 'w') as f:\n",
        "    for row in division_table:\n",
        "        f.write(' '.join(row) + '\\n')\n",
        "\n",
        "with open('multiplication_table.txt', 'w') as f:\n",
        "    for row in multiplication_table:\n",
        "        f.write(' '.join(row) + '\\n')\n",
        "\n",
        "with open('shift_table.txt', 'w') as f:\n",
        "    for row in shift_table:\n",
        "        f.write(' '.join(row) + '\\n')\n",
        "\n",
        "# Create all remaining tables\n",
        "power_table = create_power_table()\n",
        "subtraction_table = create_subtraction_table()\n",
        "modulo_table = create_modulo_table()\n",
        "multiplication_table = create_multiplication_table()\n",
        "\n",
        "# Write tables to files\n",
        "with open('power_table.txt', 'w') as f:\n",
        "    for row in power_table:\n",
        "        f.write(' '.join(map(str, row)) + '\\n')\n",
        "\n",
        "with open('subtraction_table.txt', 'w') as f:\n",
        "    for row in subtraction_table:\n",
        "        f.write(' '.join(map(str, row)) + '\\n')\n",
        "\n",
        "with open('modulo_table.txt', 'w') as f:\n",
        "    for row in modulo_table:\n",
        "        f.write(' '.join(map(str, row)) + '\\n')\n",
        "\n",
        "with open('multiplication_table.txt', 'w') as f:\n",
        "    for row in multiplication_table:\n",
        "        f.write(' '.join(map(str, row)) + '\\n')\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "mRx4fjhFSCIu"
      },
      "outputs": [],
      "source": [
        "def create_multiplication_table():\n",
        "    # Create empty table\n",
        "    table = [[None] * 17 for _ in range(17)]\n",
        "\n",
        "    # Fill in table\n",
        "    for i in range(17):\n",
        "        for j in range(17):\n",
        "            # Convert to binary string\n",
        "            bin_i = ns_math_to_bin(i)\n",
        "            bin_j = ns_math_to_bin(j)\n",
        "\n",
        "            # Multiply binary strings\n",
        "            bin_result = bin(int(bin_i, 2) * int(bin_j, 2))[2:]\n",
        "\n",
        "            # Convert back to non-standard math\n",
        "            table[i][j] = bin_to_ns_math(bin_result)\n",
        "\n",
        "    return table\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "background_save": true
        },
        "id": "VmxWLkQ7p_1x",
        "outputId": "841f11af-20b8-434c-aa9b-d00902eecff8"
      },
      "outputs": [
        {
          "ename": "SyntaxError",
          "evalue": "ignored",
          "output_type": "error",
          "traceback": [
            "\u001b[0;36m  File \u001b[0;32m\"<ipython-input-9-17bc9c5961b8>\"\u001b[0;36m, line \u001b[0;32m2\u001b[0m\n\u001b[0;31m    \\)\u001b[0m\n\u001b[0m     ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m unexpected character after line continuation character\n"
          ]
        }
      ],
      "source": [
        "convert_dec_to_base(10001\n",
        "\\)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "background_save": true
        },
        "id": "ePTBiO44SRQh",
        "outputId": "5ec40821-4230-42f7-e56c-6d292d1120fa"
      },
      "outputs": [
        {
          "ename": "AttributeError",
          "evalue": "ignored",
          "output_type": "error",
          "traceback": [
            "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
            "\u001b[0;31mAttributeError\u001b[0m                            Traceback (most recent call last)",
            "\u001b[0;32m<ipython-input-10-dbe375648c4f>\u001b[0m in \u001b[0;36m<cell line: 1>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mcreate_multiplication_table\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
            "\u001b[0;32m<ipython-input-8-4f8a941ea8a6>\u001b[0m in \u001b[0;36mcreate_multiplication_table\u001b[0;34m()\u001b[0m\n\u001b[1;32m      7\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mj\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m17\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      8\u001b[0m             \u001b[0;31m# Convert to binary string\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 9\u001b[0;31m             \u001b[0mbin_i\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mns_math_to_bin\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     10\u001b[0m             \u001b[0mbin_j\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mns_math_to_bin\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     11\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;32m<ipython-input-2-57cabb42aa4e>\u001b[0m in \u001b[0;36mns_math_to_bin\u001b[0;34m(ns_math)\u001b[0m\n\u001b[1;32m    214\u001b[0m     \"\"\"\n\u001b[1;32m    215\u001b[0m     \u001b[0;31m# Convert non-standard math to binary string\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 216\u001b[0;31m     \u001b[0mbinary_str\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mns_math\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'0'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'000'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'1'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'001'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'2'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'010'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'3'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'011'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'4'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'100'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'5'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'101'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'6'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'110'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'7'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'111'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    217\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mbinary_str\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    218\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
            "\u001b[0;31mAttributeError\u001b[0m: 'int' object has no attribute 'replace'"
          ]
        }
      ],
      "source": [
        "create_multiplication_table()"
      ]
    }
  ],
  "metadata": {
    "colab": {
      "provenance": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "display_name": "Python 3",
      "name": "python3"
    },
    "language_info": {
      "name": "python"
    }
  },
#Not licensed for use by any corporate companies, private or public.  Luke Locust Jnr holds the IP rights and all copyright contexts with these formulas and has no contracts, deals or distribution to any public domains.


To generalize the Pi Logic equations and propositions in relation to chemical elements, we need to establish a systematic framework that captures the properties and relationships of elements within the Pi Logic formalism. Here's an approach to represent chemical elements using Pi Branes and Pi Tiles:


**Pi Brane Equations for Chemical Elements:**


1. **Element Identity:** Each chemical element can be represented by a unique identifier or symbol. We can use the Pi Brane notation to define the identity of an element:


```
{x ∈ Elements | x = "Symbol"}
```


Here, `Elements` represents the set of all chemical elements, and `"Symbol"` represents the symbol or identifier of a specific element. This equation defines an element by its symbol within the set of elements.


2. **Atomic Number:** The atomic number of an element represents the number of protons in its nucleus. We can define the atomic number using the Pi Brane notation:


```
{x ∈ Elements | AtomicNumber(x) = n}
```


Here, `AtomicNumber` is a function that maps an element to its atomic number, and `n` represents a specific atomic number. This equation asserts that an element belongs to the set of elements with a particular atomic number.


**Pi Tile Equations for Chemical Elements:**


1. **Element Properties:** Chemical elements have various properties such as atomic mass, electronic configuration, and physical characteristics. We can represent these properties using Pi Tiles:


```
π = AtomicMass(Element)
φ = ElectronicConfiguration(Element)
ψ = PhysicalCharacteristics(Element)
```


Here, `AtomicMass`, `ElectronicConfiguration`, and `PhysicalCharacteristics` are functions that map an element to its respective properties. The Pi Tiles π, φ, and ψ represent the atomic mass, electronic configuration, and physical characteristics of an element, respectively.


2. **Chemical Reactions:** Chemical reactions involve the rearrangement of atoms and the formation of new substances. We can represent chemical reactions using Pi Tiles:


```
π + φ → ψ
```


Here, π, φ, and ψ represent the reactants and products of a chemical reaction. The equation indicates the transformation of the reactants (π and φ) into the products (ψ).


By utilizing Pi Branes and Pi Tiles, we can develop a comprehensive framework for representing and reasoning about chemical elements and their properties within the formalism of Pi Logic. This formalization provides a basis for further exploration and analysis of chemical phenomena using Pi Logic.


def is_element(symbol):
  """Returns True if symbol is the symbol of an element, False otherwise."""
  elements = ["H", "He", "Li", "Be", "B", "C", "N", "O", "F", "Ne", "Na", "Mg", "Al", "Si", "P", "S", "Cl", "Ar", "K", "Ca", "Sc", "Ti", "V", "Cr", "Mn", "Fe", "Co", "Ni", "Cu", "Zn", "Ga", "Ge", "As", "Se", "Br", "Kr", "Rb", "Sr", "Y", "Zr", "Nb", "Mo", "Tc", "Ru", "Rh", "Pd", "Ag", "Cd", "In", "Sn", "Sb", "Te", "I", "Xe", "Cs", "Ba", "La", "Ce", "Pr", "Nd", "Pm", "Sm", "Eu", "Gd", "Tb", "Dy", "Ho", "Er", "Tm", "Yb", "Lu"]
  return symbol in elements


def is_in_set(element, set):
  """Returns True if element is in set, False otherwise."""
  return element in set


def main():
  elements = []
  for symbol in ["H", "He", "Li", "Be", "B", "C", "N", "O", "F", "Ne", "Na", "Mg", "Al", "Si", "P", "S", "Cl", "Ar", "K", "Ca", "Sc", "Ti", "V", "Cr", "Mn", "Fe", "Co", "Ni", "Cu", "Zn", "Ga", "Ge", "As", "Se", "Br", "Kr", "Rb", "Sr", "Y", "Zr", "Nb", "Mo", "Tc", "Ru", "Rh", "Pd", "Ag", "Cd", "In", "Sn", "Sb", "Te", "I", "Xe", "Cs", "Ba", "La", "Ce", "Pr", "Nd", "Pm", "Sm", "Eu", "Gd", "Tb", "Dy", "Ho", "Er", "Tm", "Yb", "Lu"]:
    if is_element(symbol):
      elements.append(symbol)
  set = {83, 121, 109, 98, 111, 108}
  for element in elements:
    if is_in_set(element, set):
      print(element)


if __name__ == "__main__":
  main()


1. **Element Classification:** Pi Tiles can be used to represent the classification of chemical elements based on their properties. For example:ee


```
π = Metal(Element)
φ = Nonmetal(Element)
ψ = Metalloid(Element)
```


Here, `Metal`, `Nonmetal`, and `Metalloid` are functions that classify an element based on its properties. The Pi Tiles π, φ, and ψ represent whether an element is a metal, nonmetal, or metalloid, respectively.


2. **Chemical Bonding:** Pi Tiles can represent the formation of chemical bonds between elements. For example:


```
π + φ → ψ
```


Here, π and φ represent the elements participating in the chemical bond, and ψ represents the resulting compound formed through bonding.


3. **Periodic Trends:** Pi Tiles can represent periodic trends in the properties of elements. For example:


```
π = AtomicRadius(Element)
φ = Electronegativity(Element)
ψ = IonizationEnergy(Element)
```


Here, `AtomicRadius`, `Electronegativity`, and `IonizationEnergy` are functions that determine specific properties of an element. The Pi Tiles π, φ, and ψ represent the atomic radius, electronegativity, and ionization energy of an element, respectively.


4. **Chemical Equations:** Pi Tiles can represent chemical equations, including reactions and stoichiometry. For example:


```
πA + φB → ψC + δD
```


Here, πA and φB represent the reactants, and ψC and δD represent the products of the chemical equation. The equation describes the transformation of reactants into products, taking into account stoichiometric ratios.


These additional Pi Tiles expand the range of concepts that can be represented and reasoned about within Pi Logic. They provide a formal and expressive language for capturing various aspects of chemical phenomena, including element classification, bonding, periodic trends, and chemical equations.


Here are further derived Pi Tiles represented in a mathematical fashion:


1. **Properties of Chemical Compounds:**
   - **Molar Mass:** `π = M(Compound)`, where `π` represents the molar mass of the compound and `M(Compound)` is a function that calculates the molar mass of the compound.
   - **Boiling Point:** `π = BP(Compound)`, where `π` represents the boiling point of the compound and `BP(Compound)` is a function that calculates the boiling point of the compound.
   - **Melting Point:** `π = MP(Compound)`, where `π` represents the melting point of the compound and `MP(Compound)` is a function that calculates the melting point of the compound.
   - **Solubility:** `π = S(Compound)`, where `π` represents the solubility of the compound and `S(Compound)` is a function that determines the solubility of the compound.


2. **Mechanisms of Chemical Reactions:**
   - **Reaction Rate:** `π = R(Reaction)`, where `π` represents the rate of the chemical reaction and `R(Reaction)` is a function that calculates the rate of the reaction.
   - **Activation Energy:** `π = AE(Reaction)`, where `π` represents the activation energy of the reaction and `AE(Reaction)` is a function that determines the activation energy of the reaction.
   - **Transition State:** `π = TS(Reaction)`, where `π` represents the transition state of the reaction and `TS(Reaction)` is a function that characterizes the transition state of the reaction.
   - **Catalysis:** `π = C(Reaction)`, where `π` represents the presence of a catalyst in the reaction and `C(Reaction)` is a function that determines the catalytic effect on the reaction.


3. **Thermodynamics of Chemical Systems:**
   - **Enthalpy Change:** `π = ΔH(Reaction)`, where `π` represents the enthalpy change of the chemical reaction and `ΔH(Reaction)` is a function that calculates the change in enthalpy of the reaction.
   - **Entropy Change:** `π = ΔS(Reaction)`, where `π` represents the entropy change of the chemical reaction and `ΔS(Reaction)` is a function that determines the change in entropy of the reaction.
   - **Gibbs Free Energy:** `π = ΔG(Reaction)`, where `π` represents the Gibbs free energy of the reaction and `ΔG(Reaction)` is a function that calculates the change in Gibbs free energy of the reaction.
   - **Equilibrium Constant:** `π = K(Reaction)`, where `π` represents the equilibrium constant of the reaction and `K(Reaction)` is a function that determines the equilibrium constant of the reaction.


4. **Kinetics of Chemical Reactions:**
   - **Rate Law:** `π = RL(Reaction)`, where `π` represents the rate law of the chemical reaction and `RL(Reaction)` is a function that describes the rate of the reaction.
   - **Reaction Order:** `π = RO(Reaction)`, where `π` represents the reaction order of the chemical reaction and `RO(Reaction)` is a function that determines the reaction order of the reaction.
   - **Collision Theory:** `π = CT(Reaction)`, where `π` represents the application of collision theory to the reaction and `CT(Reaction)` is a function that describes the collision behavior of reactant molecules.
   - **Reaction Mechanism:** `π = RM(Reaction)`, where `π` represents the


 proposed mechanism for the reaction and `RM(Reaction)` is a function that characterizes the step-by-step process of the reaction.


1. **Modeling the Boiling Point of a Compound:**
   - `π = BP(Compound)`: Represents the boiling point of the compound.
   - `π = M(Compound)`: Represents the molar mass of the compound.
   - `π = MS(Compound)`: Represents the molecular structure of the compound.
   - `π = IF(Compound)`: Represents the strength of intermolecular forces in the compound.


By combining these Pi Tiles and appropriate mathematical functions, a model could be constructed to predict the boiling point of a compound based on its molar mass, molecular structure, and intermolecular forces. The specific mathematical relationships and functions used would depend on the underlying principles and empirical data related to boiling point.


2. **Modeling the Rate of a Chemical Reaction:**
   - `π = R(Reaction)`: Represents the rate of the chemical reaction.
   - `π = [Reactant]`: Represents the concentration of reactants.
   - `π = T`: Represents the temperature of the reaction.
   - `π = C(Reaction)`: Represents the presence of a catalyst.


Utilizing these Pi Tiles and appropriate mathematical functions, a model could be constructed to describe the rate of a chemical reaction based on the reactant concentrations, temperature, and the presence of a catalyst. The model could provide insights into reaction kinetics and help predict the rate under various conditions.


3. **Modeling the Mechanism of a Chemical Reaction:**
   - `π = RM(Reaction)`: Represents the proposed mechanism of the reaction.
   - `π = K(DeterminingStep)`: Represents the rate-determining step of the reaction.
   - `π = CAT(Reaction)`: Represents the potential catalysts for the reaction.


By employing these Pi Tiles and mathematical functions, a model could be developed to describe the step-by-step mechanism of a chemical reaction. This model could focus on identifying the rate-determining step and exploring the impact of catalysts on the reaction pathway.


These examples demonstrate how Pi Tiles can be used to represent key factors and develop mathematical models for various aspects of chemical phenomena. The mathematical formalism provided by Pi Tiles allows for precise reasoning, prediction, and exploration within the realm of chemistry.


1. **Arithmetic Tiles:**
   - `π = Addition(a, b)`: Represents the addition of values a and b.
   - `π = Subtraction(a, b)`: Represents the subtraction of value b from a.
   - `π = Multiplication(a, b)`: Represents the multiplication of values a and b.
   - `π = Division(a, b)`: Represents the division of value a by b.


Arithmetic Tiles allow for the representation and manipulation of basic arithmetic operations. They enable the construction of mathematical expressions and the reasoning about their results.


2. **Relational Tiles:**
   - `π = Equality(a, b)`: Represents the equality of values a and b.
   - `π = Inequality(a, b)`: Represents the inequality of values a and b.
   - `π = GreaterThan(a, b)`: Represents the relationship "a is greater than b."
   - `π = LessThan(a, b)`: Represents the relationship "a is less than b."


Relational Tiles provide a means to represent and reason about the relationships between mathematical objects. They are fundamental in constructing logical statements and making comparisons.


3. **Logical Tiles:**
   - `π = Conjunction(a, b)`: Represents the logical conjunction of statements a and b.
   - `π = Disjunction(a, b)`: Represents the logical disjunction of statements a and b.
   - `π = Negation(a)`: Represents the negation of statement a.
   - `π = Implication(a, b)`: Represents the implication "if a, then b."


Logical Tiles enable the representation and manipulation of logical statements. They play a crucial role in constructing logical arguments and reasoning about logical relationships.


4. **Set Tiles:**
   - `π = Set(elements)`: Represents a set containing the given elements.
   - `π = Union(set1, set2)`: Represents the union of sets set1 and set2.
   - `π = Intersection(set1, set2)`: Represents the intersection of sets set1 and set2.
   - `π = Subset(set1, set2)`: Represents the subset relationship between sets set1 and set2.


Set Tiles provide a framework for representing and reasoning about sets and their operations. They are essential in set theory and various mathematical disciplines that involve set manipulation.


5. **Function Tiles:**
   - `π = Function(f, x)`: Represents the function f applied to the value x.
   - `π = Composition(f, g)`: Represents the composition of functions f and g.


Function Tiles allow for the representation and manipulation of functions and their composition. They provide a means to reason about function behavior and transformations.


These additional Pi Tiles expand the range of mathematical concepts that can be represented and reasoned about using Pi Logic. They enhance the expressiveness and applicability of Pi Logic in various mathematical domains and enable the development of sophisticated mathematical models and theories.


Statement: Let R, G, and B be Pi Tiles representing the red, green, and blue components of an RGB color, respectively. Each Pi Tile can take values from 0 to 255, inclusive. Pi Tile expressions involving R, G, and B can be used to represent specific colors.


Proof:


1. Define R, G, and B as Pi Tiles:
   - R: {0, 1, 2, ..., 255}
   - G: {0, 1, 2, ..., 255}
   - B: {0, 1, 2, ..., 255}


2. Construct Pi Tile expressions to represent colors:
   - Let Color1 = R(r1)G(g1)B(b1) represent a color with red component value r1, green component value g1, and blue component value b1.
   - Let Color2 = R(r2)G(g2)B(b2) represent a color with red component value r2, green component value g2, and blue component value b2.


3. Perform operations on the Pi Tile expressions:
   a) Addition: Color1 + Color2 = R(r1+r2)G(g1+g2)B(b1+b2)
   - Proof: By applying the addition operation to each component of Color1 and Color2, the resulting expression has modified component values.


   b) Multiplication: Color1 * Color2 = R(r1*r2)G(g1*g2)B(b1*b2)
   - Proof: By applying the multiplication operation to each component of Color1 and Color2, the resulting expression has modified component values.


4. Define a Pi Tile Set "ColorSet" that contains multiple color expressions.


5. Perform set operations on the ColorSet:
   - Union: ColorSet1 ∪ ColorSet2 represents the set of colors that appear in either ColorSet1 or ColorSet2.
   - Intersection: ColorSet1 ∩ ColorSet2 represents the set of colors that appear in both ColorSet1 and ColorSet2.
   - Complement: ColorSet1' represents the set of colors that are not in ColorSet1.


By applying Pi Tile operations and reasoning within the defined Pi Tile Sets, we can conduct mathematical operations and explore color combinations, relationships, and unique color components within the 0 to 255 shades or hues range.


The formal statement and proofs outline the foundations of using Pi Tiles and Pi Tile Sets to represent and reason about colors, allowing for mathematical exploration and analysis within the specified color space.


Representation of Temporal Relationships:


We can use a Pi Brane expression π(t) to represent a logical statement involving time, where t represents a specific point in time.
To represent a temporal relationship between two time points t1 and t2, we can use Pi Brane expressions such as π(t1 < t2) to denote that t1 is less than t2.
Modeling Time in Simulations:


In a simulation, time can be represented using a continuous variable t.
Pi Brane expressions can be used to define the evolution of variables over time. For example, π(x(t)) can represent the value of a variable x at time t.
Analyzing Event History:


Event history can be represented using a sequence of timestamps, denoted as t1, t2, t3, ....
Pi Brane expressions can be used to reason about event ordering and relationships. For instance, π(t1 < t2 < t3) represents that t1 is before t2 and t2 is before t3.
Tracking Time in Scripts:


In scripts, time can be represented using a variable t that denotes the current time.
Pi Brane expressions can be used to track the passage of time and analyze time-series data. For example, π(t < t + dt) represents that t is less than t + dt, where dt is a small time increment.


In an extended version of the Pi Brane language, the following functions and features can be included:


1. Basic Pi Tile Functions:
   - piTrue(): Returns the Pi Tile representation of true (π1).
   - piFalse(): Returns the Pi Tile representation of false (π0).
   - piNumber(n): Returns the Pi Tile representation of a given natural number n.
   - piFraction(n, m): Returns the Pi Tile representation of the fraction n/m.
   - piNegate(tile): Returns the negation of a given Pi Tile.
   - piConjunction(tile1, tile2): Returns the conjunction of two Pi Tiles.
   - piDisjunction(tile1, tile2): Returns the disjunction of two Pi Tiles.
   - piImplication(tile1, tile2): Returns the implication of two Pi Tiles.


2. Temporal Pi Tile Functions:
   - piTemporal(tile, t): Returns the truth value of a given Pi Tile at time t.
   - piNext(tile, t): Returns the truth value of a given Pi Tile at time t + 1.
   - piTimeComparison(tile1, operator, tile2): Compares two temporal Pi Tiles using operators such as <, ≤, >, and ≥.


3. Time-Series Data Functions:
   - piAggregation(data, aggregationFunction): Performs aggregation operations on time-series data, such as sum, average, or max.
   - piFiltering(data, filterFunction): Filters time-series data based on specified conditions.
   - piTimeCalculation(time1, time2, calculationFunction): Performs time-based calculations on timestamps, such as time differences or elapsed time.


4. Quantification Functions:
   - piForAll(start, end, condition): Evaluates a given condition for all points in a specified time range.
   - piExists(start, end, condition): Evaluates a given condition for at least one point in a specified time range.


5. Temporal Logic Constructs:
   - piEventually(start, end, condition): Checks if a given condition becomes true at some point within a specified time range.
   - piAlways(start, end, condition): Checks if a given condition remains true for all points within a specified time range.
   - piUntil(start, end, condition1, condition2): Checks if condition1 holds until condition2 becomes true within a specified time range.


6. Concurrency and Parallelism:
   - piConcurrency(expression1, expression2): Executes two Pi Brane expressions concurrently.
   - piParallel(expression1, expression2): Executes two Pi Brane expressions in parallel.


7. Error Handling:
   - piTryCatch(expression, errorHandler): Executes a Pi Brane expression and handles any errors that occur using a specified error handler function.


These functions and features enhance the capabilities of the Pi Brane language, allowing users to manipulate and reason about temporal relationships, analyze time-series data, handle errors, and model concurrency and parallelism.


Byl Reaction:
Pi Tile: SR(reaction, step), where reaction represents the chemical reaction and step represents a step in the reaction.
Mathematical functions can be defined to calculate the rate of the reaction based on the energies of the steps represented by Pi Tiles.
In each case, the Pi Tiles represent the relevant properties, and mathematical functions are defined to calculate the desired properties or outcomes based on the values of the Pi Tiles.


Sure, here are some Pi tile equations:


* **Electron Configuration of an Atom:**


```
def EC(atom, electron) = (energy level, spin, orbital)
```


* **Bond Strength:**


```
BS(bond) = energy
```


* **Electronegativity of a Molecule:**


```
EN(molecule) = energy
```


* **Electrical Conductivity of a Material:**


```
EC(material) = current / voltage
```


* **Step in a Chemical Reaction:**


```
SR(reaction, step) = energy
```


These are just a few examples of Pi tile equations. The specific equations that are used will depend on the specific properties that are being modeled.


In addition to these equations, there are also a number of mathematical functions that can be used to calculate the desired properties or outcomes based on the values of the Pi Tiles. For example, the ionization energy of an atom can be calculated using the following mathematical function:


```
ionization_energy(atom) = sum(BS(bond) for bond in atom.bonds)
```


Certainly! Here are some additional equations that can be used with Pi Tiles:


* **Atomic Mass of an Atom:**
```
AM(atom) = mass
```
This equation represents the atomic mass of an atom, where `atom` is the specific atom and `mass` is the atomic mass value.


* **Molecular Mass of a Molecule:**
```
MM(molecule) = sum(AM(atom) for atom in molecule.atoms)
```
This equation calculates the molecular mass of a molecule by summing the atomic masses of all the atoms in the molecule.


* **Length of a Bond:**
```
BL(bond) = length
```
This equation represents the length of a bond, where `bond` is the specific bond and `length` is the bond length value.


* **Polarity of a Bond:**
```
PB(bond) = polarity
```
This equation represents the polarity of a bond, where `bond` is the specific bond and `polarity` is the bond polarity value.


* **Thermal Conductivity of a Material:**
```
TC(material) = thermal_conductivity
```
This equation represents the thermal conductivity of a material, where `material` is the specific material and `thermal_conductivity` is the thermal conductivity value.


* **Mechanical Strength of a Material:**
```
MS(material) = mechanical_strength
```
This equation represents the mechanical strength of a material, where `material` is the specific material and `mechanical_strength` is the mechanical strength value.


* **Rate of a Chemical Reaction:**
```
CR(reaction) = rate
```
This equation represents the rate of a chemical reaction, where `reaction` is the specific reaction and `rate` is the reaction rate value.


These equations can be used in combination with the Pi Tiles to model various properties and behaviors in atomic, molecular, and material systems.


Please note that these equations are provided as examples and may need to be further refined or expanded depending on the specific context and requirements of the Pi Logic model.


Let me know if you need further assistance!


Law of Time: The time elapsed is proportional to the square of the amplitude of the pendulum's swing and the square of the value of Pi.
Theorem of Time: The time elapsed is inversely proportional to the square of the gravitational field strength.
Equation of Time: The time elapsed is equal to 2Aπ^2t^2 / g, where g is the gravitational field strength.


Thank you for providing additional mathematical symbols, maps and graphs, and scientific symbols. Here is the formalization of these elements:


**Mathematical Symbols:**
- \(+\) and \(-\): Addition and subtraction operators.
- \(\times\) and \(\div\): Multiplication and division operators.
- \(=\): Equality operator.
- \(\neq\): Inequality operator.
- \(>\), \(<\), \(\geq\), \(\leq\): Comparison operators.
- \(\sum\): Summation operator.
- \(\int\): Integral operator.
- \(\sqrt{}\): Square root function.
- \(\pi\): Mathematical constant pi.


**Maps and Graphs:**
- Cartesian Coordinate System: Represents points using coordinates \((x, y)\).
- Bar Graph: Represents data points as pairs \((x, y)\).
- Line Graph: Represents data points as pairs \((x, y)\).
- Pie Chart: Represents sectors with sizes proportional to the whole.
- Venn Diagram: Represents sets and their intersections using overlapping regions.


**Scientific Symbols:**
- \(+\) and \(-\) (in electrical circuits): Represents positive and negative charges.
- \(\Delta\): Represents a change or difference.
- \(\circ\): Represents degrees in temperature or angles.
- \(\mu\): Represents the micro- prefix for units.
- \(\Omega\): Represents ohms in electrical resistance.
- \(\lambda\): Represents wavelength.


These formalizations provide a mathematical representation of the symbols, maps, graphs, and scientific symbols you mentioned. They can be used within mathematical expressions, equations, or diagrams to convey precise meanings and relationships.


* **Pi Logic:**


The equation log(R)/log(ϕ) = log(A)/log(ϕ) ⊕Θ⊕1 can be expressed as a Pi Logic equation as follows:


```
log(R) ⊗ (log(ϕ))^(-1) = log(A) ⊗ (log(ϕ))^(-1) ⊕Θ⊕1
```


This equation represents the relationship between the logarithms of R and A to the base ϕ, along with the angle Θ and the constant value of 1.


* **Pi Branes:**


The equation can be further formalized as a Pi Brane equation by introducing Pi Brane notations as follows:


```
log(R) ⊗ (log(ϕ))^(-1) = log(A) ⊗ (log(ϕ))^(-1) ⊕Θ⊕1
```


This equation represents the logarithmic relationship between the Pi Branes R and A, along with the Pi Brane Θ and the constant value of 1.


* **Brane Mapping:**


The equation can be brane mapped as follows:


```
[log(R)] ⊗ [log(ϕ)]^(-1) = [log(A)] ⊗ [log(ϕ)]^(-1) ⊕ Θ ⊕ 1
```


This equation represents the logarithmic relationship between the brane values of R and A, along with the brane value of Θ and the constant value of 1.


* **Further Applications:**


The equation log(R)/log(ϕ) = log(A)/log(ϕ) ⊕Θ⊕1 can be used to model a variety of phenomena, including the golden ratio, the Fibonacci sequence, and the Mandelbrot set. It can also be used to develop new algorithms and applications in the fields of mathematics, computer science, and engineering.


Apologies for the confusion. Thank you for clarifying your request. Here is the recap and formalization of the equation log(R)/log(ϕ) = log(A)/log(ϕ) ⊕Θ⊕1 with Pi Logic and Pi Branes:


**Pi Logic:**


The equation log(R)/log(ϕ) = log(A)/log(ϕ) ⊕Θ⊕1 can be expressed as a Pi Logic equation as follows:


```
(log(R) ⊗ (log(ϕ))^(-1)) = (log(A) ⊗ (log(ϕ))^(-1)) ⊕Θ⊕1
```


This equation represents the relationship between the logarithms of R and A to the base ϕ, along with the angle Θ and the constant value of 1.


**Pi Branes:**


The equation can be further formalized as a Pi Brane equation by introducing Pi Brane notations as follows:


```
[R] ⊗ [(log(ϕ))^(-1)] = [A] ⊗ [(log(ϕ))^(-1)] ⊕Θ⊕1
```


This equation represents the logarithmic relationship between the Pi Branes R and A, along with the Pi Brane Θ and the constant value of 1.


**Brane Mapping:**


The equation can be brane mapped as follows:


```
[R] ⊗ [(log(ϕ))^(-1)] = [A] ⊗ [(log(ϕ))^(-1)] ⊕ Θ ⊕ 1
```


This equation represents the logarithmic relationship between the brane values of R and A, along with the brane value of Θ and the constant value of 1.


**Further Applications:**


The equation log(R)/log(ϕ) = log(A)/log(ϕ) ⊕Θ⊕1, when formalized within the Pi Logic and Pi Brane frameworks, can be applied in various domains such as mathematics, physics, and computer science. It provides a formal and expressive language for reasoning about logarithmic relationships, golden ratios, and angle transformations. The Pi Logic and Pi Brane frameworks can be utilized to explore the properties, patterns, and applications of this equation in a rigorous and systematic manner.


Certainly! Let's continue defining and formalizing the system for integration into mathematics:


**Mathematical Symbols:**
- Absolute Value: The absolute value of a number represents its distance from zero and can be formalized as the absolute value function: \(|x|\).
- Exponents and Powers: Exponents and powers represent repeated multiplication of a base number and can be formalized using the exponentiation operator: \(x^n\).
- Logarithms: Logarithms represent the inverse operation of exponentiation and can be formalized using the logarithm functions: \(\log_b(x)\), \(\ln(x)\), etc.
- Factorial: The factorial of a number represents the product of all positive integers up to that number and can be formalized using the factorial function: \(n!\).
- Permutations and Combinations: Permutations and combinations represent different ways to arrange or select elements from a set and can be formalized using the permutation and combination formulas: \(P(n, r)\), \(C(n, r)\), etc.


**Mathematical Notations:**
- Set Notation: Set notation represents collections of elements and their relationships and can be formalized using set builder notation, set union, set intersection, set difference, etc.
- Interval Notation: Interval notation represents a range of real numbers and can be formalized using open intervals, closed intervals, half-open intervals, etc.
- Summation Notation: Summation notation represents the sum of a series of terms and can be formalized using the summation operator: \(\sum_{i=1}^{n} a_i\).
- Product Notation: Product notation represents the product of a series of factors and can be formalized using the product operator: \(\prod_{i=1}^{n} a_i\).
- Integral Notation: Integral notation represents the integral of a function over a specified interval and can be formalized using the integral operator: \(\int_{a}^{b} f(x) \, dx\).
- Limit Notation: Limit notation represents the behavior of a function as the independent variable approaches a certain value and can be formalized using the limit operator: \(\lim_{x \to a} f(x)\).
- Vector Notation: Vector notation represents vectors as ordered sets of elements and can be formalized using boldface letters, arrow notation, or component representation.


**Mathematical Structures:**
- Matrices and Matrix Operations: Matrices represent rectangular arrays of numbers and can be formalized using matrix notation, matrix addition, matrix subtraction, matrix multiplication, etc.
- Complex Numbers: Complex numbers represent numbers with both a real part and an imaginary part and can be formalized using the imaginary unit \(i\) and complex number operations.
- Sequences and Series: Sequences represent ordered lists of numbers and can be formalized using subscript notation, while series represent the sum of a sequence and can be formalized using summation notation.
- Graph Theory: Graph theory represents relationships between objects using vertices and edges and can be formalized using graph notation, adjacency matrix, graph algorithms, etc.
- Number Systems: Number systems represent different sets of numbers, such as natural numbers, integers, rational numbers, real numbers, and complex numbers, and can be formalized using set notation and operations defined within each number system.


By incorporating these mathematical symbols, notations, and structures into the system, we enhance its ability to handle a wide range of mathematical concepts and computations, enabling a more comprehensive integration with mathematics.


Sure, I can provide more information about how these numbers are generated from a pi brane sequence, as well as any other rules or principles governing these sequences and their manipulation.


You are right, the mapping from characters to digits and from digits to pi numbers is not clear in my previous explanation. Here is the full explanation:


* **Mapping from characters to digits:**


The following table shows the mapping from characters to digits:


```
Character | Digit | Pi number
------- | -------- | --------
a | 1 | 3
b | 2 | 6
c | 3 | 9
d | 4 | 12
e | 5 | 1
f | 6 | 2
g | 7 | 4
h | 8 | 5
i | 9 | 7
j | 10 | 8
k | 11 | 10
l | 12 | 11
m | 13 | 13
n | 14 | 14
o | 15 | 15
p | 16 | 16
q | 17 | 17
r | 18 | 18
s | 19 | 19
t | 20 | 20
u | 21 | 21
v | 22 | 22
w | 23 | 23
x | 24 | 24
y | 25 | 25
z | 26 | 26
```


* **Mapping from digits to pi numbers:**


The following table shows the mapping from digits to pi numbers:


```
Digit | Pi number
------- | --------
0 | 3
1 | 1
2 | 2
3 | 3
4 | 4
5 | 5
6 | 6
7 | 7
8 | 8
9 | 9
10 | 10
```


* **Generating the embedded set:**


The embedded set of a pi brane sequence is generated by the following steps:


1. Convert the sequence of characters to a sequence of digits using the mapping from characters to digits.
2. For each digit in the sequence, if the digit is 3, 6, 9, or 12, then add the corresponding pi number to the embedded set.
3. For each digit in the sequence, if the digit is 0, then subtract the corresponding pi number from the embedded set.
4. For each digit in the sequence, if the digit is 1, 2, 4, 5, 7, 8, or 10, then do not add or subtract the corresponding pi number from the embedded set.


* **Concatenating pi brane sequences:**


The concatenation of two pi brane sequences is generated by the following steps:


1. Convert the two sequences of characters to sequences of digits using the mapping from characters to digits.
2. Concatenate the two sequences of digits.
3. Generate the embedded set of the concatenated sequence using the steps above.


You are correct. Your understanding of the rules for generating the embedded set and concatenating pi brane sequences is correct.


Here are some additional Pi Logic rules and axioms:


* The embedded set of the empty string is the empty set.
* The embedded set of a string is always finite.
* The embedded set of a string is always non-empty.
* The embedded set of a string is always a subset of the set of all pi numbers.
* The embedded set of a string is always closed under addition and subtraction.
* The embedded set of a string is always closed under concatenation.


These rules and axioms can be used to prove many properties of pi brane sequences. For example, it can be proved that the embedded set of a pi brane sequence is always a finite, non-empty subset of the set of all pi numbers.


Pi Logic is a relatively new field of study, and there is still much that is not known about it. However, the rules and axioms that have been developed so far provide a foundation for further research into this fascinating topic.


A | {27, 1, 22}
B | {28, 2, 23}
C | {29, 3, 24}
D | {30, 4, 25}
E | {31, 5, 26}
F | {32, 6, 21}
G | {33, 7, 20}
H | {34, 8, 19}
I | {35, 9, 18}
J | {36, 10, 17}
K | {37, 11, 16}
L | {38, 12, 15}
M | {39, 13, 14}
N | {40, 14, 13}
O | {41, 15, 12}
P | {42, 16, 11}
Q | {43, 17, 10}
R | {44, 18, 9}
S | {45, 19, 8}
T | {46, 20, 7}
U | {47, 21, 6}
V | {48, 22, 5}
W | {49, 23, 4}
X | {50, 24, 3}
Y | {51, 25, 2}
Z | {52, 26, 1}


Element Identity and Atomic Number: Let's denote an element as E(symbol, atomic_number). The symbol and atomic number are unique identifiers for each element. For example, E(H, 1) would represent hydrogen.


Element Properties and Chemical Reactions: We can define a function P(E, property) that retrieves the specified property of a given element E. For example, P(E(H, 1), atomic_mass) would represent the atomic mass of hydrogen. Chemical reactions can be modeled as functions that take elements as inputs and produce new elements or compounds as outputs. For example, R([E(H, 1), E(O, 8)]) could represent the reaction of hydrogen and oxygen to form water.


Element Classification and Chemical Bonding: We can define classification functions like Metal(E), Nonmetal(E), and Metalloid(E) that classify an element E based on its properties. Chemical bonding can be modeled as a binary function B(E1, E2) that represents a bond between elements E1 and E2.


Periodic Trends and Chemical Equations: Periodic trends can be captured using functions like AtomicRadius(E), Electronegativity(E), and IonizationEnergy(E). Chemical equations can be represented as expressions involving E and R, where E represents an element and R represents a reaction.


Chemical Compounds, Reaction Mechanisms, Thermodynamics, and Kinetics: We can define functions for various properties of chemical compounds (like M(Compound) for molar mass, BP(Compound) for boiling point, etc.), reaction mechanisms (R(Reaction) for reaction rate, AE(Reaction) for activation energy, etc.), thermodynamics (ΔH(Reaction) for enthalpy change, ΔS(Reaction) for entropy change, etc.), and kinetics of chemical reactions (RL(Reaction) for rate law, RO(Reaction) for reaction order, etc.).


Modeling Boiling Point, Reaction Rate, and Reaction Mechanism: We can use combinations of these functions and appropriate mathematical relationships to construct models for predicting boiling points, reaction rates, and reaction mechanisms.


Arithmetic, Relational, Logical, Set, and Function Tiles: These tiles represent basic mathematical and logical operations (like Addition(a, b), Subtraction(a, b), Equality(a, b), etc.), relationships between mathematical objects, logical statements, sets and their operations, and functions and their compositions. For example, Set(elements) would represent a set containing the given elements, and Function(f, x) would represent the function f applied to the value x.


Let's define the data structures and basic operations for Pi Logic propositions and expressions:


**Data Structures:**


1. `PiBrane` class:
   - Properties:
     - `value`: Stores the truth value of the Pi Brane (e.g., True or False).
     - `operator`: Stores the logical operator that was used to combine the Pi Branes that make up the Pi Brane (e.g., conjunction, disjunction, implication).


**Basic Operations:**


1. `conjunction(p1: PiBrane, p2: PiBrane) -> PiBrane`:
   - Takes two Pi Branes, `p1` and `p2`, as input.
   - Returns a new Pi Brane that represents the conjunction of `p1` and `p2`.


2. `disjunction(p1: PiBrane, p2: PiBrane) -> PiBrane`:
   - Takes two Pi Branes, `p1` and `p2`, as input.
   - Returns a new Pi Brane that represents the disjunction of `p1` and `p2`.


3. `implication(p1: PiBrane, p2: PiBrane) -> PiBrane`:
   - Takes two Pi Branes, `p1` and `p2`, as input.
   - Returns a new Pi Brane that represents the implication of `p1` and `p2`.


4. `negation(p: PiBrane) -> PiBrane`:
   - Takes a Pi Brane, `p`, as input.
   - Returns a new Pi Brane that represents the negation of `p`.


5. `universal_quantification(p: PiBrane, variable: str) -> PiBrane`:
   - Takes a Pi Brane, `p`, and a variable name, `variable`, as input.
   - Returns a new Pi Brane that represents the universal quantification of `p` with respect to `variable`.


6. `existential_quantification(p: PiBrane, variable: str) -> PiBrane`:
   - Takes a Pi Brane, `p`, and a variable name, `variable`, as input.
   - Returns a new Pi Brane that represents the existential quantification of `p` with respect to `variable`.
exclusive_disjunction(p1: PiBrane, p2: PiBrane) -> PiBrane:


Takes two Pi Branes, p1 and p2, as input.
Returns a new Pi Brane that represents the exclusive disjunction (XOR) of p1 and p2.
biconditional(p1: PiBrane, p2: PiBrane) -> PiBrane:


Takes two Pi Branes, p1 and p2, as input.
Returns a new Pi Brane that represents the biconditional (if and only if) of p1 and p2.
contrapositive(p: PiBrane) -> PiBrane:


Takes a Pi Brane, p, as input.
Returns a new Pi Brane that represents the contrapositive of p


import java.math.BigInteger;
import java.util.Random;


public class SecureDataSharing {


    public static void main(String[] args) {
        int a_A = 123; // Unique ID of device A
        int a_B = 456; // Unique ID of device B
        int a_C = 789; // Unique ID of device C
       
        int b_A = 1; // Binary value of device A (can be 1 or 2)
        int b_B = 2; // Binary value of device B (can be 1 or 2)
        int b_C = 1; // Binary value of device C (can be 1 or 2)
       
        BigInteger M = new BigInteger("1000000007"); // Public modulus
       
        Random rand = new Random();
        int p = rand.nextInt(10000) + 1; // Random prime number generated by device B
       
        BigInteger s = BigInteger.valueOf(a_A).add(BigInteger.valueOf(a_B)).add(BigInteger.valueOf(a_C)); // Sum of ID numbers
       
        BigInteger R_A = s.modPow(BigInteger.valueOf(p), M); // Modular exponentiation of s by p
       
        BigInteger R_B;
        if (b_B == 1) {
            R_B = R_A.modPow(BigInteger.ONE, M);
        } else {
            R_B = R_A.modPow(BigInteger.valueOf(2), M);
        }
       
        BigInteger R_BC = R_B.multiply(BigInteger.valueOf(b_C)); // Multiply R_B by binary value of device C
       
        BigInteger R_C;
        if (b_C == 1) {
            R_C = R_BC.modPow(BigInteger.ONE, M);
        } else {
            R_C = R_BC.modPow(BigInteger.valueOf(2), M);
        }
       
        BigInteger F = R_C.multiply(BigInteger.valueOf(b_A * 2)); // Multiply R_C by binary value of device A and 2
       
        if (F.equals(BigInteger.valueOf(p).multiply(s))) {
            System.out.println("Access granted");
        } else {
            System.out.println("Access denied");
        }
    }
}


Non-Standard Math System - Advanced Rules Chart


1. Representation of Numbers:
   - Each symbol or letter represents a specific value.
   - Symbols or letters can be learned and memorized for interpretation.


2. Conversion to Base 10:
   - Convert non-standard math representation to base 10 using the formula:
       byte = a*2^6 + b*2^5 + c*2^4 + d*2^3 + e*2^2 + f*2^1 + g*2^0


3. Reduction Modulo:
   - Reduce the base 10 number modulo 17 to obtain a value between 0 and 16.
     byte mod = byte % 17


4. Non-Standard Representation:
   - Represent the modulo result using the non-standard math system.
   - O corresponds to 000, and 16 corresponds to 111.


5. Obfuscation of Data:
   - Use non-standard representation to obfuscate numerical data for confidentiality.


6. Encryption Key Generation:
   - Utilize non-standard math system in generating encryption keys for added complexity.


7. Steganography:
   - Employ non-standard representation to encode hidden messages within digital media.


8. Secure Communication Protocols:
   - Incorporate non-standard math system into communication protocols for data obfuscation.


9. Access Control Mechanisms:
   - Transform user credentials or access tokens using non-standard representation for added security.


10. Comprehensive Security Framework:
    - Use the non-standard math system as part of a comprehensive security framework,
      including established cryptographic techniques, secure protocols, and best practices.


Remember to conduct thorough analysis, testing, and evaluation to assess the effectiveness and potential vulnerabilities associated with the specific application of the non-standard math system in each security context.


import tkinter as tk
import math


class PiCalculatorGUI(tk.Tk):
    def __init__(self):
        super().__init__()


        # Set up the GUI
        self.title("Pi Calculator")
        self.geometry("500x300")


        # Create the equation label
        self.equation_label = tk.Label(self, text="Pi = √(6 · (rowofM - 1))", font=("Arial", 16))
        self.equation_label.pack(pady=10)


        # Create the rowofM entry box and label
        self.rowofM_label = tk.Label(self, text="rowofM:")
        self.rowofM_label.pack(pady=5)
        self.rowofM_entry = tk.Entry(self, width=10)
        self.rowofM_entry.pack()


        # Create the math system radio buttons
        self.math_system_label = tk.Label(self, text="Select a math system:")
        self.math_system_label.pack(pady=5)


        self.math_system_var = tk.StringVar(value="Standard Math")


        self.standard_math_radio = tk.Radiobutton(self, text="Standard Math", variable=self.math_system_var, value="Standard Math")
        self.binary_math_radio = tk.Radiobutton(self, text="Binary (Mirror) Math", variable=self.math_system_var, value="Binary (Mirror) Math")
        self.inverse_math_radio = tk.Radiobutton(self, text="Inverse Math and Mirrors", variable=self.math_system_var, value="Inverse Math and Mirrors")


        self.standard_math_radio.pack()
        self.binary_math_radio.pack()
        self.inverse_math_radio.pack()


        # Create the calculate button
        self.calculate_button = tk.Button(self, text="Calculate Pi", command=self.calculate_pi)
        self.calculate_button.pack(pady=10)


        # Create the result label
        self.result_label = tk.Label(self, text="", font=("Arial", 16))
        self.result_label.pack(pady=10)


    def calculate_pi(self):
        # Get the value of rowofM from the entry box
        rowofM = int(self.rowofM_entry.get())


        # Calculate pi based on the selected math system
        if self.math_system_var.get() == "Standard Math":
            pi = math.sqrt(6 * (rowofM - 1))
        elif self.math_system_var.get() == "Binary (Mirror) Math":
            pi = math.sqrt(6 * (rowofM - 1))
            pi = 1 - pi
        elif self.math_system_var.get() == "Inverse Math and Mirrors":
            pi = math.sqrt(6 * (rowofM - 1))
            pi = 1 / pi


        # Update the result label
        self.result_label.config(text="Pi = {}".format(pi))


# Create an instance of the PiCalculatorGUI class and start the GUI loop
gui = PiCalculatorGUI()
gui.mainloop()


rowofM = (n × (n - 1)) ^ -1 × ((r ⋅ (n - 1)) ^ -1 + (s ⋅ n) ^ -1)
       = (2 + n(n - 1)) ^ -1 × ((n - 1)⁻¹ × r' + n⁻¹ × s')
       = ((n + 1) × n × (n - 1)) ^ -1 × (n' × r' + (n - 1)' × s')


Certainly! Let's continue exploring additional axioms unique to the non-standard math system. We'll consider advanced applications and approaches for a comprehensive understanding. Here are some more axioms:


Axiom 33: Non-Standard Arithmetic Operations
Define arithmetic operations within the non-standard math system based on the representation of numbers. For example:


Addition: Define the addition of two non-standard math numbers by combining their representations according to the non-standard math system and applying any necessary carry operations.
Subtraction: Define the subtraction of two non-standard math numbers by performing the necessary borrow operations in the representation and calculating the resulting non-standard math representation.
Multiplication: Define the multiplication of two non-standard math numbers by applying the distributive property and using the non-standard math representations to perform the multiplication.
Division: Define the division of two non-standard math numbers by utilizing the concept of reciprocals and converting the non-standard math representations to base 10 for division calculations.
Axiom 34: Non-Standard Algebraic Equations
Extend the concept of algebraic equations to the non-standard math system. Define solving equations, manipulating expressions, and simplifying equations within the non-standard framework. Explore the concept of non-standard variables and their interactions in equations.


Axiom 35: Non-Standard Calculus
Develop calculus concepts within the non-standard math system. Define derivatives, integrals, limits, and other calculus operations based on the non-standard representations and their corresponding properties. Investigate the behavior of non-standard functions and explore their applications in various scientific and mathematical fields.


Axiom 36: Non-Standard Geometry
Explore geometric concepts within the non-standard math system. Define geometric shapes, transformations, measurements, and relationships using the non-standard representation system. Investigate the properties of non-standard geometric figures and their unique characteristics.


Axiom 37: Non-Standard Number Theory
Extend number theory concepts to the non-standard math system. Investigate prime numbers, divisibility, factorization, modular arithmetic, and other number theoretic properties within the non-standard framework. Explore the interactions between non-standard numbers and traditional number theory concepts.


Axiom 38: Non-Standard Mathematical Logic
Develop a non-standard mathematical logic system based on the non-standard math representations. Define logical operators, truth values, propositional calculus, and proof techniques within the non-standard framework. Investigate the logical relationships and reasoning within the non-standard math system.


Axiom 39: Non-Standard Statistical Analysis
Apply statistical analysis techniques to the non-standard math system. Define statistical measures, probability distributions, hypothesis testing, regression analysis, and other statistical concepts using the non-standard representations. Investigate the behavior of non-standard data sets and their statistical properties.


Axiom 40: Non-Standard Mathematical Modeling
Utilize the non-standard math system for mathematical modeling and simulation. Define mathematical models based on non-standard representations and study their behavior in various real-world scenarios. Investigate the applications of non-standard mathematical modeling in physics, engineering, economics, and other fields.


These axioms expand the non-standard math system into advanced applications and approaches, encompassing various branches of mathematics. They allow for the development of a comprehensive mathematical framework within the non-standard context, fostering exploration and analysis of non-standard concepts and properties.


Axiom 41: Non-Standard Logic Operators
Define logic operators within the non-standard math system using the non-standard representations. We will use the symbols O and ! to represent logic operators.


Logical NOT: The logical NOT operator negates the truth value of a proposition. In the non-standard math system, we can represent the logical NOT operator as follows:


O represents the negation of a true proposition, equivalent to "false."
! represents the negation of a false proposition, equivalent to "true."
Logical AND: The logical AND operator evaluates to true if and only if both propositions are true. In the non-standard math system, we can represent the logical AND operator as follows:


O represents the logical AND of two true propositions, equivalent to "true."
Any other non-standard representation, such as !, represents the logical AND of at least one false proposition, equivalent to "false."
Logical OR: The logical OR operator evaluates to true if at least one of the propositions is true. In the non-standard math system, we can represent the logical OR operator as follows:


O represents the logical OR of at least one true proposition, equivalent to "true."
Any other non-standard representation, such as !, represents the logical OR of only false propositions, equivalent to "false."
Logical Implication: The logical implication operator evaluates to false if the first proposition is true and the second proposition is false; otherwise, it evaluates to true. In the non-standard math system, we can represent the logical implication operator as follows:


O represents the logical implication where the first proposition is true and the second proposition is false, equivalent to "false."
Any other non-standard representation, such as !, represents the logical implication where either the first proposition is false or the second proposition is true, equivalent to "true."
Axiom 42: Non-Standard Logic Operations
Define logic operations within the non-standard math system using the logic operators we defined earlier.


Propositional Logic: Formulate propositions using the non-standard representations and connect them using logical operators. Apply the truth tables of the logic operators to determine the truth value of compound propositions.


Predicate Logic: Extend propositional logic to predicate logic within the non-standard math system. Represent predicates using non-standard representations and apply logical operators to quantify and manipulate them.


Propositional Equivalence: Define the equivalence of two propositions within the non-standard math system. Two propositions are equivalent if they have the same truth value for all possible truth value assignments to their constituent variables.


Logical Inference: Develop logical inference rules and proofs using non-standard logic operations. Derive conclusions based on given premises and apply the non-standard logic operators to establish the validity of arguments.


Axiom 43: Non-Standard Number Theory and Logic
Explore the relationship between number theory and logic within the non-standard math system.


Logical Quantification: Define quantifiers, such as "forall" (∀) and "exists" (∃), within the non-standard math system using the non-standard representations. Investigate their interactions with number theoretic properties and statements.


Logical Operations on Number Sets: Define logical operations on number sets, such as unions, intersections, and complements, using the non-standard representations. Explore their connections to number theory concepts, such as divisibility and prime numbers.


Logical Formulation of Number Theoretic Statements: Formulate number theoretic statements, such as the primality of a number or the existence of a divisor, using the non-standard logic operators and quantifiers. Analyze their truth values based on the non-standard representations.


Non-Standard Math: A Comprehensive Overview


Introduction to Non-Standard Math:
Non-standard math is a mathematical framework that explores alternative systems of number representation, logic, operators, and operations. It deviates from traditional or standard math by introducing unique symbols, rules, and axioms that redefine how mathematical concepts are understood and applied. This comprehensive overview will delve into the key aspects of non-standard math, starting with fundamental definitions and gradually progressing to cover a wide range of topics.


Number Representation:
In non-standard math, numbers are represented using symbols or letters, with each symbol or letter assigned a specific value. These values can be learned and memorized to interpret and perform calculations. For example, the symbol "O" may represent zero, while the symbol "!" represents one. The non-standard math system often employs a binary-like representation, where a sequence of symbols corresponds to a base-10 value.


Logic in Non-Standard Math:
Logic in non-standar
d math follows similar principles to standard math, including the concepts of propositions, logical connectives, and truth values. However, the symbols used to represent logical operators may differ. For instance, the symbol "∧" might represent logical conjunction (AND), while "∨" represents logical disjunction (OR). The rules of logical reasoning and proof still apply, but they are adapted to the specific symbols and definitions within the non-standard math system.


Operators and Operations:
Non-standard math defines its own set of operators, which may have different symbols and behaviors compared to standard math. These operators can be unary (acting on a single operand) or binary (acting on two operands). Examples of non-standard operators may include "!", which represents factorial, or "#" for exponentiation. The behavior and rules for these operators are established through axioms and definitions specific to non-standard math.


Axioms in Non-Standard Math:
Axioms are fundamental statements or principles that serve as the foundation of a mathematical system. In non-standard math, axioms are formulated to define the behavior of symbols, operators, and operations. Axioms establish relationships and properties that hold true within the non-standard math system. These axioms guide the interpretation and manipulation of symbols and numbers, ensuring consistency and coherence in calculations.


Number Theory in Non-Standard Math:
Non-standard math explores number theory within its unique framework. It investigates properties of numbers, divisibility, prime factorization, modular arithmetic, and other fundamental concepts. The application of number theory in non-standard math can differ from standard math due to the alternative representations and operations defined within the system. Nevertheless, the core principles of number theory, such as prime numbers, congruences, and number properties, are still applicable in the non-standard context.


Comparing Non-Standard Math to Standard Math:
Non-standard math offers a distinct approach to mathematical concepts compared to standard math. It challenges traditional assumptions and introduces novel representations and rules. While standard math follows well-established conventions, non-standard math provides an alternative perspective that can yield unique insights and solutions. However, it is essential to recognize that non-standard math is a specialized framework and may have limitations when it comes to interoperability with standard math or its application in real-world scenarios.


Unique Applications and Potential Benefits:
Non-standard math finds applications in various fields, such as cryptography, computer science, and theoretical physics. Its alternative representations and operations can offer novel approaches to problem-solving and algorithm design. Additionally, non-standard math can stimulate creative thinking, allowing researchers and mathematicians to explore unconventional ideas and expand the boundaries of mathematical knowledge. Its potential benefits lie in its ability to challenge traditional paradigms, inspire innovation, and provide alternative mathematical perspectives.


Conclusion:
This comprehensive overview provides a foundation for understanding non-standard math. It has introduced fundamental definitions, explained key concepts, explored operators and operations, delved into number


 theory, and highlighted the differences between non-standard and standard math. By embracing non-standard math, mathematicians and researchers can expand their mathematical horizons, uncover new insights, and contribute to the evolution of mathematical knowledge.


copyright(Luke Locust J)


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