User:Salpynx/Abysys

Abysys(?) is a tag-system comprising 8 symbols in the abelian group C₂xC₂xC₂.

Inspired by Abysys, A bliss-pit, Grill Tag, and searching for a mechanism to embed a tag-system inside a group, ideally abelian. Is this meaningful, or possible? Turns out: maybe and yes.

The system operates through the composition of symbols in two strings: instruction, and state, each a sequence of Trigrams within the C₂³ group.

Group table

☷ ☶ ☵ ☴ ☳ ☲ ☱ ☰ 
☶ ☷ ☴ ☵ ☲ ☳ ☰ ☱ 
☵ ☴ ☷ ☶ ☱ ☰ ☳ ☲ 
☴ ☵ ☶ ☷ ☰ ☱ ☲ ☳ 
☳ ☲ ☱ ☰ ☷ ☶ ☵ ☴ 
☲ ☳ ☰ ☱ ☶ ☷ ☴ ☵ 
☱ ☰ ☳ ☲ ☵ ☴ ☷ ☶ 
☰ ☱ ☲ ☳ ☴ ☵ ☶ ☷ 

☷ ((U+2637) Earth) is the identity element: ground. The table above lacks separate headings, but the first row and column represent the identity, so the result of composing two symbols is found by finding the intersection of two symbols by row and column. Order of operation does not matter as this group is abelian, which means the table is symmetrical about the diagonal.

Execution

Each instruction symbol, i, is read in order and composed with the current head, h, of the state string. Instructions are read in a loop. Execution halts when the data string is empty since two symbols are required for composition.

i ∘ h = r

• If the top line is unbroken, the head (leftmost) symbol of the state string is deleted
• If the middle line is unbroken, append (add to right) r ∘ r (i.e. ☷ (Earth)) to the state-string (this anchors the following appendant)
• If the lower line is also unbroken (anchored by the middle line), append r ∘ i (i.e. h) to the state-string

Computational class and relation to other tag systems

This group contains a subset of symbols that can directly implement Grill Tag.

Converting Grill Tag:

• 10 (command): Use ☶ (Mountain)
• 11 (command): Use ☳ (Thunder)
• 0 (data): Use ☷ (Earth)
• 1 (data): Use ☵ (Water)

Examples

Test Grill_Tag#Example, demonstrating equivalent output in parentheses: 1011101110 => ☶☳☶☳☶, data: 110110 => ☵☵☷☵☵☷

./abysys.py ☶☳☶☳☶ ☵☵☷☵☵☷
☵☵☷☵☵☷		(110110)
☵☷☵☵☷☷		(101100)
☵☷☵☵☷☷☷☵	(10110001)
☷☵☵☷☷☷☵☷	(01100010)
☷☵☵☷☷☷☵☷	(01100010)
☵☵☷☷☷☵☷		(1100010)
☵☷☷☷☵☷☷		(1000100)
☵☷☷☷☵☷☷☷☵	(100010001)
☷☷☷☵☷☷☷☵☷	(000100010)
☷☷☷☵☷☷☷☵☷	(000100010)
☷☷☵☷☷☷☵☷	(00100010)
☷☵☷☷☷☵☷		(0100010)
☷☵☷☷☷☵☷		(0100010)
☵☷☷☷☵☷		(100010)
☵☷☷☷☵☷☷☵	(10001001)
☷☷☷☵☷☷☵☷	(00010010)
☷☷☵☷☷☵☷		(0010010)
☷☷☵☷☷☵☷		(0010010)
☷☵☷☷☵☷		(010010)
☷☵☷☷☵☷		(010010)
☵☷☷☵☷		(10010)
☷☷☵☷☷		(00100)
☷☷☵☷☷		(00100)
☷☵☷☷		(0100)
☷☵☷☷		(0100)
☵☷☷		(100)
☷☷☷		(000)
☷☷☷		(000)
☷☷		(00)
☷☷		(00)
☷		(0)

Interpreter

#!/usr/bin/env python3
import sys

EARTH = 0x2637  # Unicode Trigram For Earth
WATER = EARTH - 2

# Convert to Grill tag data string:
def gt(s):
    return s.replace(chr(EARTH), '0').replace(chr(WATER), '1')

# C2xC2xC2 group operator: XOR
def xor(a, b):
    return (EARTH - ord(a)) ^ (EARTH - ord(b))


def main():
    commands, data = sys.argv[1:]
    ip = 0
    while data:
        print(f'{data}\t({gt(data)})')
        cmd, head = commands[ip], data[0]
        r = xor(cmd, head)
        data = (data[r&1:] +          # delete head?
               chr(EARTH)*(r>>1&1) +  # append earth?
               head*(r>5))            # append head?
        ip = (ip + 1) % len(commands)


if __name__ == '__main__':
    main()