Abysys

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

Inspired by User:Miui's art-lang seeds: Abysys, A bliss-pit, and also Grill Tag. An experiment in embedding tag-systems inside groups, ideally abelian. Is this meaningful, or even 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 to form r, the resultant.

i ∘ h = r

The resultant trigram modifies the state string as follows:

• Top line unbroken: delete head
• Middle line unbroken: append r ∘ r
• Lower line also unbroken: append r ∘ i

Instructions are read in a loop. Execution halts when the data string is empty since two symbols are required for composition.

Computational class and relation to other tag systems

This group contains a subset of symbols that can directly implement Grill Tag. Translation is given in trigrams as well as equivalent ASCII options for convenience, since in practice the alphabet is any Unicode codepoint Mod 8.

	Grill Tag	Trigram	ASCII
command	10	☶ (Mountain)	6 . ^
	11	☳ (Thunder)	3 + #
data	0	☷ (Earth)	7 o O
	1	☵ (Water)	5 = -

Cyclic Tag

Attempts to map Cyclic Tag rules into Abysys highlight a fundamental friction between tag systems and group theory. Tag systems rely on asymmetric (re)writing, whereas C₂³ is an elementary Abelian group where every element is self-inverse. Because bitwise XOR defines the C₂³ group over 3-bit strings, we can force a Grill Tag evaluator which uses XOR, somewhat theatrically, but only by isolating and operating within an asymmetric subset of C₂³. Translating general Cyclic Tag is constrained by the uniform symmetry of C₂³.

	Cyclic Tag	Trigram(s)	ASCII
command	0
	1
	0;
	1;
data	0	☷ (Earth)	7 o O
	1	☵ (Water)	5 = -

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&6)//6)))      # append head?
        ip = (ip + 1) % len(commands)


if __name__ == '__main__':
    main()