A rule R is reversible if there exists a rule R^-1 such that for all a, b, c, d, e, f, g we see the following evolution (where . stands for "don't care"):

    a  b  c  d  e  f  g

       Apply rule R

    .  b' c' d' e' f' .
       
       Apply rule R^-1

    .  .  c  d  e  .  .

In equations, it means there must exist some R^-1 such that for all a, b, c, d, e, f, g the following holds:

    c = R^-1(R(a, b, c), R(b, c, d), R(c, d, e))
    d = R^-1(R(b, c, d), R(c, d, e), R(d, e, f))
    e = R^-1(R(c, d, e), R(d, e, f), R(e, f, g))

Writing a quick program that checks all possible combinations of R and R^-1 (https://play.rust-lang.org/?version=stable&mode=debug&editio...) we find the following inverse pairs:

   0x33 <-> 0x33
   0x55 <-> 0x0f
   0xcc <-> 0xcc
   0xf0 <-> 0xaa

That is, the following two rules are self-inverses (they're the NOT and IDENTITY gates on the center cell respectively):

    111 110 101 100 011 010 001 000
     0   0   1   1   0   0   1   1
     1   1   0   0   1   1   0   0

We have the left <-> right moving pair you identified:

    111 110 101 100 011 010 001 000
     1   0   1   0   1   0   1   0

    111 110 101 100 011 010 001 000
     1   1   1   1   0   0   0   0

And there's just one more, which is the same as the above but it also inverts the output (move left and invert is the inverse of move right and invert):

    111 110 101 100 011 010 001 000
     0   1   0   1   0   1   0   1

    111 110 101 100 011 010 001 000
     0   0   0   0   1   1   1   1

> My question is how can we test which of the 256 rules are reversible and how do they pair up?

I wondered this too. It feel like a generalization of the well-studied deconvolution problem, but thankfully without noise.

I haven't had my coffee yet, but some quick googling any thinking didn't deliver an answer, so I'd just try enumerating all the rules and seeing which ones invert each other.

Or, take all permutations of five bits, and find the central three bits in the next generation. If-and-only-if all patterns resulting in the three bits have the same central bit in the original pattern, then the rule is invertible.

Check out the Hisense Canvas.

I've sometimes thought it might be reasonable to think of a single cell in the system as an 'automaton' which might make it somewhat ok to call the whole collection of cells 'automata'.

I accept that there's two levels here though and sometimes people refer to the system as a whole as being a single automaton even if there's loads of cells.

I was considering posting this, but then I realised the hypocrisy in my use of the word "data" :)

The post describes multiple automata though?

  >multiple automata

i wrote something similar a real long time ago, but it was a pig, super slow. So i upgraded to winpro or whatever and had copilot figure out what was slowing it down, and it was a simple fix, and i fleshed out the UI a bit. I forget where i first saw the rules, but when i went and fixed it up i used the wolframalpha reference[0]

Adjust the screen rez. i am not a software developer and i put stuff on github as additional backups.

[0]https://mathworld.wolfram.com/ElementaryCellularAutomaton.ht...

my code: https://github.com/genewitch/opensource/blob/master/elementa... it uses pygame you can do python -m venv cellular; scripts\activate.[bat|ps1|sh]; pip install pygame; python elementalautomata.py

the rules are the same numbers as OP used, so given the same seeds the monochrome output should be identical (this is a deterministic automaton) I forget which rules were my favorite, but i constantly test 110 and 73 (73 is psuedorandom but doesn't look like it is, it looks like it repeats.) - my code checks for loops, but it defaults to 1 screen-fulls of checking, you can press H to increase this by 1 screen-full per press. this will slow my code down to a crawl, but you can find loops in the longer period automata.

i was going to do a version similar in godot but i got distracted by ipv6 routing.

edit:

the part you'd need to adapt to add color in my code is just this one part, i think:

        for pixel in range(1, w):

            if lineOut[pixel] == 1:

                world.set_at((pixel, h-1), (0, 0, 0))

            else:
                world.set_at((pixel, h-1), (255, 255, 255))