Universal binary CA commutative with XOR

[mk248269]

Hi!

I wonder if there exists a universal binary cellular automaton, whose transition function is commutative with XOR, i.e. if I do two experiments:
1) first flip the states of all the cells in the CA and then perform one step of simulation
2) first perform one step of simulation and then flip the states of all the cells in the CA
then I get equal results, for all initial configurations. And moreover it'd be best if such a CA could be 1-dimensional with range 1, but this is not a must (here there would be only 16 CAs to consider, but I don't know if any of them is capable of universal computations).

I'd guess that such a CA does not exist, but I've decided to ask you guys. Does anyone know anything about such a CA? Or maybe anyone knows a proof, why my requirements are not satisfiable?

Thanks!

Marian Marek Kędzierski <><

[Wojowu]

Such automata are called self-complementary. There is 16 such 1D range 1 automata, and almost surely none of them is universal. Great example of self-complementary automaton is B3678/S34678, called Day & Night. Such 2D automata almost certainly contain some universal ones. It can be hard to prove, but if so many simple systems where proven universal, it wouldn't be impossible.