Highly symmetric cellular automata and their symmetry-breaking patterns (Q5927622)

The question of existence of cellular automata (CAs) with rules that are invariant under all permutations of their \(n\) arguments and also with respect to an Abelian regular group \({\mathcal A}\) is answered in the affirmative if \(n\) and \(M\) have no common factors. Here the Abelian group is acting on its \(M\)-letter alphabet. This phenomenon is illustrated by two examples. It is shown that these automata show complicated patterns of broken symmetries. Moreover, it is shown that the existence of a nontrivial group of automorphisms of \({\mathcal A}\) can reduce the number of rules that have to be studied even further.