Highly symmetric cellular automata and their symmetry-breaking patterns

A highly symmetric cellular automaton (in one dimension) is defined by two properties: (i) its local rule is a completely symmetric function of the n cells of its surrounding and (ii) it is invariant with respect to a regular, Abelian group acting on its M-letter alphabet, which can then be identified with this group. The motivation for studying such systems is provided by the unsolved problem of partial symmetry breaking in spin models with such an Abelian symmetry. It is shown that the symmetries (i) and (ii) do not contradict each other if and only if n and M have no common factor. As examples, the four-letter alphabets, which may be identified either with the cyclic group C(4) or with the Klein group K(4)≅S(2)⊗S(2) are studied for the standard n=3 surrounding. It is shown that these automata show complicated patterns of broken symmetries. These give information on the corresponding spin models, the chiral 4-state clock model and the (symmetric and general) Ashkin–Teller models, respectively.