Groups and Monoids of Cellular Automata

Abstract

We prove that the group of reversible cellular automata (RCA), on any alphabet A, contains a perfect subgroup generated by six involutions which contains an isomorphic copy of every finitely-generated group of RCA on any alphabet B. This result follows from a case study of groups of RCA generated by symbol permutations and partial shifts with respect to a fixed Cartesian product decomposition of the alphabet. For prime alphabets, we show that this group is virtually cyclic, and that for composite alphabets it is non-amenable. For alphabet size four, it is a linear group. For non-prime non-four alphabets, it contains copies of all finitelygenerated groups of RCA. We also obtain that RCA of biradius one on all large enough alphabets generate copies of all finitely-generated groups of RCA. We ask a long list of questions.

A note on subgroups of automorphism groups of full shifts

“…Another one can be found by a direct application of the ideas of [1,5], namely closure under direct sums, and we show this in Theorem 1. Our main result is that this set of groups is also closed under free products, Theorem 2, which answers a question I myself asked in [9].…”

Cellular automata and finite groups

“…Let CA(G; A) be the set of all cellular automata over A G ; by [6,Corollary 1.4.11], this set equipped with the composition of functions is a monoid. Although results on monoids of CA have appeared in the literature before (see [4,14,19]), the algebraic structure of CA(G; A) remains basically unknown. In particular, the study of CA(G; A), when G and A are both finite, has been generally disregarded (except for the case when G = Z n , which is the study of one-dimensional CA on periodic points).…”

Universal groups of cellular automata

“…For any countable subshift X, Aut(X) is elementarily amenable by [48], thus cannot contain a free group, thus cannot contain every finitely generated subgroup of Aut(A Z ) for any non-trivial alphabet A. That paper is unpublished, but the case of countable sofics can be obtained by adapting [42,Proposition 2].…”