Universal groups of cellular automata

Abstract

We prove that the group of reversible cellular automata (RCA), on any alphabet A, contains a subgroup generated by three 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 (equivalently, partitioned cellular automata) 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 finitely generated groups of RCA. We also prove this property for the group generated by RCA of biradius one on any full shift with large enough alphabet, and also for some perfect finitely generated groups of RCA.

Conjugacy of reversible cellular automata and one-head machines

“…Groups of reversible cellular automata have been studied at least in [65,11,2,46,13], and more recently in at least [69,27,72,74,75]. Automorphism groups of other subshifts have been of much recent interest [62,77,71,19,15,18,23,22,17,5,33].…”

“…In this paper, we show that reversible cellular automata give rise to new f.g. groups with undecidable conjugacy problem. A particularly simple example (obtained from combining our results with [74]) is described in Figure 1 (see Corollary 4 for details). One could quite easily imagine bumping into such groups without any a priori interest in their computability properties -this is what happened.…”

“…The concept of f.g.-universality, on top of which we build here, arose in [74] from studying the groups of RCA that can be built from what we considered the simplest imagineable building blocks among elements of Aut(A Z ), namely partial shifts and symbol permutations, see Figure 1. These automorphisms are classical in the theory of cellular automata: a composition of a partial shift with a symbol permutation is often called a partitioned CA.…”

Universal CA groups with few generators

“…We let B with |B| ≥ 2 be arbitrary and C = {0, 1} and use the alphabet A = B × C, with B Z the "top track" and C Z the "bottom track". By [3], there exists a finitely-generated group H of cellular automata containing a copy of every finitely-generated group of cellular automata. By Lemma 7 in [3] (more precisely, its proof), for any large enough ℓ and unbordered word |w| = ℓ, if a group G ≤ RCA(B × C) contains π| [w]i and π| [ww]i for all π ∈ Alt({0, 1} ℓ ) and all i ∈ Z, then G contains a copy of H. The notation π| [u]i is as in Definition 2 of [3], and means that we apply π on the second track if and only if u appears on the first track, with offset i.…”

“…Thinking of x ∈ (B ′ × B × C) Z as having three binary tracks, and writing σ 0 and σ 1 for the shifts on the first two tracks, it is easy to see that σ −1 0 × σ 1 is the composition of two involutions, say σ −1 0 × σ 1 = a • b. In the proof of universality in [3], the shift on the first (B-)track is only used to construct the generators of an arbitrary f.g. group, but total sum of shifts is 0 in the elements giving the embedding. Thus, G = a, b, f 0 , where f 0 is as above but ignores the B ′ -track, is clearly f.g.-universal, and a quotient of G ′ .…”