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Natural Computing

Cellular automata and finite groups

  • 30-09-2017
Published in:
Authors
Alonso Castillo-Ramirez
Maximilien Gadouleau
Published in
Natural Computing | Issue 3/2019

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Abstract

For a finite group G and a finite set A, we study various algebraic aspects of cellular automata over the configuration space \(A^G\). In this situation, the set \(\mathrm {CA}(G;A)\) of all cellular automata over \(A^G\) is a finite monoid whose basic algebraic properties had remained unknown. First, we investigate the structure of the group of units \(\mathrm {ICA}(G;A)\) of \(\mathrm {CA}(G;A)\). We obtain a decomposition of \(\mathrm {ICA}(G;A)\) into a direct product of wreath products of groups that depends on the numbers \(\alpha _{[H]}\) of periodic configurations for conjugacy classes [H] of subgroups of G. We show how the numbers \(\alpha _{[H]}\) may be computed using the Möbius function of the subgroup lattice of G, and we use this to improve the lower bound recently found by Gao, Jackson and Seward on the number of aperiodic configurations of \(A^G\). Furthermore, we study generating sets of \(\mathrm {CA}(G;A)\); in particular, we prove that \(\mathrm {CA}(G;A)\) cannot be generated by cellular automata with small memory set, and, when all subgroups of G are normal, we determine the relative rank of \(\mathrm {ICA}(G;A)\) on \(\mathrm {CA}(G;A)\), i.e. the minimal size of a set \(V \subseteq \mathrm {CA}(G;A)\) such that \(\mathrm {CA}(G;A) = \langle \mathrm {ICA}(G;A) \cup V \rangle\).

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previous article Remarks on the cellular automaton global synchronisation problem: deterministic versus stochastic models
next article Cellular automata on group sets and the uniform Curtis–Hedlund–Lyndon theorem
Title
Cellular automata and finite groups
Authors
Alonso Castillo-Ramirez
Maximilien Gadouleau
Publication date
30-09-2017
Publisher
Springer Netherlands
Published in
Natural Computing / Issue 3/2019
Print ISSN: 1567-7818
Electronic ISSN: 1572-9796
DOI
https://doi.org/10.1007/s11047-017-9640-3
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