On Finite Monoids of Cellular Automata

Castillo-Ramirez, Alonso; Gadouleau, Maximilien

Abstract

For any group G and set A, a cellular automaton over G and A is a transformation τ:AG→AGτ:AG→AG defined via a finite neighbourhood S⊆GS⊆G (called a memory set of ττ) and a local function μ:AS→Aμ:AS→A. In this paper, we assume that G and A are both finite and study various algebraic properties of the finite monoid CA(G,A)CA(G,A) consisting of all cellular automata over G and A. Let ICA(G;A)ICA(G;A) be the group of invertible cellular automata over G and A. In the first part, using information on the conjugacy classes of subgroups of G, we give a detailed description of the structure of ICA(G;A)ICA(G;A) in terms of direct and wreath products. In the second part, we study generating sets of CA(G;A)CA(G;A). In particular, we prove that CA(G,A)CA(G,A) cannot be generated by cellular automata with small memory set, and, when G is finite abelian, we determine the minimal size of a set V⊆CA(G;A)V⊆CA(G;A) such that CA(G;A)=⟨ICA(G;A)∪V⟩CA(G;A)=⟨ICA(G;A)∪V⟩.

Citation

Castillo-Ramirez, A., & Gadouleau, M. (2016, June). On Finite Monoids of Cellular Automata. Presented at International workshop on cellular automata and discrete complex systems, Zurich, Switzerland

• Presentation Conference Type: Conference Paper (published)
• Conference Name: International workshop on cellular automata and discrete complex systems
• Start Date: Jun 15, 2016
• End Date: Jun 17, 2016
• Acceptance Date: Mar 20, 2016
• Online Publication Date: Jun 2, 2016
• Publication Date: Jun 2, 2016
• Deposit Date: Nov 28, 2016
• Publicly Available Date: Jun 2, 2017
• Print ISSN: 0302-9743
• Pages: 90-104
• Series Title: Lecture notes in computer science
• Series Number: 9664
• Series ISSN: 0302-9743,1611-3349
• Book Title: Cellular automata and discrete complex systems : 22nd IFIP WG 1.5 International Workshop, AUTOMATA 2016, Zurich, Switzerland, June 15-17, 2016. Proceedings.
• ISBN: 9783319392998
• DOI: https://doi.org/10.1007/978-3-319-39300-1_8
• Public URL: https://durham-repository.worktribe.com/output/1149278