We use cookies to ensure that we give you the best experience on our website. You can change your cookie settings at any time. Otherwise, we'll assume you're OK to continue.

Durham University

Computer Science


Publication details for Dr Maximilien Gadouleau

Castillo-Ramirez, Alonso & Gadouleau, Maximilien (2016), On Finite Monoids of Cellular Automata, in Cook, Matthew & Neary, Turlough eds, Lecture Notes in Computer Science, 9664 International workshop on cellular automata and discrete complex systems. Zurich, Switzerland, Springer, 90-104.

Author(s) from Durham


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⟩.