Ranks of finite semigroups of one-dimensional cellular automata

TL;DR

This paper investigates the minimal number of cellular automata needed to generate the entire semigroup of cellular automata over finite cyclic groups, providing exact ranks for specific cases and bounds for others.

Contribution

It introduces a finite semigroup perspective to cellular automata, determining the rank of the automata semigroup for certain cyclic groups and establishing bounds in general.

Findings

Exact rank for prime, power of two, and mixed cyclic groups.

Upper and lower bounds for the semigroup rank in general cases.

Connections established between cellular automata and semigroup theory.

Abstract

Since first introduced by John von Neumann, the notion of cellular automaton has grown into a key concept in computer science, physics and theoretical biology. In its classical setting, a cellular automaton is a transformation of the set of all configurations of a regular grid such that the image of any particular cell of the grid is determined by a fixed local function that only depends on a fixed finite neighbourhood. In recent years, with the introduction of a generalised definition in terms of transformations of the form $\tau : A^G \to A^G$ (where $G$ is any group and $A$ is any set), the theory of cellular automata has been greatly enriched by its connections with group theory and topology. In this paper, we begin the finite semigroup theoretic study of cellular automata by investigating the rank (i.e. the cardinality of a smallest generating set) of the semigroup $\text{CA}(\mathbb{Z}_n; A)$ consisting of all cellular automata over the cyclic group $\mathbb{Z}_n$ and a finite set $A$. In particular, we determine this rank when $n$ is equal to $p$, $2^k$ or $2^kp$, for any odd prime $p$ and $k \geq 1$, and we give upper and lower bounds for the general case.