On the minimal memory set of cellular automata

TL;DR

This paper investigates the properties of minimal memory sets in cellular automata, establishing conditions under which the minimal set equals the original memory set or differs by a single element, advancing theoretical understanding.

Contribution

It provides the first general theoretical results linking minimal memory sets with generating patterns and local maps in cellular automata.

Findings

If |S| ≥ 2 and |P| is not a multiple of |A|, then the minimal memory set is S.

When |P| = |A| and |S| ≥ 3, the minimal set is either S or S minus one element.

These results deepen the theoretical understanding of memory sets in cellular automata.

Abstract

For a group $G$ and a finite set $A$, a cellular automaton (CA) is a transformation $\tau : A^G \to A^G$ defined via a finite memory set $S \subseteq G$ and a local map $\mu : A^S \to A$. Although memory sets are not unique, every CA admits a unique minimal memory set, which consists on all the essential elements of $S$ that affect the behavior of the local map. In this paper, we study the links between the minimal memory set and the generating patterns $\mathcal{P}$ of $\mu$; these are the patterns in $A^S$ that are not fixed when the cellular automaton is applied. In particular, we show that when $\vert S \vert \geq 2$ and $\vert \mathcal{P} \vert$ is not a multiple of $\vert A \vert$, then the minimal memory set must be $S$ itself. Moreover, when $\vert \mathcal{P} \vert = \vert A \vert$, $\vert S \vert \geq 3$, and the restriction of $\mu$ to these patterns is well-behaved, then the minimal memory set must be $S$ or $S \setminus \{s\}$, for some $s \in S \setminus \{e\}$. These are some of the first general theoretical results on the minimal memory set of a cellular automaton.