On the minimal memory set of cellular automata
Alonso Castillo-Ramirez, Eduardo Veliz-Quintero
TL;DR
This paper investigates the minimal memory set (MMS) of a cellular automaton (CA) defined on a group $G$ with finite alphabet $A$, focusing on how the MMS relates to the generating patterns $\mathcal{P}$ of the local map $\mu$ through a generating pair $(\mathcal{P}, f)$. It proves precise conditions under which the MMS must equal the memory set $S$ or a variation $S\setminus\{s\}$, depending on the size and structure of $\mathcal{P}$, the well-behavedness of $f$, and the cardinalities $|A|$ and $|S|$, including a critical case where $|\mathcal{P}|=|A|^{|S|}-|A|^{|S|-1}$ allowing MMS to be any proper subset. The results connect the MMS to the generating patterns via the subshift of finite type $X_{\mathcal{P}}$ and establish how the fixed-point set of the CA aligns with $X_{\mathcal{P}}$, providing both theoretical insight and practical guidance for computing MMS. Collectively, these findings constitute foundational theoretical progress on MMS in CA and offer a pathway to more efficient analyses of pattern-generated automata.
Abstract
For a group $G$ and a finite set $A$, a cellular automaton (CA) is a transformation $τ: A^G \to A^G$ defined via a finite memory set $S \subseteq G$ and a local map $μ: A^S \to A$. 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 $μ$; 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 $μ$ 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.