Connections between the minimal neighborhood and the activity value of cellular automata

TL;DR

The paper studies cellular automata on a group $G$ with a finite alphabet $A$, focusing on the minimal neighborhood $\mathrm{MN}(\mu)$ of a local map and the activity value $\alpha(\mu)$, the count of active transitions. It proves that the activity is tightly constrained by the minimal neighborhood: if the identity lies in the minimal neighborhood, $\alpha(\mu)$ is a multiple of $|A|^{|S\setminus\mathrm{MN}(\mu)|}$ (via reduction to $S_0=\mathrm{MN}(\mu)$); if not, $\alpha(\mu)=|A|^{|S|}-|A|^{|S|-1}$. The work also shows that every feasible activity value $k$ with $0<k<|A|^{|S|}$ can be realized by choosing a local map with prescribed MN, and that when $|A|\ge 3$ one can realize $\alpha(\mu)=|A|^{|S|}$. Symmetry results demonstrate that both $\alpha(\mu)$ and $\mathrm{MN}(\mu)$ are preserved under alphabet bijections and group automorphisms, and the binary alphabet case yields further structural simplifications. Overall, the paper provides a complete explanation of how the size and structure of the minimal neighborhood constrain the dynamical activity of local rules, with constructive realizations and clear avenues for future work including composition, inversion, and per-site activity generalizations.

Abstract

For a group $G$ and a finite set $A$, a cellular automaton is a transformation of the configuration space $A^G$ defined via a finite neighborhood and a local map. Although neighborhoods are not unique, every CA admits a unique minimal neighborhood, which consists on all the essential cells in $G$ that affect the behavior of the local map. An active transition of a cellular automaton is a pattern that produces a change on the current state of a cell when the local map is applied. In this paper, we study the links between the minimal neighborhood and the number of active transitions, known as the activity value, of cellular automata. Our main results state that the activity value usually imposes several restrictions on the size of the minimal neighborhood of local maps.

Figures (2)

• Figure 1: Size of MN vs. Activity value for local functions with $\vert A \vert = 2$ and $\vert S \vert = 5$.
• Figure 2: Size of MN vs. Activity value for local functions with $\vert A \vert = 3$ and $\vert S \vert = 3$.

Theorems & Definitions (40)

• Definition 1
• Theorem 1
• Example 1
• Definition 2
• Proposition 1
• Lemma 1
• Definition 3
• Example 2
• Proposition 2: c.f. Exercise 1.24 in ExCA
• proof