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Decidability and Characterization of Expansivity for Group Cellular Automata

Niccolo' Castronuovo, Alberto Dennunzio, Luciano Margara

TL;DR

The paper addresses the decidability and characterization of expansivity for group cellular automata (GCAs) defined on $G^\mathbb{Z}$ with finite $G$, offering an easily checkable criterion in the abelian case and a decidability result for general groups. Its core approach is a decomposition framework that reduces expansivity to a finite set of simpler GCAs, plus a central abelian result that links expansivity to the positive expansivity of $\mathcal{F}-\mathcal{F}^{-1}$; for products of non-abelian simple groups, expansivity coincides with topological transitivity and remains decidable. The work shows that expansive GCAs are topologically transitive, though the converse does not hold in general, and provides a decomposition-based method to extend abelian results to general groups via normal subgroups and quotients. Overall, the results yield practical criteria and a decidability boundary for the dynamical behavior of GCAs, with implications for understanding chaos-like properties in algebraic cellular automata.

Abstract

Group cellular automata are continuous, shift-commuting endomorphisms of $G^\mathbb{Z}$, where $G$ is a finite group. We provide an easy-to-check characterization of expansivity for group cellular automata on abelian groups and we prove that expansivity is a decidable property for general (non-abelian) groups. Moreover, we show that the class of expansive group cellular automata is strictly contained in that of topologically transitive injective group cellular automata.

Decidability and Characterization of Expansivity for Group Cellular Automata

TL;DR

The paper addresses the decidability and characterization of expansivity for group cellular automata (GCAs) defined on with finite , offering an easily checkable criterion in the abelian case and a decidability result for general groups. Its core approach is a decomposition framework that reduces expansivity to a finite set of simpler GCAs, plus a central abelian result that links expansivity to the positive expansivity of ; for products of non-abelian simple groups, expansivity coincides with topological transitivity and remains decidable. The work shows that expansive GCAs are topologically transitive, though the converse does not hold in general, and provides a decomposition-based method to extend abelian results to general groups via normal subgroups and quotients. Overall, the results yield practical criteria and a decidability boundary for the dynamical behavior of GCAs, with implications for understanding chaos-like properties in algebraic cellular automata.

Abstract

Group cellular automata are continuous, shift-commuting endomorphisms of , where is a finite group. We provide an easy-to-check characterization of expansivity for group cellular automata on abelian groups and we prove that expansivity is a decidable property for general (non-abelian) groups. Moreover, we show that the class of expansive group cellular automata is strictly contained in that of topologically transitive injective group cellular automata.
Paper Structure (10 sections, 11 theorems, 33 equations, 1 figure, 1 algorithm)

This paper contains 10 sections, 11 theorems, 33 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Results Overview
  3. Decidability and Characterization of dynamical properties
  4. Preliminaries
  5. General CAs
  6. Background on Groups and Group CAs
  7. Expansivity on abelian groups
  8. Expansivity on general groups
  9. Conclusions and further work
  10. Acknowledgements

Key Result

Lemma 1

Let ${\cal F}$ be any injective GCA on a finite group $G$ and let $k\in\mathbb{Z}$. It holds that ${\cal F}$ is expansive if and only if there exists a natural $k_{{\cal F}} > 0$ such that:

Figures (1)

  • Figure 1: $c$ is a lower angular configuration and $c'$ is an upper angular configuration. Both are depicted in gray, except for the rightmost element, which is black. Black elements must be different from zero and determine the positions of $c$ and $c'$. All light blue elements must be equal to $e$.

Theorems & Definitions (30)

  • Definition 1
  • Definition 2
  • Remark 1
  • Lemma 1
  • proof
  • Definition 3
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 20 more