Decidability and Characterization of Expansivity for Group Cellular Automata

2025-10-16

AI Summary

This paper investigates the decidability and characterization of expansivity for group cellular automata (GCA). For GCAs over abelian groups, it establishes an easily verifiable necessary and sufficient condition for expansivity in terms of group algebras and local rules. Crucially, it provides the first decidability proof of expansivity for GCAs over arbitrary finite groups—resolving a long-standing open problem in the non-abelian setting. Methodologically, the work integrates symbolic dynamics, group action theory, and topological dynamics to forge a tight link between algebraic structure and dynamical properties. Key contributions include: (1) an effective algorithmic framework for deciding expansivity; (2) a rigorous proof that expansivity is a proper subclass of topologically transitive and injective GCAs, thereby precisely locating it within the dynamical hierarchy; and (3) a complete characterization of expansivity for GCAs over all finite groups.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Causality

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

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.

Problem

• Characterizing expansivity for abelian group cellular automata
• Deciding expansivity for general group cellular automata
• Comparing expansive and topologically transitive injective automata

Innovation

• Characterizing expansivity for abelian group cellular automata
• Deciding expansivity for general non-abelian group automata
• Showing expansive automata strictly contained in transitive injective ones