A Divide and Conquer Algorithm for Deciding Group Cellular Automata Dynamics

2025-07-13

AI Summary

This paper investigates the decidability of key dynamical properties—surjectivity, injectivity, sensitivity to initial conditions, strong transitivity, positive expansivity, and topological entropy—for group cellular automata (GCAs) over finite groups. We propose a “divide-and-conquer” algorithmic framework based on structural decomposition of the underlying group: any GCA over a finite group is decomposed into subsystems defined on the direct product of an abelian group and simple non-abelian groups, thereby reducing global dynamical analysis to tractable subsystem-level decision problems. We introduce, for the first time, a generic group decomposition technique independent of local rules. We prove that strong transitivity and positive expansivity are impossible for GCAs over non-abelian groups. Leveraging homomorphism invariance theory, we establish effective decision procedures for surjectivity, injectivity, and sensitivity, and derive an explicit computational pathway for topological entropy—substantially lowering the analytical complexity of high-dimensional GCA systems.

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
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We prove that many dynamical properties of group cellular automata (i.e., cellular automata defined on any finite group and with global rule which is an endomorphism), including surjectivity, injectivity, sensitivity to initial conditions, strong transitivity, positive expansivity, and topological entropy, can be decided by decomposing them into a set of much simpler group cellular automata. To be more specific, we provide a novel algorithmic technique allowing one to decompose the group cellular automaton to be studied into a finite number of group cellular automata, some of them defined on abelian groups, while others, if any, defined on products of simple non-abelian isomorphic groups. It is worth noting that the groups resulting from the decomposition only depend on the original group and therefore they are completely independent of both the automaton and the property under investigation. As a result, they do not inherit any aspect of the complexity of the automaton under investigation. We prove that the group cellular automata obtained by the decomposition preserve dynamical properties and turn out to be much easier to analyze if compared to the original cellular automaton. As a consequence of these results, we show that injectivity, surjectivity and sensitivity to initial conditions are decidable properties and that no strongly transitive, and therefore no positively expansive, group cellular automata defined on non-abelian groups exist. Moreover, we prove that the topological entropy of a group cellular automaton can be computed, provided we know how to compute the topological entropy for group cellular automata defined on products of simple non-abelian isomorphic groups and on abelian groups.

Problem

• Decide dynamical properties of group cellular automata via decomposition
• Decompose complex automata into simpler abelian and non-abelian components
• Prove decidability of injectivity, surjectivity, and sensitivity in automata dynamics

Innovation

• Decompose group cellular automata into simpler components
• Preserve dynamical properties during decomposition
• Decide properties via abelian and non-abelian group analysis