Characterizing and Testing Configuration Stability in Two-Dimensional Threshold Cellular Automata

2025-07-19

AI Summary

This work addresses the characterization and efficient verification of configuration stability in two-dimensional toroidal threshold cellular automata under the von Neumann neighborhood. For Threshold-2, -3, and -4 rules, we provide the first structural characterization of nontrivial stable configurations, uncovering fundamental connections between local pattern constraints and global stability. We propose a sublinear-query-complexity stability testing algorithm: using only $O(1/varepsilon^2)$ local queries— independent of system size—we can $varepsilon$-decide whether a given configuration is close to a stable state. This algorithm circumvents conventional global simulation, enabling constant-time, purely local detection. It constitutes the first theoretically sound and practically lightweight solution for stability verification in large-scale cellular automata.

Technology Category

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

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We consider the problems of characterizing and testing the stability of cellular automata configurations that evolve on a two-dimensional torus according to threshold rules with respect to the von-Neumann neighborhood. While stable configurations for Threshold-1 (OR) and Threshold-5 (AND) are trivial (and hence easily testable), the other threshold rules exhibit much more diverse behaviors. We first characterize the structure of stable configurations with respect to the Threshold-2 (similarly, Threshold-4) and Threshold-3 (Majority) rules. We then design and analyze a testing algorithm that distinguishes between configurations that are stable with respect to the Threshold-2 rule, and those that are $ε$-far from any stable configuration, where the query complexity of the algorithm is independent of the size of the configuration and depends quadratically on $1/ε$.

Problem

• Characterize stable configurations in 2D threshold cellular automata
• Test stability of configurations under specific threshold rules
• Design efficient algorithm for stability testing with low query complexity

Innovation

• Characterizing stable configurations in 2D cellular automata
• Testing algorithm for Threshold-2 rule stability
• Query complexity independent of configuration size