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

Fuente: arXiv

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Detalles Bibliográficos
Autores principales:	Nakar, Yonatan, Ron, Dana
Formato:	Preprint
Publicado:	2025
Materias:	Data Structures and Algorithms
Acceso en línea:	Acceder al recurso
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_version_	1866916851897860096
author	Nakar, Yonatan Ron, Dana
author_facet	Nakar, Yonatan Ron, Dana
contents	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 $1/ε$.
format	Preprint
id	arxiv_https___arxiv_org_abs_2507_14569
institution	arXiv
publishDate	2025
record_format	arxiv
spellingShingle	Characterizing and Testing Configuration Stability in Two-Dimensional Threshold Cellular Automata Nakar, Yonatan Ron, Dana Data Structures and Algorithms 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 $1/ε$.
title	Characterizing and Testing Configuration Stability in Two-Dimensional Threshold Cellular Automata
topic	Data Structures and Algorithms
url	https://arxiv.org/abs/2507.14569

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