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

TL;DR

This paper studies the stability of configurations in two-dimensional threshold cellular automata, characterizes stable states for various rules, and develops an efficient testing algorithm with query complexity independent of the configuration size.

Contribution

It provides a detailed characterization of stable configurations for multiple threshold rules and introduces a novel, size-independent testing algorithm for stability.

Findings

Characterized stable configurations for Threshold-2, Threshold-3, and Threshold-4 rules.

Designed a testing algorithm with quadratic dependence on 1/ε.

Algorithm's query complexity is independent of configuration size.

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 $\epsilon$-far from any stable configuration, where the query complexity of the algorithm is independent of the size of the configuration and depends quadratically on $1/\epsilon$.