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

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 $1/ε$.