Testing Dynamic Environments: Back to Basics

TL;DR

This paper studies the complexity of testing dynamic environments modeled by cellular automata with threshold rules, introducing a meta-algorithm that efficiently tests their evolution with poly(1/ε) queries.

Contribution

It identifies conditions on local rules for cellular automata that enable efficient, non-adaptive testing algorithms with poly(1/ε) complexity.

Findings

Threshold rules satisfy the conditions for testability.

The proposed meta-algorithm is non-adaptive with one-sided error.

All threshold rules are poly(1/ε)-testable.

Abstract

We continue the line of work initiated by Goldreich and Ron (Journal of the ACM, 2017) on testing dynamic environments and propose to pursue a systematic study of the complexity of testing basic dynamic environments and local rules. As a first step, in this work we focus on dynamic environments that correspond to elementary cellular automata that evolve according to threshold rules. Our main result is the identification of a set of conditions on local rules, and a meta-algorithm that tests evolution according to local rules that satisfy the conditions. The meta-algorithm has query complexity poly$ (1/\epsilon) $, is non-adaptive and has one-sided error. We show that all the threshold rules satisfy the set of conditions, and therefore are poly$ (1/\epsilon) $-testable. We believe that this is a rich area of research and suggest a variety of open problems and natural research directions that may extend and expand our results.