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/ε), 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/ε)-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.

In this work we study the testability of a family of graph partition properties that generalizes a family previously studied by Goldreich, Goldwasser, and Ron (Journal of the ACM, 1998 ). While the family studied by Goldreich, Goldwasser, and Ron includes a variety of natural properties, such as k-colorability and containing a large cut, it does not include other properties of interest, such as split graphs, and more generally (p,q)-colorable graphs. The generalization we consider allows us to impose constraints on the edge-densities within and between parts (relative to the sizes of the parts). We denote the family studied in this work by GPP. We first show that all properties in GPP have a testing algorithm whose query complexity is polynomial in 1/epsilon, where epsilon is the given proximity parameter (and there is no dependence on the size of the graph). As the testing algorithm has two-sided error, we next address the question of which properties in GPP can be tested with one-sided error and query complexity polynomial in 1/epsilon. We answer this question by establishing a characterization result. Namely, we define a subfamily GPP_{0,1} of GPP and show that a property P in GPP is testable by a one-sided error algorithm that has query complexity poly(1/epsilon) if and only if P in GPP_{0,1}.