Regular Languages Are Testable with a Constant Number of Queries

We continue the study of combinatorial property testing, initiatedby Goldreich, Goldwasser and Ron in [7]. The subject of this paper is testing regular languages. Our main result is as follows. For a regular language L ∈ {0, 1}∗ and an integer n there exists a randomized algorithm which always accepts a word w of length n if w ∈ L, and rejects it with high probability if w has to be modified in at least ǫn positions to create a word in L. The algorithm queries Õ(1/ǫ) bits of w. This query complexity is shown to be optimal up to a factor poly-logarithmic in 1/ǫ. We also discuss testability of more complex languages and show, in particular, that the query complexity required for testing contextfree languages cannot be bounded by any function of ǫ. The problem of testing regular languages can be viewed as a part of a very general approach, seeking to probe testability of properties defined by logical means.