Regular languages are testable with a constant number of queries

We continue the study of combinatorial property testing, initiated by 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 qq {0.1}^* and an integer n there exists a randomized algorithm which always accepts a word w of length n if w qq L, and rejects it with high probability if w has to be modified in at least qqn positions to create a word on L. The algorithm queries qq(1/qq) bits of w. This query complexity is shown to be optimal up to a factory poly-logarithmic in 1/qq. We also discuss testability of more complex languages and show, in particular, that the query complexity required for testing context free languages cannot be bounded by any function of qq. 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.