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 [J. ACM, 45 (1998), pp. 653-750]. The subject of this paper is testing regular languages. Our main result is as follows. For a regular language L is an element of {0,1}* and an integer n there exists a randomized algorithm which always accepts a word w of length n if w is an element of L and rejects it with high probability if w has to be modified in at least epsilonn positions to create a word in L. The algorithm queries (O) over tilde (1/epsilon) bits of w. This query complexity is shown to be optimal up to a factor polylogarithmic in 1/epsilon. We also discuss the 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 epsilon. 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.