Regular Languages are Testable with a Constant Number of Queries

Abstract

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 $L\in \{0,1\}^*$ and an integer n there exists a randomized algorithm which always accepts a word w of length n if $w\in L$ and rejects it with high probability if w has to be modified in at least $\epsilon n$ positions to create a word in L. The algorithm queries $\tilde{O}(1/\epsilon)$ 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.