 
Summary: Regular Languages are Testable with a Constant Number of Queries
Noga Alon
Michael Krivelevich
Ilan Newman §
Mario Szegedy ¶
Abstract
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 {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 n positions to create a word in L. The algorithm queries ~O(1/ ) bits of w. This
query complexity is shown to be optimal up to a factor polylogarithmic in 1/ . 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 . 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.
1 Introduction
Property testing deals with the question of deciding whether a given input x satisfies a prescribed
property P or is "far" from any input satisfying it. Let P be a property, i.e. a nonempty family of
