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 2 f0; 1g and an integer n there exists a randomized algorithm which always accepts a word w of length n if w 2 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 poly-logarithmic 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

The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomial-time algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the

Automata provide an effective mechanization of decision procedures for Presburger arithmetic. However, only crude lower and upper bounds are known on the sizes of the automata produced by this approach. In this paper, we prove that the number of states of the minimal deterministic automaton for a Presburger arithmetic formula is triple exponentially bounded in the length of the formula. This upper bound is established by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier elimination method. We also show that this triple exponential bound is tight (even for nondeterministic automata). Moreover, we provide optimal automata constructions for linear equations and inequations.

We give in the present work a new methodology that allows to give isoperimetric proofs, for Kneser’s Theorem and Kemperman’s structure Theory and most sophisticated results of this type. As an illustration we present a new proof of Kneser’s Theorem. 1

Let G be a group and let X be a finite subset. The isoperimetric method investigates the objective function |(XB)\X|, defined on the subsets X with |X | ≥ k and |G\(XB) | ≥ k. In this paper we present all the basic facts about the isoperimetric method. We improve some of our previous results and obtain generalizations and short proofs for several known results. We also give some new applications. Some of the results obtained here will be used in coming papers [25, 26, 27] to improve Kempermann structure Theory. 1