ω-Regular Languages Are Testable with a Constant Number of Queries

Abstract

We continue the study of combinatorial property testing. For a property ψ, an ɛ-test for ψ, for 0 < ɛ ≤ 1, is a randomized algorithm that given an input x, returns “yes” if x satisfies ψ, and returns “no” with high probability if x is ɛ-far from satisfying ψ, where ɛ-far essentially means that an ɛ-fraction of x needs to be changed in order for it to satisfy ψ. In [AKNS99], Alon et al. show that regular languages are ɛ-testable with a constant (depends on ψ and ɛ and independent of x) number of queries. We extend the result in [AKNS99] to ω-regular languages: given a nondeterministic Büchi automaton A on infinite words and a small ɛ > 0, we describe an algorithm that gets as input an infinite lasso-shape word of the form x · y ω, for finite words x and y, samples only a constant number of letters in x and y, returns “yes” if w ∈ L(A), and returns “no” with probability 2/3 if w is ɛ-far from L(A). We also discuss the applicability of property testing to formal verification, where ω-regular languages are used for the specification of the behavior of nonterminating reactive systems, and computations correspond to lasso-shape words.KeywordsModel CheckRegular LanguageQuery ComplexityProperty TestingInput WordThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.