Piecewise Testable Languages and Nondeterministic Automata

A regular language is k-piecewise testable if it is a finite boolean combination of languages of the form Σa1Σ · · ·ΣanΣ, where ai ∈ Σ and 0 ≤ n ≤ k. Given a DFA A and k ≥ 0, it is an NLcomplete problem to decide whether the language L(A) is piecewise testable and, for k ≥ 4, it is coNP-complete to decide whether the language L(A) is k-piecewise testable. It is known that the depth of the minimal DFA serves as an upper bound on k. Namely, if L(A) is piecewise testable, then it is k-piecewise testable for k equal to the depth of A. In this paper, we show that some form of nondeterminism does not violate this upper bound result. Specifically, we define a class of NFAs, called ptNFAs, that recognize piecewise testable languages and show that the depth of a ptNFA provides an (up to exponentially better) upper bound on k than the minimal DFA. We provide an application of our result, discuss the relationship between k-piecewise testability and the depth of NFAs, and study the complexity of k-piecewise testability for ptNFAs. 1998 ACM Subject Classification F.1.1 Models of Computation, F.4.3 Formal Languages