Piecewise Testable Languages and Nondeterministic Automata

Abstract: A regular language is $k$-piecewise testable if it is a finite boolean combination of languages of the form $$\Sigma^* a_1 \Sigma^* \cdots \Sigma^* a_n \Sigma<sup>*$$, where $a_i\in\Sigma$ and $0\le n \le k$. Given a DFA $A$ and $k\ge 0$, it is an NL-complete problem to decide whether the language $L(A)$ is piecewise testable and, for $k\ge 4$, 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.