Piecewise Testable Languages and Nondeterministic Automata

TL;DR

This paper investigates the relationship between nondeterministic automata and piecewise testable languages, showing that certain nondeterministic automata can provide tighter bounds on the complexity of testing piecewise testability.

Contribution

It introduces ptNFAs, a class of nondeterministic automata recognizing piecewise testable languages, and demonstrates that their depth offers a significantly better upper bound on $k$ than minimal DFAs.

Findings

ptNFAs recognize piecewise testable languages.

Depth of ptNFAs bounds $k$-piecewise testability more tightly.

Complexity results for $k$-piecewise testability in ptNFAs.

Abstract

A regular language is $k$-piecewise testable if it is a finite boolean combination of languages of the form $\Sigma^* a_1 \Sigma^* \cdots \Sigma^* a_n \Sigma^*$, 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.