Universality of Confluent, Self-Loop Deterministic Partially Ordered NFAs is Hard

An automaton is partially ordered if the only cycles in its transition diagram are self-loops. The expressivity of partially ordered NFAs (poNFAs) can be characterized by the Straubing-Th\'erien hierarchy. Level 3/2 is recognized by poNFAs, level 1 by confluent, self-loop deterministic poNFAs as well as by confluent poDFAs, and level 1/2 by saturated poNFAs. We study the universality problem for confluent, self-loop deterministic poNFAs. It asks whether an automaton accepts all words over its alphabet. Universality for both NFAs and poNFAs is a PSpace-complete problem. For confluent, self-loop deterministic poNFAs, the complexity drops to coNP-complete if the alphabet is fixed but is open if the alphabet may grow. We solve this problem by showing that it is PSpace-complete if the alphabet may grow polynomially. Consequently, our result provides a lower-bound complexity for some other problems, including inclusion, equivalence, and $k$-piecewise testability. Since universality for saturated poNFAs is easy, confluent, self-loop deterministic poNFAs are the simplest and natural kind of NFAs characterizing a well-known class of languages, for which deciding universality is as difficult as for general NFAs.Comment: arXiv admin note: text overlap with arXiv:1609.0346

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 $\Sigma^* a_1 \Sigma^* \cdots \Sigma^* a_n \Sigma^*$, 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.Comment: Corrections in section 4.