Saturation Problems for Families of Automata

TL;DR

This paper studies saturation in families of automata representing regular omega-languages, providing polynomial-time algorithms for saturation and regularity, and establishing complexity results for related properties.

Contribution

It introduces polynomial-time algorithms for deciding saturation in FDFA and FDWA, and improves the complexity bounds for related properties in automata theory.

Findings

Deciding saturation of FDFA is in P, reducing the complexity from PSPACE.

Deciding almost saturation is PSPACE-complete.

Saturation for FDWA is decidable in polynomial time, and FDWAs always define regular omega-languages.

Abstract

Families of deterministic finite automata (FDFA) represent regular $\omega$-languages through their ultimately periodic words (UP-words). An FDFA accepts pairs of words, where the first component corresponds to a prefix of the UP-word, and the second component represents a period of that UP-word. An FDFA is termed saturated if, for each UP-word, either all or none of the pairs representing that UP-word are accepted. We demonstrate that determining whether a given FDFA is saturated can be accomplished in polynomial time, thus improving the known PSPACE upper bound by an exponential. We illustrate the application of this result by presenting the first polynomial learning algorithms for representations of the class of all regular $\omega$-languages. Furthermore, we establish that deciding a weaker property, referred to as almost saturation, is PSPACE-complete. Since FDFAs do not necessarily define regular $\omega$-languages when they are not saturated, we also address the regularity problem and show that it is PSPACE-complete. Finally, we explore a variant of FDFAs called families of deterministic weak automata (FDWA), where the semantics for the periodic part of the UP-word considers $\omega$-words instead of finite words. We demonstrate that saturation for FDWAs is also decidable in polynomial time, that FDWAs always define regular $\omega$-languages, and we compare the succinctness of these different models.