A novel family of finite automata for recognizing and learning $\omega$-regular languages

TL;DR

This paper introduces a new family of finite automata called limit FDFAs for recognizing and learning omega-regular languages, offering more succinct representations and better checking capabilities than existing automata types.

Contribution

The paper proposes limit FDFAs, a novel automaton family that is more succinct than syntactic FDFAs and exponentially coarser than periodic FDFAs, enhancing omega-language recognition and automata learning.

Findings

Limit FDFAs can efficiently check omega-language regularity.

They can determine acceptance by deterministic Büchi automata.

Combining limit and recurrent FDFAs improves automata efficiency.

Abstract

Families of DFAs (FDFAs) have recently been introduced as a new representation of $\omega$-regular languages. They target ultimately periodic words, with acceptors revolving around accepting some representation $u\cdot v^\omega$. Three canonical FDFAs have been suggested, called periodic, syntactic, and recurrent. We propose a fourth one, limit FDFAs, which can be exponentially coarser than periodic FDFAs and are more succinct than syntactic FDFAs, while they are incomparable (and dual to) recurrent FDFAs. We show that limit FDFAs can be easily used to check not only whether {\omega}-languages are regular, but also whether they are accepted by deterministic B\"uchi automata. We also show that canonical forms can be left behind in applications: the limit and recurrent FDFAs can complement each other nicely, and it may be a good way forward to use a combination of both. Using this observation as a starting point, we explore making more efficient use of Myhill-Nerode's right congruences in aggressively increasing the number of don't-care cases in order to obtain smaller progress automata. In pursuit of this goal, we gain succinctness, but pay a high price by losing constructiveness.