Families of DFAs as Acceptors of $\omega$-Regular Languages

TL;DR

This paper investigates the computational complexity, succinctness, and translation properties of Families of DFAs (FDFAs) for recognizing omega-regular languages, showing their advantages and limitations compared to other automata models.

Contribution

It provides a detailed complexity analysis of Boolean operations and decision problems on FDFAs, and compares their succinctness with standard omega-automata, including polynomial translations and exponential blowups.

Findings

FDFAs enable Boolean operations and decision problems in nondeterministic logarithmic space.

Polynomial translations exist from deterministic Büchi and co-Büchi automata to FDFAs.

Translation from nondeterministic Büchi automata to FDFAs may involve exponential blowup.

Abstract

Families of DFAs (FDFAs) provide an alternative formalism for recognizing $\omega$-regular languages. The motivation for introducing them was a desired correlation between the automaton states and right congruence relations, in a manner similar to the Myhill-Nerode theorem for regular languages. This correlation is beneficial for learning algorithms, and indeed it was recently shown that $\omega$-regular languages can be learned from membership and equivalence queries, using FDFAs as the acceptors. In this paper, we look into the question of how suitable FDFAs are for defining omega-regular languages. Specifically, we look into the complexity of performing Boolean operations, such as complementation and intersection, on FDFAs, the complexity of solving decision problems, such as emptiness and language containment, and the succinctness of FDFAs compared to standard deterministic and nondeterministic $\omega$-automata. We show that FDFAs enjoy the benefits of deterministic automata with respect to Boolean operations and decision problems. Namely, they can all be performed in nondeterministic logarithmic space. We provide polynomial translations of deterministic B\"uchi and co-B\"uchi automata to FDFAs and of FDFAs to nondeterministic B\"uchi automata (NBAs). We show that translation of an NBA to an FDFA may involve an exponential blowup. Last, we show that FDFAs are more succinct than deterministic parity automata (DPAs) in the sense that translating a DPA to an FDFA can always be done with only a polynomial increase, yet the other direction involves an inevitable exponential blowup in the worst case.