FAMILIES OF DFAS AS ACCEPTORS OF ω-REGULAR LANGUAGES

Families of DFAS (FDFA5) 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 FDFA5 as the acceptors. In this paper, we look into the question of how suitable FDFA5 are for defining omega-regular languages. Specifically, we look into the complexity of performing Boolean operations, such as complementation and intersection, on FDFA5, the complexity of solving decision problems, such as emptiness and language containment, and the succinctness of FDFA5 compared to standard deterministic and nondeterministic omega-automata. We show that FDFA5 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 Bfichi and co-Bfichi automata to FDFA5 and of FDFA5 to nondeterministic Bfichi automata (NBA5). We show that translation of an NBA to an FDFA may involve an exponential blowup. Last, we show that FDFA5 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.