Minimization of Automata for Liveness Languages

While the minimization problem for deterministic B <spacing diaeresis>uchi word automata is known to be NP-complete, several fundamental problems around it are still open. This includes the complexity of minimzation for transition-based automata, where acceptance is defined with respect to the set of transitions that a run traverses infinitely often, and minimization for good-for-games (GFG) automata, where nondeterminism is allowed, yet has to be resolved in a way that only depends on the past. Of special interest in formal verification are liveness properties, which state that something "good" eventually happens. Liveness languages constitute a strict fragment of.-regular languages, which suggests that minimization of automata recognizing liveness languages may be easier, as is the case for languages recognizable by weak automata, in particular safety languages. We define three classes of liveness, and study the minimization problem for automata recognizing languages in the classes. Our results refer to the basic minimization problem as well as to its extension to transition-based and GFG automata. In some cases, we provide bounds, and in others we provide connections between the different settings. Thus, our results are of practical interest and also improve our understanding of the (still very mysterious) minimization problem.