Promptness in ω-regular Automata

Liveness properties of on-going reactive systems assert that something good will happen eventually. In satisfying liveness properties, there is no bound on the “wait time”, namely the time that may elapse until an eventuality is fulfilled. The traditional “unbounded” semantics of liveness properties nicely corresponds to the classical semantics of automata on infinite objects. Indeed, acceptance is defined with respect to the set of states the run visits infinitely often, with no bound on the number of transitions taken between successive visits. In many applications, it is important to bound the wait time in liveness properties. Bounding the wait time by a constant is not always possible, as the bound may not be known in advance. It may also be very large, resulting in large specifications. Researchers have studied prompt eventualities, where the wait time is bounded, but the bound is not known in advance. We study the automata-theoretic counterpart of prompt eventually. In a prompt-Büchi automaton, a run r is accepting if there exists a bound k such that r visits an accepting state every at most k transitions. We study the expressive power of nondeterministic and deterministic prompt-Büchi automata, their properties, and decision problems for them. In particular, we show that regular nondeterministic prompt Büchi automata are exactly as expressive as nondeterministic co-Büchi automata.