We give efficient algorithms, beating or matching optimal known bounds, for computing a variety of simulation relations on the state space of a Buechi automaton. Our algorithms are derived via a unified and simple parity-game framework. This framework incorporates previously studied notions like fair and direct simulation, but our main motivation is state space reduction, and for this purpose we introduce a new natural notion of simulation, called delayed simulation. We show that, unlike fair simulation, delayed simulation preserves the automaton language upon quotienting, and that it allows substantially better state reduction than direct simulation. We use the parity-game approach, based on a recent algorithm by Jurdzinski, to efficiently compute all the above simulation relations. In particular, we obtain an O(mn~3)-time and O(mn)-space algorithm for computing both the delayed and fair simulation relations. The best prior algorithm for fair simulation requires time O(n~6) ([HKR97]). Our framework also allows one to compute bisimulations efficiently: we compute the fair bisimulation relation in O(mn~3) time and O(mn) space, whereas the best prior algorithm for fair bisimulation requires time O(n~(10)) ([HR00]).
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