Minimization and Canonization of GFG Transition-Based Automata
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While many applications of automata in formal methods can use nondeterministic automata, some applications, most notably synthesis, need deterministic or good-for-games(GFG) automata. The latter are nondeterministic automata that can resolve their nondeterministic choices in a way that only depends on the past. The minimization problem for deterministic Büchi and co-Büchi word automata is NP-complete. In particular, no canonical minimal deterministic automaton exists, and a language may have different minimal deterministic automata. We describe a polynomial minimization algorithm for GFG co-Büchi word automata with transition-based acceptance. Thus, a run is accepting if it traverses a set α of designated transitions only finitely often. Our algorithm is based on a sequence of transformations we apply to the automaton, on top of which a minimal quotient automaton is defined. We use our minimization algorithm to show canonicity for transition-based GFG co-Büchi word automata: all minimal automata have isomorphic safe components (namely components obtained by restricting the transitions to these not in α) and once we saturate the automata with α-transitions, we get full isomorphism.
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