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Register a new userMinimising Good-for-Games automata is NP complete
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Sven Schewe
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Authors
Sven Schewe
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This paper discusses the hardness of finding minimal good-for-games (GFG) Buchi, Co-Buchi, and parity automata with state based acceptance. The problem appears to sit between finding small deterministic and finding small nondeterministic automata, where minimality is NP-complete and PSPACE-complete, respectively. However, recent work of Radi and Kupferman has shown that minimising Co-Buchi automata with transition based acceptance is tractable, which suggests that the complexity of minimising GFG automata might be cheaper than minimising deterministic automata. We show for the standard state based acceptance that the minimality of a GFG automaton is NP-complete for Buchi, Co-Buchi, and parity GFG automata. The proofs are a surprisingly straight forward generalisation of the proofs from deterministic Buchi automata: they use a similar reductions, and the same hard class of languages.
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In GFG automata, it is possible to resolve nondeterminism in a way that only depends on the past and still accepts all the words in the language. The motivation for GFG automata comes from their adequacy for games and synthesis, wherein general nondeterminism is inappropriate. We continue the ongoing effort of studying the power of nondeterminism in GFG automata. Initial indications have hinted that every GFG automaton embodies a deterministic one. Today we know that this is not the case, and in fact GFG automata may be exponentially more succinct than deterministic ones. We focus on the typeness question, namely the question of whether a GFG automaton with a certain acceptance condition has an equivalent GFG automaton with a weaker acceptance condition on the same structure. Beyond the theoretical interest in studying typeness, its existence implies efficient translations among different acceptance conditions. This practical issue is of special interest in the context of games, where the Buchi and co-Buchi conditions admit memoryless strategies for both players. Typeness is known to hold for deterministic automata and not to hold for general nondeterministic automata. We show that GFG automata enjoy the benefits of typeness, similarly to the case of deterministic automata. In particular, when Rabin or Streett GFG automata have equivalent Buchi or co-Buchi GFG automata, respectively, then such equivalent automata can be defined on a substructure of the original automata. Using our typeness results, we further study the place of GFG automata in between deterministic and nondeterministic ones. Specifically, considering automata complementation, we show that GFG automata lean toward nondeterministic ones, admitting an exponential state blow-up in the complementation of a Streett automaton into a Rabin automaton, as opposed to the constant blow-up in the deterministic case.
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