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.

中文翻译：

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最小化 Good-for-Games 自动机是 NP 完成的

本文讨论了寻找具有基于状态的接受度的最小对游戏 (GFG) Buchi、Co-Buchi 和奇偶自动机的难度。问题似乎位于寻找小型确定性自动机和寻找小型非确定性自动机之间，其中最小值分别是 NP 完全和 PSPACE 完全。然而，Radi 和 Kupferman 最近的工作表明，通过基于转移的接受来最小化 Co-Buchi 自动机是容易处理的，这表明最小化 GFG 自动机的复杂性可能比最小化确定性自动机更便宜。我们证明了基于标准状态的接受，GFG 自动机的极小性对于 Buchi、Co-Buchi 和奇偶 GFG 自动机是 NP 完全的。这些证明是对确定性 Buchi 自动机的证明的令人惊讶的直接概括：它们使用类似的归约，