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 B\"uchi and co-B\"uchi 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-B\"uchi word automata with transition-based acceptance. Thus, a run is accepting if it traverses a set $\alpha$ 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-B\"uchi word automata: all minimal automata have isomorphic safe components (namely components obtained by restricting the transitions to these not in $\alpha$) and once we saturate the automata with $\alpha$-transitions, we get full isomorphism.

中文翻译：

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基于 GFG 转换的自动机的最小化和标准化

虽然形式方法中自动机的许多应用程序可以使用非确定性自动机，但一些应用程序，最显着的是合成，需要确定性或适合游戏 (GFG) 自动机。后者是非确定性自动机，可以以仅依赖于过去的方式解决其非确定性选择。确定性 B\"uchi 和 co-B\"uchi 词自动机的最小化问题是 NP 完全的。特别是，不存在规范的最小确定自动机，并且一种语言可能具有不同的最小确定自动机。我们描述了 GFG co-B\"uchi 词自动机的多项式最小化算法，具有基于转换的接受。因此，如果运行仅有限地遍历指定转换的集合 $\alpha$，则运行是接受的。我们的算法基于我们应用于自动机的转换序列，在其上定义了一个最小商自动机。我们使用我们的最小化算法来显示基于转换的 GFG co-B\"uchi 字自动机的规范性：所有最小自动机都具有同构安全组件（即通过将转换限制为不在 $\alpha$ 中的这些组件而获得的组件）并且一旦我们饱和具有 $\alpha$-transitions 的自动机，我们得到完全同构。