Minimization of deterministic automata on finite words results in a {\em canonical\/} automaton. For deterministic automata on infinite words, no canonical minimal automaton exists, and a language may have different minimal deterministic B\"uchi (DBW) or co-B\"uchi (DCW) automata. In recent years, researchers have studied {\em good-for-games\/} (GFG) automata -- nondeterministic automata that can resolve their nondeterministic choices in a way that only depends on the past. Several applications of automata in formal methods, most notably synthesis, that are traditionally based on deterministic automata, can instead be based on GFG automata. The {\em minimization\/} problem for DBW and DCW is NP-complete, and it stays NP-complete for GFG B\"uchi and co-B\"uchi automata. On the other hand, minimization of GFG co-B\"uchi automata with {\em transition-based\/} acceptance (GFG-tNCWs) can be solved in polynomial time. In these automata, acceptance is defined by a set $\alpha$ of transitions, and a run is accepting if it traverses transitions in $\alpha$ only finitely often. This raises the question of canonicity of minimal deterministic and GFG automata with transition-based acceptance. In this paper we study this problem. We start with GFG-tNCWs and show that the safe components (that is, these obtained by restricting the transitions to these not in $\alpha$) of all minimal GFG-tNCWs are isomorphic, and that by saturating the automaton with transitions in $\alpha$ we get isomorphism among all minimal GFG-tNCWs. Thus, a canonical form for minimal GFG-tNCWs can be obtained in polynomial time. We continue to DCWs with transition-based acceptance (tDCWs), and their dual tDBWs. We show that here, while no canonical form for minimal automata exists, restricting attention to the safe components is useful, and implies that the only minimal tDCWs that have no canonical form are these for which the transition to the GFG model results in strictly smaller automaton, which do have a canonical minimal form.

中文翻译：

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GFG 和基于转换的自动机中的规范性

对有限词的确定性自动机的最小化导致 {\em canonical\/} 自动机。对于无限词的确定性自动机，不存在规范的最小自动机，并且一种语言可能具有不同的最小确定性 B\"uchi (DBW) 或 co-B\"uchi (DCW) 自动机。近年来，研究人员研究了 {\em good-for-games\/} (GFG) 自动机——非确定性自动机，可以以仅依赖于过去的方式解决他们的非确定性选择。自动机在形式方法中的几种应用，最显着的是合成，传统上基于确定性自动机，可以改为基于 GFG 自动机。DBW 和 DCW 的{\em 最小化\/}问题是 NP 完全的，而对于 GFG B\"uchi 和 co-B\"uchi 自动机，它保持 NP 完全。另一方面，最小化 GFG co-B\" 具有 {\em transition-based\/} 接受度的 uchi 自动机 (GFG-tNCWs) 可以在多项式时间内求解。在这些自动机中，接受是由一组 $\alpha$ 转换定义的，如果一次运行仅有限频繁地遍历 $\alpha$ 中的转换，则它是接受的。这提出了具有基于转换的接受的最小确定性和 GFG 自动机的规范性问题。在本文中，我们研究这个问题。我们从 GFG-tNCWs 开始，并表明所有最小 GFG-tNCWs 的安全组件（即，通过将转换限制为不在 $\alpha$ 中的这些组件而获得的组件）是同构的，并且通过在 $\alpha$ 中使用转换使自动机饱和\alpha$ 我们在所有最小 GFG-tNCW 之间得到同构。因此，可以在多项式时间内获得最小 GFG-tNCW 的规范形式。我们继续使用基于过渡的验收 (tDCW) 的 DCW，和他们的双 tDBW。我们在这里表明，虽然不存在最小自动机的规范形式，但将注意力限制在安全组件上是有用的，并且暗示唯一没有规范形式的最小 tDCW 是那些过渡到 GFG 模型导致严格更小的自动机，它们确实具有规范的最小形式。