Most slowly synchronizing automata over binary alphabets are circular, i.e., containing a letter permuting the states in a single cycle, and their set of synchronizing words has maximal state complexity, which also implies complete reachability.Here, we take a closer look at generalized circular and completely reachable automata. We derive that over a binary alphabet every completely reachable automaton must be circular, a consequence of a structural result stating that completely reachable automata over strictly less letters than states always contain permutational letters. We state sufficient conditions for the state complexity of the set of synchronizing words of a generalized circular automaton to be maximal. We apply our main criteria to the family $\mathscr K_n$ of automata that was previously only conjectured to have this property.

中文翻译：

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圆形自动机和二进制自动机上同步词集的状态复杂度

在二进制字母表上最慢的同步自动机是循环的，即包含一个在单个周期内排列状态的字母，并且它们的同步词集具有最大的状态复杂度，这也意味着完全的可达性。在这里，我们仔细研究一下广义循环完全可以达到的自动机 我们得出结论，在二进制字母上，每个完全可到达的自动机都必须是圆形的，这是结构性结果的结果，该结果表明，在比州少得多的字母下，完全可到达的自动机总是包含置换字母。我们陈述了充分的条件，以使广义圆形自动机的同步字集合的状态复杂度最大。我们将主要标准应用于以前仅被推测具有此属性的自动机家族$ \ mathscr K_n $。