For general input automata, there exist regular constraint languages such that asking if a given input automaton admits a synchronizing word in the constraint language is PSPACE-complete or NP-complete. Here, we investigate this problem for commutative automata over an arbitrary alphabet and automata with simple idempotents over a binary alphabet as input automata. The latter class contains, for example, the \v{C}ern\'y family of automata. We find that for commutative input automata, the problem is always solvable in polynomial time, for every constraint language. For input automata with simple idempotents over a binary alphabet and with a constraint language given by a partial automaton with up to three states, the constrained synchronization problem is also solvable in polynomial time.

中文翻译：

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交换自动机和具有简单幂等的自动机的约束同步

对于一般输入自动机，存在规则约束语言，使得询问给定的输入自动机是否承认约束语言中的同步词是 PSPACE-complete 或 NP-complete。在这里，我们研究了任意字母表上的交换自动机和具有简单幂等的自动机作为输入自动机的二进制字母表上的这个问题。例如，后一类包含\v{C}ern\'y 自动机系列。我们发现，对于可交换输入自动机，对于每种约束语言，问题始终可以在多项式时间内解决。对于在二进制字母表上具有简单幂等性的输入自动机以及由最多三个状态的部分自动机给出的约束语言，约束同步问题也可以在多项式时间内解决。