Here, we study the question if synchronizing words exist that belong to some fixed constraint language, given by some partial finite automaton called constraint automaton. We strengthen a previous result by giving a complete classification of the computational complexity landscape for constraint automata with two states and an arbitrary alphabet. We also give a classification for three state automata with a binary alphabet, for the class of automata such that the initial state is connected with at most one other state. Among them, we find constraint automata with three states and a binary alphabet, for which the problem is PSPACE-complete. We conclude that, for a binary alphabet, we need at least three states to realise PSPACE-hard problems. As it turns out, the three state constraint automata for which the problem is NP-complete are quite rare. To derive our results, we generalize the known polynomial time algorithm from the unconstrained setting to broaden the range of constraint problems that could be solved in PTIME.

中文翻译：

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与两个或三个状态部分约束自动机的约束同步

在这里，我们研究是否存在属于某种固定约束语言的同步词，由称为约束自动机的某些部分有限自动机给出。我们通过对具有两个状态和任意字母表的约束自动机的计算复杂度进行完整分类来加强先前的结果。我们还对具有二进制字母表的三状态自动机进行了分类，对于自动机类，初始状态至多与一个其他状态连接。其中，我们找到了具有三个状态和一个二进制字母表的约束自动机，其问题是 PSPACE 完全的。我们得出结论，对于二进制字母表，我们至少需要三个状态来实现 PSPACE-hard 问题。事实证明，问题是 NP 完全的三状态约束自动机非常罕见。