The constrained synchronization problem (CSP) asks for a synchronizing word of a given input automaton contained in a regular set of constraints. It could be viewed as a special case of synchronization of a discrete event system under supervisory control. Here, we study the computational complexity of this problem for the class of sparse regular constraint languages. We give a new characterization of sparse regular sets, which equal the bounded regular sets, and derive a full classification of the computational complexity of CSP for letter-bounded regular constraint languages, which properly contain the strictly bounded regular languages. Then, we introduce strongly self-synchronizing codes and investigate CSP for bounded languages induced by these codes. With our previous result, we deduce a full classification for these languages as well. In both cases, depending on the constraint language, our problem becomes NP-complete or polynomial time solvable.

中文翻译：

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稀疏规则约束下同步的计算复杂度

约束同步问题 (CSP) 要求包含在一组常规约束中的给定输入自动机的同步字。它可以被视为在监督控制下的离散事件系统同步的特殊情况。在这里，我们针对稀疏正则约束语言类研究该问题的计算复杂度。我们给出了稀疏正则集的新特征，它等于有界正则集，并推导出了字母有界正则约束语言的 CSP 计算复杂度的完整分类，其中正确包含了严格有界的正则语言。然后，我们引入了强自同步代码，并研究了由这些代码引起的有界语言的 CSP。根据我们之前的结果，我们也推导出了这些语言的完整分类。