The intersection problem for finite semigroups asks, given a set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. In previous work, it was shown that is problem is [Formula: see text]-complete. We introduce compressibility measures as a useful tool to classify the complexity of the intersection problem for certain classes of finite semigroups. Using this framework, we obtain a new and simple proof that for groups and for commutative semigroups, the problem (as well as the variant where the languages are represented by finite automata) is contained in [Formula: see text]. We uncover certain structural and non-structural properties determining the complexity of the intersection problem for varieties of semigroups containing only trivial submonoids. More specifically, we prove [Formula: see text]-hardness for classes of semigroups having a property called unbounded order and for the class of all nilpotent semigroups of bounded order. On the contrary, we show that bounded order and commutativity imply decidability in poly-logarithmic time on alternating random-access Turing machines with a single alternation. We also establish connections to the monoid variant of the problem.

中文翻译：

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有限半群的交集问题

有限半群的交集问题问，给定一组正则语言，以识别有限半群的态射为代表，它们的交集中是否存在一个词。在以前的工作中，表明问题是[公式：见文本]-完成。我们引入可压缩性度量作为一种有用的工具，用于对某些有限半群类的交集问题的复杂性进行分类。使用这个框架，我们获得了一个新的简单证明，即对于群和交换半群，问题（以及语言由有限自动机表示的变体）包含在 [公式：参见文本] 中。我们揭示了某些结构和非结构性质，这些性质决定了仅包含平凡子类的各种半群的交集问题的复杂性。更具体地说，我们证明了具有称为无界阶性质的半群类和所有有界阶幂等半群类的[公式：见文本]-硬度。相反，我们证明了有界顺序和交换性意味着多对数时间在具有单次交替的交替随机访问图灵机上的可判定性。我们还建立了与问题的幺半群变体的连接。