A leaf path language is a Boolean combination of sets of the form $\mathsf{{}^mE}^k L$, with $k \ge 1$ and $L$ a regular word language, which consist of those forests where the node labels in at least $k$ leaf-to-root paths make up a word that belongs to $L$. We look at the class $\mathsf{*D}$ of the languages recognized by iterated wreath products of syntactic algebras of leaf path languages. We prove the existence of an algorithm that, given a regular forest language, returns in finite time a sequence of such algebras; their wreath product is divided by the language's syntactic algebra if, and only if this language belongs to $\mathsf{*D}$, which makes membership in this class a decidable question. The result also applies to the subclasses $\mathsf{PDL}$ and $\mathsf{CTL^*}$.

中文翻译：

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通过计算最大路径定义的林语言

叶路径语言是$ \ mathsf {{} ^ mE} ^ k L $形式的集合的布尔组合，其中$ k \ ge 1 $和$ L $是常规单词语言，其中包括至少$ k $叶到根路径中的节点标签组成了一个属于$ L $的单词。我们来看叶子路径语言的句法代数的迭代花圈产品所识别的语言的$ \ mathsf {* D} $类。我们证明了存在一种算法，该算法在给定常规森林语言的情况下，会在有限时间内返回此类代数的序列。仅当该语言属于$ \ mathsf {* D} $时，它们的花圈乘积才被该语言的句法代数所除，这使该类的成员资格成为一个可商定的问题。结果也适用于子类$ \ mathsf {PDL} $和$ \ mathsf {CTL ^ *} $。