We classify the regular languages using an operator $\mathcal{C}\mapsto TL(\mathcal{C})$. For each input class of languages $\mathcal{C}$, it builds a larger class $TL(\mathcal{C})$ consisting of all languages definable in a variant of unary temporal logic whose future/past modalities depend on $\mathcal{C}$. This defines the temporal hierarchy of basis $\mathcal{C}$: level $n$ is built by applying this operator $n$ times to $\mathcal{C}$. This hierarchy is closely related to another one, the concatenation hierarchy of basis $\mathcal{C}$. In particular, the union of all levels in both hierarchies is the same. We focus on bases $\mathcal{G}$ of group languages and natural extensions thereof, denoted $\mathcal{G}^+$. We prove that the temporal hierarchies of bases $\mathcal{G}$ and $\mathcal{G}^+$ are strictly intertwined, and we compare them to the corresponding concatenation hierarchies. Furthermore, we look at two standard problems on classes of languages: membership (decide if an input language is in the class) and separation (decide, for two input regular languages $L_1,L_2$, if there is a language $K$ in the class with $L_1 \subseteq K$ and $L_2 \cap K = \emptyset$). We prove that if separation is decidable for $\mathcal{G}$, then so is membership for level two in the temporal hierarchies of bases $\mathcal{G}$ and $\mathcal{G}^+$. Moreover, we take a closer look at the case where $\mathcal{G}$ is the trivial class $ST=\{\emptyset,A^*\}$. The levels one in the hierarchies of bases $ST$ and $ST^+$ are the standard variants of unary temporal logic while the levels two were considered recently using alternate definitions. We prove that for these two bases, level two has decidable separation. Combined with earlier results about the operator $\mathcal{G}\mapsto TL(\mathcal{G})$, this implies that the levels three have decidable membership.

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常规语言的时间层次结构

我们使用运算符 $\mathcal{C}\mapsto TL(\mathcal{C})$ 对常规语言进行分类。对于语言的每个输入类$\mathcal{C}$，它构建一个更大的类$TL(\mathcal{C})$，其中包含可在一元时序逻辑的变体中定义的所有语言，其未来/过去模态取决于$\数学{C}$。这定义了基础 $\mathcal{C}$ 的时间层次结构：级别 $n$ 是通过将此运算符 $n$ 次应用于 $\mathcal{C}$ 来构建的。该层次结构与另一个层次结构密切相关，即基础 $\mathcal{C}$ 的串联层次结构。特别是，两个层次结构中所有级别的联合是相同的。我们关注群体语言的基 $\mathcal{G}$ 及其自然扩展，表示为 $\mathcal{G}^+$。我们证明了基 $\mathcal{G}$ 和 $\mathcal{G}^+$ 的时间层次结构是严格交织在一起的，并将它们与相应的串联层次结构进行比较。此外，我们研究语言类的两个标准问题：成员资格（决定输入语言是否在该类中）和分离（决定，对于两个输入常规语言 $L_1,L_2$，如果存在语言 $K$具有 $L_1 \subseteq K$ 和 $L_2 \cap K = \emptyset$ 的类）。我们证明，如果 $\mathcal{G}$ 的分离是可判定的，那么基 $\mathcal{G}$ 和 $\mathcal{G}^+$ 的时间层次结构中第二级的成员资格也是可判定的。此外，我们仔细研究 $\mathcal{G}$ 是平凡类 $ST=\{\emptyset,A^*\}$ 的情况。基 $ST$ 和 $ST^+$ 的层次结构中的第一级是一元时序逻辑的标准变体，而第二级最近被考虑使用替代定义。我们证明，对于这两个基地，第二级具有可判定的分离。结合之前关于运算符 $\mathcal{G}\mapsto TL(\mathcal{G})$ 的结果，这意味着三个级别具有可判定的成员资格。