正则语言的导航层次
Navigational hierarchies of regular languages
AI总结：
本文研究无星语言的导航层次，通过算子 $C\mapsto TL(C)$ 构建替代层次结构，证明基 $G$ 和 $G^+$ 的导航层次严格交织，并解决相关逻辑变体的可判定性问题与猜想。
摘要：
我们研究无星语言类。一个长期存在的目标是根据其描述的复杂性对它们进行分类。最具影响力的研究工作涉及拼接层次，它衡量“补”与“并加拼接”之间的交替。我们探索同样对无星语言进行分层的替代层次。它们通过算子 $C\mapsto TL(C)$ 构建。从输入类 $C$ 出发，它产生一个更大的类 $TL(C)$，由在一元时序逻辑变体中可定义的所有语言组成，其中时态模态依赖于 $C$。基 $C$ 的导航层次中的第 $n$ 层是通过将该算子应用于 $C$ $n$ 次来构造的。作为基 $G$，我们关注群语言及其自然扩展，记为 $G^+$。我们证明基 $G$ 和 $G^+$ 的导航层次严格交织，并深入研究了它们与拼接层次的关系。我们还考察了语言类上的两个问题：成员资格（判定一个语言是否在该类中）和分离（对于两个语言 $L_1,L_2$，判定该类中是否存在语言 $K$ 满足 $L_1\subseteq K$ 且 $L_2\cap K=\emptyset$）。我们证明如果 $G$ 的分离是可判定的，那么基 $G$ 和 $G^+$ 的导航层次中第二层的成员资格也是可判定的。我们考察了平凡类 $ST=\{\emptyset,A^*\}$。对于基 $ST$ 和 $ST^+$，第一层是一元时序逻辑的标准变体。第二层对应于二变量逻辑的变体，最近由 Krebs、Lodaya、Pandya 和 Straubing 进行了研究。我们解决了他们的一个猜想。我们还证明对于这两个基，第二层具有可判定的分离。结合关于算子 $C\mapsto TL(C)$ 的早期结果，这蕴含第三层具有可判定的成员资格。
英文摘要：
We study the class of star-free languages. A long-standing goal is to classify them by the complexity of their descriptions. The most influential research effort involves concatenation hierarchies, which measure alternations between ``complement'' and ``union plus concatenation''. We explore alternative hierarchies that also stratify star-free languages. They are built with an operator $C\mapsto TL(C)$. From an input class $C$, it produces a larger one $TL(C)$, consisting of all languages definable in a variant of unary temporal logic, where temporal modalities depend on $C$. Level $n$ in the navigational hierarchy of basis $C$ is constructed by applying this operator $n$ times to $C$. As bases $G$, we focus on group languages and natural extensions thereof, denoted $G^+$. We prove that the navigational hierarchies of bases $G$ and $G^+$ are strictly intertwined and conduct a thorough investigation of their relationships with concatenation hierarchies. We also look at two problems on classes of languages: membership (decide if a language is in the class) and separation (decide, for two 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 $G$, then so is membership for level \emph{two} in the navigational hierarchies of bases $G$ and $G^+$. We take a look at the trivial class $ST=\{\emptyset,A^*\}$. For the bases $ST$ and $ST^+$, the levels \emph{one} are standard variants of unary temporal logic. The levels \emph{two} correspond to variants of two-variable logic, investigated recently by Krebs, Lodaya, Pandya and Straubing. We solve one of their conjectures. We also prove that for these two bases, level \emph{two} has decidable \emph{separation}. Combined with earlier results on the operator $C\mapsto TL(C)$, this implies that level \emph{three} has decidable membership.
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