Navigational hierarchies of regular languages

Abstract

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.