Temporal hierarchies of regular languages

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 $L1,L2$, if there is a language $K$ in the class with $L1 \subseteq K$ and $L2 \cap K = \emptyset$). 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.