On All Things Star-Free

We investigate the star-free closure, which associates to a class of languages its closure under Boolean operations and marked concatenation. We prove that the star-free closure of any finite class and of any class of groups languages with decidable separation (plus mild additional properties) has decidable separation. We actually show decidability of a stronger property, called covering. This generalizes many results on the subject in a unified framework. A key ingredient is that star-free closure coincides with another closure operator where Kleene stars are also allowed in restricted contexts

A generic characterization of generalized unary temporal logic and two-variable first-order logic

We investigate an operator on classes of languages. For each class $C$, it outputs a new class $FO^2(I_C)$ associated with a variant of two-variable first-order logic equipped with a signature$I_C$ built from $C$. For $C = \{\emptyset, A^*\}$, we get the variant $FO^2(<)$ equipped with the linear order. For $C = \{\emptyset, \{\varepsilon\},A^+, A^*\}$, we get the variant $FO^2(<,+1)$, which also includes the successor. If $C$ consists of all Boolean combinations of languages $A^*aA^*$ where $a$ is a letter, we get the variant $FO^2(<,Bet)$, which also includes ``between relations''. We prove a generic algebraic characterization of the classes $FO^2(I_C)$. It smoothly and elegantly generalizes the known ones for all aforementioned cases. Moreover, it implies that if $C$ has decidable separation (plus mild properties), then $FO^2(I_C)$ has a decidable membership problem. We actually work with an equivalent definition of \fodc in terms of unary temporal logic. For each class $C$, we consider a variant $TL(C)$ of unary temporal logic whose future/past modalities depend on $C$ and such that $TL(C) = FO^2(I_C)$. Finally, we also characterize $FL(C)$ and $PL(C)$, the pure-future and pure-past restrictions of $TL(C)$. These characterizations as well imply that if \Cs is a class with decidable separation, then $FL(C)$ and $PL(C)$ have decidable membership

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 $L_1,L_2$, 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

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems