About the type of modal logics for the unification problem

Dans cette thèse, nous étudierons le problème de l'unification dans les logiques modales ordinaires, les fusions de deux logiques modales et les logiques épistémiques multi-modales. Relativement à une logique propositionnelle L, étant donnée une formule A, nous devons trouver des substitutions s telle que s(A) est dans L. Lorsqu'elles existent, ces substitutions sont appelées unifieurs de A dans L. Nous étudions différentes méthodes pour construire des ensembles minimaux complets d'unifieurs d'une formule donnée A et, en fonction de la cardinalité des ces ensembles minimaux complets, nous discutons du type de l'unification de A. Enfin, nous déterminons les types de l'unification de plusieurs logiques propositionnelles.In this thesis, we shall investigate on the unification problem in ordinary modal logics, fusions of two modal logics and multi-modal epistemic logics. With respect to a propositional logic L, given a formula A, we have to find substitutions s such that s(A) is in L. When they exist, these substitutions are called unifiers of A in L. We study different methods for the construction of minimal complete sets of unifiers of a given formula A and according to the cardinality of these minimal complete sets, we shall discuss on the unification type of A. Then, we determine the unification types of several propositional logics

Restricted Unification in the DL FL₀: Extended Version

Unification in the Description Logic (DL) FL₀ is known to be ExpTimecomplete, and of unification type zero. We investigate in this paper whether a lower complexity of the unification problem can be achieved by either syntactically restricting the role depth of concepts or semantically restricting the length of role paths in interpretations. We show that the answer to this question depends on whether the number formulating such a restriction is encoded in unary or binary: for unary coding, the complexity drops from ExpTime to PSpace. As an auxiliary result, which is however also of interest in its own right, we prove a PSpace-completeness result for a depth-restricted version of the intersection emptiness problem for deterministic root-to-frontier tree automata. Finally, we show that the unification type of FL₀ improves from type zero to unitary (finitary) for unification without (with) constants in the restricted setting

A propos du type de logiques modales pour le problème d'unification

In this thesis, we shall investigate the unification problem in ordinary modal logics, fusions of two modal logics, and multi-modal epistemic logics. With respect to a propositional logic L, given a formula A, we have to find substitutions s such that s(A) is in L. When they exist, these substitutions are called unifiers of A in L. We study different methods for the construction of minimal complete sets of unifiers of a given formula A and according to the cardinality of these minimal complete sets, we shall discuss the unification type of A. Then, we determine the unification types of several propositional logics.Dans cette thèse, nous étudierons le problème de l’unification dans les logiques modales ordinaires, les fusions de deux logiques modales et les logiques épistémiques multi-modales. Relativement à une logique propositionnelle L, étant donnée une formule A, nous devons trouver des substitutions s telles que s(A) est dans L. Lorsqu’elles existent, ces substitutions sont appelées unifieurs de A dans L. Nous étudions différentes méthodes pour construire des ensembles minimaux complets d’unifieurs d’une formule donnée A et, en fonction de la cardinalité des ces ensembles minimaux complets, nous discutons du type de l’unification de A. Enfin, nous déterminons les types de l’unification de plusieurs logiques propositionnelle

About the unification type of fusions of modal logics

International audienceIn a modal logic L, a unifier of a formula ϕ is a substitution σ such that σ(ϕ) is in L. When unifiable formulas have no minimal complete sets of unifiers, they are nullary. Otherwise, they are either infinitary, or finitary, or unitary depending on the cardinality of their minimal complete sets of unifiers. In this paper, we prove that if the fusion L 1 ⊗ L 2 is unitary then L 1 and L 2 are unitary and if the fusion L 1 ⊗ L 2 is finitary then L 1 and L 2 are either unitary, or finitary. We also prove that the fusion of arbitrary consistent extensions of S5 is nullary when these extensions are different from Triv

Restricted Unification in the DL FL₀: Extended Version

Unification in the Description Logic (DL) FL₀ is known to be ExpTimecomplete, and of unification type zero. We investigate in this paper whether a lower complexity of the unification problem can be achieved by either syntactically restricting the role depth of concepts or semantically restricting the length of role paths in interpretations. We show that the answer to this question depends on whether the number formulating such a restriction is encoded in unary or binary: for unary coding, the complexity drops from ExpTime to PSpace. As an auxiliary result, which is however also of interest in its own right, we prove a PSpace-completeness result for a depth-restricted version of the intersection emptiness problem for deterministic root-to-frontier tree automata. Finally, we show that the unification type of FL₀ improves from type zero to unitary (finitary) for unification without (with) constants in the restricted setting

Restricted Unification in the DL FL₀: Extended Version

Unification in the Description Logic (DL) FL₀ is known to be ExpTimecomplete, and of unification type zero. We investigate in this paper whether a lower complexity of the unification problem can be achieved by either syntactically restricting the role depth of concepts or semantically restricting the length of role paths in interpretations. We show that the answer to this question depends on whether the number formulating such a restriction is encoded in unary or binary: for unary coding, the complexity drops from ExpTime to PSpace. As an auxiliary result, which is however also of interest in its own right, we prove a PSpace-completeness result for a depth-restricted version of the intersection emptiness problem for deterministic root-to-frontier tree automata. Finally, we show that the unification type of FL₀ improves from type zero to unitary (finitary) for unification without (with) constants in the restricted setting

Remarks about the unification types of some locally tabular normal modal logics

International audienceIt is already known that unifiable formulas in normal modal logic K + Box Box ⊥ are either finitary, or unitary and unifiable formulas in normal modal logic Alt1 + Box Box ⊥ are unitary. In this paper, we prove that for all d≥3, unifiable formulas in normal modal logic K + Box^{d} ⊥ are either finitary, or unitary and unifiable formulas in normal modal logic Alt1 + Box^{d} ⊥ are unitary

About the unification type of K + [][]⊥

International audienc

A gentle introduction to unification in modal logics

International audienceUnification in propositional logics is an active research area. In this paper, we introduce the results we have obtained within the context of modal logics and epistemic logics and we present some of the open problems whose solution will have an important impact on the future of the area.L'unification dans les logiques propositionnelles est un domaine de recherche actif. Dans cet article, nous présentons les résultats que nous avons obtenus dans le cadre des logiques modales et des logiqueś epistémiques et nous introduisons quelques uns des problèmes ouverts dont la résolution aura un impact important sur l'avenir du domaine