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

Labelled Modal Tableaux

Labelled tableaux are extensions of semantic tableaux with annotations (labels, indices) whose main function is to enrich the modal object language with semantic elements. This paper consists of three parts. In the first part we consider some options for labels: simple constant labels vs labels with free variables, logic depended inference rules vs labels manipulation based on a label algebra. In the second and third part we concentrate on a particular labelled tableaux system called KEM using free variable and a specialised label algebra. Specifically in the second part we show how labelled tableaux (KEM) can account for different types of logics (e.g., non-normal modal logics and conditional logics). In the third and final part we investigate the relative complexity of labelled tableaux systems and we show that the uses of KEM's label algebra can lead to speed up on proofs

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

Stone-Type Dualities for Separation Logics

Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because -- in addition to elegant abstraction -- they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar BI and Boolean BI (BBI), and extending to both classical and intuitionistic Separation Logic. We demonstrate the uniformity and modularity of this analysis by additionally capturing the bunched logics obtained by extending BI and BBI with modalities and multiplicative connectives corresponding to disjunction, negation and falsum. This includes the logic of separating modalities (LSM), De Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as corollaries soundness and completeness theorems for the specific Kripke-style models of these logics as presented in the literature: for DMBI, the sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene BI (connecting our work to Concurrent Separation Logic), this is the first time soundness and completeness theorems have been proved. We thus obtain a comprehensive semantic account of the multiplicative variants of all standard propositional connectives in the bunched logic setting. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualizes the "resource semantics" interpretation underpinning Separation Logic amongst them

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book