In this paper we investigate those extensions of the bimodal provability logic C⃗ SM0 (alias P⃗ RL1 or F⃗ −) which are subframe logics, i.e. whose general frames are closed under a certain type of substructures. Most bimodal provability logics are in this class. The main result states that all finitely axiomatizable subframe logics containing C⃗ SM0 are decidable. We note that, as a rule, interesting systems in this class do not have the finite model property and are not even complete with respect to Kripke semantics.

This is a study of the semantics and proof theory of the logic of bunched implications (BI), which is promoted as a logic of (computational) resources, and is a foundational component of separation logic, an approach to program analysis. BI combines an additive, or intuitionistic, fragment with a multiplicative fragment. The additive fragment has full use of the structural rules of weakening and contraction, and the multiplicative fragment has none. Thus it contains two conjunctive and two implicative connectives. At various points, we illustrate a resource view of BI based upon the Kripke resource semantics. Our first original contribution is the formulation of a proof system for BI in the newly developed proof-theoretical formalism of the calculus of structures. The calculus of structures is distinguished by its employment of deep inference, but we already see deep inference in a limited form in the established proof theory for BI. We show that our system is sound with respect to the elementary Kripke resource semantics for BI, and complete with respect to a formulation of the partially-defined Kripke resource semantics. Our second contribution is the development from a semantic standpoint of preliminary ideas for a hybrid logic of bunched implications (HBI). We give a Kripke semantics for HBI in which nominal propositional atoms can be seen as names for resources, rather than as names for locations, as is the case with related proposals for BI-Loc and for intuitionistic hybrid logic.

Dans le domaine de la modélisation du raisonnement, plusieurs approches se basent sur les logiques modales qui permettent de formaliser le raisonnement sur des éléments non factuels, comme la croyance, le savoir ou encore la nécessité. Une extension pondérées de ces logiques modales permet de moduler les éléments non factuels qu'elle décrit. En particulier, nous nous intéressons à l'extension pondérée des logiques modales qui permet de formaliser des croyances graduelles : nous traitons des aspects sémantiques et axiomatiques ainsi que des aspects syntaxiques liés à la manipulations de telles croyances modulées. Ainsi, les travaux de cette thèse sont organisés en trois parties. Nous proposons, d'une part, une sémantique proportionnelle qui étend la sémantique de Kripke classiquement utilisée pour les logiques modales ; ainsi qu'une étude des axiomes modaux dans le contexte de cette sémantique des modalités pondérées. D'autre part, nous proposons un modèle ensembliste flou pour représenter et manipuler des degrés de croyances. Enfin, nous mettons en œuvre ces modèles théoriques dans deux applications : un outil de vérification de formules modales pondérées et un joueur artificiel pour le jeu coopératif Hanabi dont la prise de décision repose sur un raisonnement sur ses propres croyances. / In the field of reasoning models, many approaches are based on modal logics, which allow to formalise the non-factual reasoning, as belief, knowledge or necessity reasoning. A weighted extension for these modal logics aims at modulating the considered non-factual elements. In particular, we examine the weighted extension of modal logics for graded beliefs: we study their semantical and axiomatical issues related to manipulating such modulated beliefs. Therefore, this thesis works are organised in three parts. We first propose a proportional semantics which extends the Kripke semantics, classically used for modal logics. We also study modal axioms regarding the proposed semantics. Then, we propose a fuzzy set model for representing and manipulating belief degrees. We finally use these two formal models in two different applications: a model checking tool for weighted modal formulae and an artifical player for a cooperative game called Hanabi in which decision making is based on graded belief reasoning.