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

Logiques de Ressources Dynamiques : Modèles, Propriétés et Preuves

In computer science, the notion of resource is a central concern. We consider as a resource, any entity that can be composed or decomposed into sub-entities. Many logics were proposed to model and express properties on these resources, like BI logic, a logic about sharing and separation of resources. As the computer systems manipulate resources, a crucial issue consists in providing new models that capture the dynamics of resources, and also in verifying and proving properties on these models. In this context, we define new logics with new models and new languages allowing to respectively capture and express new properties on the dynamics of resources.Moreover, for all these logics, we also study the foundations of proof search and provide tableau methods and counter-model extraction methods.After defining new Petri nets we propose a new semantics based on such nets for BI logic, that allows us to show that BI is able to capture a kind of dynamics of resources. After observing that it is necessary to introduce new modalities in BI logic, we study successively different modal extensions of BI. We define a logic, called DBI, that allows us to model resources having dynamic properties, meaning that they evolve during the iterations of a system.Then, we define a logic, called DMBI, that allows us to model systems that manipulate/produce/consume resources.Moreover, we define a new modal logic, called LSM, having new multiplicative modalities, that deals with resources.Finally, we introduce the notion of separation in Epistemic Logic, obtaining a new logic, called ESL, that models and expresses new properties on agent knowledge.En informatique, la notion de ressource, entité pouvant être composée ou décomposée en sous-entités, est une notion centrale. Plusieurs logiques ont été proposées en vue de modéliser et d'exprimer des propriétés sur celles-ci, comme par exemple la logique BI, permettant d'exprimer des propriétés de partage et de séparation. Puisque les systèmes informatiques manipulent des ressources, la proposition de nouveaux modèles capturant la dynamique de ces ressources, ainsi que des méthodes permettant la vérification et la preuve de propriétés sur ces modèles, sont des enjeux cruciaux. Dans ce contexte, nous nous intéressons à la modélisation logique de la dynamique des ressources, avec pour objectif la proposition de nouveaux modèles et de nouveaux langages permettant l'expression de nouvelles propriétés sur la dynamique des ressources. De plus, pour les logiques que nous proposons, nous étudions la recherche de preuves en proposant des méthodes des tableaux et d'extraction de contre-modèles.Dans un premier temps, nous définissons de nouveaux réseaux de Petri et nous proposons une nouvelle sémantique à base de tels réseaux de Petri pour BI, qui capture ainsi une forme de dynamique des ressources. Après avoir analysé la nécessité d'introduire de nouvelles modalités dans BI, nous étudions successivement différentes extensions modales de BI. Nous proposons alors une première extension, nommée DBI, permettant la modélisation de ressources ayant des propriétés dynamiques, c'est-à-dire évoluant en fonction de l'état courant d'un système. Puis, nous proposons une logique, nommée DMBI, permettant la modélisation de systèmes manipulant/produisant/consommant des ressources.Par ailleurs, nous proposons une nouvelle logique modale, nommée LSM, possédant de nouvelles modalités multiplicatives, exprimant des propriétés dynamiques en lien avec les ressources. Pour finir, nous introduisons la séparation au sein des logiques épistémiques, obtenant alors une nouvelle logique, nommée ESL, en vue d'exprimer de nouvelles propriétés en lien avec la connaissance

Non-normal modalities in variants of Linear Logic

This article presents modal versions of resource-conscious logics. We concentrate on extensions of variants of Linear Logic with one minimal non-normal modality. In earlier work, where we investigated agency in multi-agent systems, we have shown that the results scale up to logics with multiple non-minimal modalities. Here, we start with the language of propositional intuitionistic Linear Logic without the additive disjunction, to which we add a modality. We provide an interpretation of this language on a class of Kripke resource models extended with a neighbourhood function: modal Kripke resource models. We propose a Hilbert-style axiomatization and a Gentzen-style sequent calculus. We show that the proof theories are sound and complete with respect to the class of modal Kripke resource models. We show that the sequent calculus admits cut elimination and that proof-search is in PSPACE. We then show how to extend the results when non-commutative connectives are added to the language. Finally, we put the logical framework to use by instantiating it as logics of agency. In particular, we propose a logic to reason about the resource-sensitive use of artefacts and illustrate it with a variety of examples

Spatial Logics for Bigraphs

Bigraphs are emerging as an interesting model for concurrent calculi, like CCS, pi-calculus, and Petri nets. Bigraphs are built orthogonally on two structures: a hierarchical place graph for locations and a link (hyper-)graph for connections. With the aim of describing bigraphical structures, we introduce a general framework for logics whose terms represent arrows in monoidal categories. We then instantiate the framework to bigraphical structures and obtain a logic that is a natural composition of a place graph logic and a link graph logic. We explore the concepts of separation and sharing in these logics and we prove that they generalise some known spatial logics for trees, graphs and tree contexts

Type-driven semantic interpretation and feature dependencies in R-LFG

Once one has enriched LFG's formal machinery with the linear logic mechanisms needed for semantic interpretation as proposed by Dalrymple et. al., it is natural to ask whether these make any existing components of LFG redundant. As Dalrymple and her colleagues note, LFG's f-structure completeness and coherence constraints fall out as a by-product of the linear logic machinery they propose for semantic interpretation, thus making those f-structure mechanisms redundant. Given that linear logic machinery or something like it is independently needed for semantic interpretation, it seems reasonable to explore the extent to which it is capable of handling feature structure constraints as well. R-LFG represents the extreme position that all linguistically required feature structure dependencies can be captured by the resource-accounting machinery of a linear or similiar logic independently needed for semantic interpretation, making LFG's unification machinery redundant. The goal is to show that LFG linguistic analyses can be expressed as clearly and perspicuously using the smaller set of mechanisms of R-LFG as they can using the much larger set of unification-based mechanisms in LFG: if this is the case then we will have shown that positing these extra f-structure mechanisms is not linguistically warranted.Comment: 30 pages, to appear in the the ``Glue Language'' volume edited by Dalrymple, uses tree-dvips, ipa, epic, eepic, fullnam