Focused Proof-search in the Logic of Bunched Implications

The logic of Bunched Implications (BI) freely combines additive and multiplicative connectives, including implications; however, despite its well-studied proof theory, proof-search in BI has always been a difficult problem. The focusing principle is a restriction of the proof-search space that can capture various goal-directed proof-search procedures. In this paper, we show that focused proof-search is complete for BI by first reformulating the traditional bunched sequent calculus using the simpler data-structure of nested sequents, following with a polarised and focused variant that we show is sound and complete via a cut-elimination argument. This establishes an operational semantics for focused proof-search in the logic of Bunched Implications.Comment: 18 pages conten

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

Structural Interactions and Absorption of Structural Rules in BI Sequent Calculus

Development of a contraction-free BI sequent calculus, be it in the sense of G3i or G4i, has not been successful in literature. We address the open problem by presenting such a sequent system. In fact our calculus involves no structural rules

Coalgebraic completeness-via-canonicity for distributive substructural logics

We prove strong completeness of a range of substructural logics with respect to a natural poset-based relational semantics using a coalgebraic version of completeness-via-canonicity. By formalizing the problem in the language of coalgebraic logics, we develop a modular theory which covers a wide variety of different logics under a single framework, and lends itself to further extensions. Moreover, we believe that the coalgebraic framework provides a systematic and principled way to study the relationship between resource models on the semantics side, and substructural logics on the syntactic side.Comment: 36 page

Structural Interactions and Absorption of Structural Rules in BI Sequent Calculus

Development of a contraction-free BI sequent calculus, be the contraction-freeness implicit or explicit, has not been successful in the literature. We address this problem by presenting such a sequent system. Our calculus involves no structural rules. It should be an insight into non-formula contraction absorption in other non-classical logics. Contraction absorption in sequent calculus is associated to simpler cut elimination and to efficient proof searches

Latin America and East Asia in the Context of an Insurance Model of Currency Crises

This paper focuses on the 1995 Latin American and 1997 East Asian crises using an insurance-based model of financial crises. First the model of Dooley (forthcoming) is described. Second, some empirical evidence for an insurance model is presented. The key variables in this approach include the ratio of foreign exchange reserves to bank loans (domestic credit) extended to the private sector, the ability of the private sector to appropriate government assets, and appropriation as measured by capital flight. We argue that the insurance model is consistent with the observed evolution of these variables in the recent crises in Latin America and Asia. Finally, we examine the statistical evidence in favor of the model using panel regressions. We find that the econometric results are consistent with the insurance model, and tend to support this approach over some competing explanations.

Ribbon Proofs - A Proof System for the Logic of Bunched Implications

Submitted for the degree of Doctor of Philosophy, Queen Mary, University of London

Un système argumentatif pour le raisonnement sur des ressources limitées

Dans cet article, nous proposons quelques bases pour l’argumentation déductive pour le raisonnement sur des ressources consommables et limitées. Nous nous appuyons sur une nouvelle logique, simple et proche du langage et des principes de la logique booléenne, permettant le raisonnement à partir de ressources consommables en quantité bornée. Une méthode des tableaux sémantiques pour cette logique est fournie. Enfin, pour prendre en compte la rareté des ressources consommables en argumentation, nous développons une approche pour le traitement du raisonnement argumentatif à partir des ressources consommables en quantité bornée