Exploring the relation between Intuitionistic BI and Boolean BI: an unexpected embedding

The logic of Bunched Implications, through both its intuitionistic version ( BI ) and one of its classical versions, called Boolean BI ( BBI ), serves as a logical basis to spatial or separation logic frameworks. In BI , the logical implication is interpreted intuitionistically whereas it is generally interpreted classically in spatial or separation logics, as in BBI . In this paper, we aim to give some new insights into the semantic relations between BI and BBI . Then we propose a sound and complete syntactic constraints based framework for the Kripke semantics of both BI and BBI , a sound labelled tableau proof system for BBI , and a representation theorem relating the syntactic models of BI to those of BBI . Finally, we deduce as our main, and unexpected, result, a sound and faithful embedding of BI into BBI .

[1]  Étienne Lozes.  Adjuncts elimination in the static ambient logic , 2003, EXPRESS.

[2]  David J. Pym,et al.  The semantics and proof theory of the logic of bunched implications , 2002, Applied logic series.

[3]  Dominique Larchey-Wendling,et al.  Expressivity properties of boolean BI through relational models , 2006 .

[4]  Anuj Dawar,et al.  Adjunct Elimination Through Games in Static Ambient Logic , 2004, FSTTCS.

[5]  Daniel Méry.  Preuves et sémantiques dans des logiques de ressources , 2004 .

[6]  David J. Pym,et al.  Systems Modelling via Resources and Processes: Philosophy, Calculus, Semantics, and Logic , 2007, Computation, Meaning, and Logic.

[7]  Luca Cardelli,et al.  Anytime, anywhere: modal logics for mobile ambients , 2000, POPL '00.

[8]  David J. Pym,et al.  The semantics of BI and resource tableaux , 2005, Mathematical Structures in Computer Science.

[9]  Luca Cardelli,et al.  Deciding validity in a spatial logic for trees , 2003, TLDI '03.

[10]  Peter W. O'Hearn,et al.  Local Reasoning about Programs that Alter Data Structures , 2001, CSL.

[11]  Nicolas Biri,et al.  Models and Separation Logics for Resource Trees , 2007, J. Log. Comput..

[12]  Christoph Kreitz,et al.  T-String Unification: Unifying Prefixes in Non-classical Proof Methods , 1996, TABLEAUX.

[13]  Didier Galmiche,et al.  Semantic Labelled Tableaux for Propositional BI , 2003, J. Log. Comput..

[14]  Luís Caires,et al.  Elimination of quantifiers and undecidability in spatial logics for concurrency , 2006, Theor. Comput. Sci..

[15]  Richard Statman,et al.  Intuitionistic Propositional Logic is Polynomial-Space Complete , 1979, Theor. Comput. Sci..

[16]  Peter W. O'Hearn,et al.  BI as an assertion language for mutable data structures , 2001, POPL '01.

[17]  David J. Pym,et al.  A Calculus and logic of resources and processes , 2006, Formal Aspects of Computing.

[18]  Melvin Fitting,et al.  First-Order Logic and Automated Theorem Proving , 1990, Graduate Texts in Computer Science.

[19]  David J. Pym,et al.  On bunched predicate logic , 1999, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158).

[20]  Philippa Gardner,et al.  Context logic as modal logic: completeness and parametric inexpressivity , 2007, POPL '07.