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

Published online by Cambridge University Press:  01 June 2009

DOMINIQUE LARCHEY-WENDLING  and
DIDIER GALMICHE
DOMINIQUE LARCHEY-WENDLING
Affiliation:
LORIA - CNRS, Campus Scientifique, BP 239, 54 506 Vandœuvre-lès-Nancy, France Email: larchey@loria.fr
DIDIER GALMICHE
Affiliation:
Université Henri Poincaré, Campus Scientifique, BP 239, 54 506 Vandœuvre-lès-Nancy, France Email: galmiche@loria.fr
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Abstract

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.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 19 , Issue 3 , June 2009 , pp. 435 - 500
Copyright
Copyright © Cambridge University Press 2009

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References

Biri, N. and Galmiche, D. (2007) Models and Separation Logics for Resource Trees. Journal of Logic and Computation 17 (4)687–726.CrossRefGoogle Scholar
Brochenin, R., Demri, S. and Lozes, E. (2008) On the Almighty Wand. In: Proceedings of the 17th EACSL Annual Conference on Computer Science Logic (CSL'08). Springer-Verlag Lecture Notes in Computer Science 5213 323–338.CrossRefGoogle Scholar
Caires, L. and Lozes, E. (2006) Elimination of Quantifiers and Undecidability in Spatial Logics for Concurrency. Theoretical Computer Science 358 (2–3)293–314.Google Scholar
Calcagno, C., Cardelli, L. and Gordon, A. D. (2005) Deciding Validity in a Spatial Logic for Trees. Journal of Functional Programming 15 (4)543–572.CrossRefGoogle Scholar
Calcagno, C., Gardner, Ph. and Zarfaty, U. (2007) Context Logic as Modal Logic: Completeness and Parametric Inexpressivity. In: Proceedings of the 34th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, ACM 123–134.Google Scholar
Cardelli, L. and Gordon, A. D. (2000) Anytime, Anywhere: Modal Logics for Mobile Ambients. In: Proceedings of the 27th ACM SIGPLAN-SIGACT Principles of Programming Languages conference, ACM 365–377.Google Scholar
Dawar, A., Gardner, Ph. and Ghelli, G. (2004) Adjunct Elimination Through Games in Static Ambient Logic. In: Proceedings of the 24th Conference on Foundations of Software Technology and Theoretical Computer Science. Springer-Verlag Lecture Notes in Computer Science 3328 211–223.Google Scholar
Fitting, M. (1990) First-Order Logic and Automated Theorem Proving, Texts and Monographs in Computer Science, Springer-Verlag.Google Scholar
Galmiche, D. and Larchey-Wendling, D. (2006) Expressivity properties of Boolean BI through Relational Models. In: Proceedings of the 26th Conference on Foundations of Software Technology and Theoretical Computer Science. Springer-Verlag Lecture Notes in Computer Science 4337 358–369.Google Scholar
Galmiche, D. and Méry, D. (2003) Semantic Labelled Tableaux for Propositional BI. Journal of Logic and Computation 13 (5)707–753.Google Scholar
Galmiche, D., Méry, D. and Pym, D. (2005) The semantics of BI and resource tableaux. Mathematical Structures in Computer Science 15 (6)1033–1088.Google Scholar
Ishtiaq, S. and O'Hearn, P. (2001) BI as an Assertion Language for Mutable Data Structures. In: Proceedings of the 28th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, ACM 14–26.Google Scholar
Lozes, E. (2004) Adjuncts elimination in the static ambient logic. Electronic Notes in Theoretical Computer Science 96 51–72.CrossRefGoogle Scholar
Méry, D. (2004) Preuves et Sémantiques dans des Logiques de Ressources, Ph.D. thesis, Université Henri Poincaré, Nancy I, France. (Available at http://www.loria.fr/~dmery/these.pdf.)Google Scholar
O'Hearn, P., Reynolds, J. and Yang, H. (2001) Local Reasoning about Programs that Alter Data Structures. In: Proceedings of the 10th EACSL Annual Conference on Computer Science Logic (CSL'01). Springer-Verlag Lecture Notes in Computer Science 2142 1–19.Google Scholar
Otten, J. and Kreitz, C. (1996) T-String Unification: Unifying Prefixes in Non-classical Proof Methods. In: Proceedings of the 5th International Workshop on Theorem Proving with Analytic Tableaux and Related Methods. Springer-Verlag Lecture Notes in Computer Science 1071 244–260.Google Scholar
Pym, D. (1999) On Bunched Predicate Logic. In: Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science, IEEE 183–192.Google Scholar
Pym, D. (2002) The Semantics and Proof Theory of the Logic of Bunched Implications. Applied Logic Series 26, Kluwer Academic Publishers. (Errata available at http://www.cs.ac.uk/~pym/pym-tofts-fac-errata.pdf.)Google Scholar
Pym, D. and Tofts, C. (2006) A Calculus and logic of resources and processes, Formal Aspects of Computing 18 (4)495–517. (Errata available at http://www.cs.ac.uk/~pym/pym-tofts-fac-errata.pdf.)CrossRefGoogle Scholar
Pym, D. and Tofts, C. (2007) Systems modelling via resources and processes: Philosophy, calculus, semantics, and logic. Electronic Notes in Theoretical Computer Science 172, 545–587. Elsevier. (Errata available at http://www.cs.ac.uk/~pym/pym-tofts-fac-errata.pdf.CrossRefGoogle Scholar
Statman, R. (1979) Intuitionistic Propositional Logic is Polynomial-Space Complete. Theoretical Computer Science 9 67–72.Google Scholar