skip to main content
10.1145/1480881.1480923acmconferencesArticle/Chapter ViewAbstractPublication PagespoplConference Proceedingsconference-collections
research-article

Classical BI: a logic for reasoning about dualising resources

Published:21 January 2009Publication History

ABSTRACT

We show how to extend O'Hearn and Pym's logic of bunched implications, BI, to classical BI (CBI), in which both the additive and the multiplicative connectives behave classically. Specifically, CBI is a non-conservative extension of (propositional) Boolean BI that includes multiplicative versions of falsity, negation and disjunction. We give an algebraic semantics for CBI that leads us naturally to consider resource models of CBI in which every resource has a unique dual. We then give a cut-eliminating proof system for CBI, based on Belnap's display logic, and demonstrate soundness and completeness of this proof system with respect to our semantics.

References

  1. Nuel D. Belnap, Jr. Display logic. Journal of Philosophical Logic, 11:375--417, 1982.Google ScholarGoogle ScholarCross RefCross Ref
  2. Josh Berdine and Peter O'Hearn. Strong update, disposal and encapsulation in bunched typing. In Proceedings of MFPS, ENTCS. Elsevier, 2006. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Patrick Blackburn, Maarten de Rijke, and Yde Venema. Modal Logic. Cambridge University Press, 2001. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. R. Bornat, C. Calcagno, P. O'Hearn, and M. Parkinson. Permission accounting in separation logic. In 32nd POPL, pp59--70, 2005. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. James Brotherston and Cristiano Calcagno. Algebraic models and complete proof calculi for classical BI. Technical Report 2008/7, Imperial College London, 2008. Available from http://www.doc.ic.ac.uk/~jbrother.Google ScholarGoogle Scholar
  6. James Brotherston and Cristiano Calcagno. Classical logic of bunched implications. In the informal proceedings of CL&C 2008, an ICALP satellite workshop; available from http://www.doc.ic.ac.uk/~jbrother, 2008.Google ScholarGoogle Scholar
  7. Cristiano Calcagno, Dino Distefano, Peter O'Hearn and Hongseok Yang. Compositional Shape Analysis by means of BI-Abduction. In Proceedings of POPL-36, 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. C. Calcagno, P. Gardner, and U. Zarfaty. Context logic as modal logic: Completeness and parametric inexpressivity. In Proceedings of POPL-34, 2007. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Cristiano Calcagno, Matthew Parkinson, and Viktor Vafeiadis. Modular safety checking for fine-grained concurrency. In SAS, 2007. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Bor-Yuh Evan Chang and Xavier Rival. Relational inductive shape analysis. In Proceedings of POPL-35, 2008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Wei-Ngan Chin, Cristina David, Huu Hai Nguyen, and Shengchao Qin. Enhancing modular OO verification with separation logic. In Proceedings of POPL-35, 2008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. Matthew Collinson, David Pym, and Edmund Robinson. Bunched polymorphism. Mathematical Structures in Computer Science, 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. D. Distefano and M. Parkinson. jStar: Towards Practical Verification for Java. In OOPSLA, 2008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Kevin Donnelly, Tyler Gibson, Neel Krishnaswami, Stephen Magill, and Sungwoo Park. The inverse method for the logic of bunched implications. In Proceedings of LPAR 2004, volume 3452 of LNAI, pages 466--480. Springer-Verlag, 2005.Google ScholarGoogle Scholar
  15. Michael Dunn. Star and perp: Two treatments of negation. Philosophical Perspectives, 7:331--357, 1993.Google ScholarGoogle ScholarCross RefCross Ref
  16. D. Galmiche, D. Mery, and D. Pym. The semantics of BI and resource tableaux. Mathematical Structures in Computer Science, 15:1033--1088, 2005. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Didier Galmiche and Dominique Larchey-Wendling. Expressivity properties of Boolean BI through relational In Proceedings of FSTTCS, 2006. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. Jean-Yves Girard. Linear logic: Its syntax and semantics.Google ScholarGoogle Scholar
  19. In J.-Y. Girard, Y. Lafont, and L. Regnier, editors, Advances in Linear Logic, pages 1--42. Cambridge University Press, 1995. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Rajeev Gore. Cut-free display calculi for relation algebras. In Proceedings of CSL'96, volume 1258 of LNCS, pages 198--210, 1997. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Rajeev Gore. Substructural logics on display. Logic Journal of the IGPL, 6(3):451--504, 1998.Google ScholarGoogle ScholarCross RefCross Ref
  22. H.Yang, O.Lee, J.Berdine, C.Calcagno, B.Cook, D.Distefano, and P.O'Hearn. Scalable shape analysis for systems code. In CAV, 2008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. Samin Ishtiaq and Peter W. O'Hearn. BI as an assertion language for mutable data structures. In Proceedings of POPL'01, January 2001. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. H.H. Nguyen and W.-N. Chin. Enhancing program verification with lemmas. In Proceedings of CAV, 2008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. P.W. O'Hearn and D. J. Pym. The logic of bunched implications. Bulletin of Symbolic Logic, 5(2):215--244, June 1999.Google ScholarGoogle ScholarCross RefCross Ref
  26. Matthew Parkinson and Gavin Bierman. Separation logic, abstraction and inheritance. In Proceedings of POPL-35, 2008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  27. David Pym. The Semantics and Proof Theory of the Logic of Bunched Implications. Applied Logic Series. Kluwer, 2002. Errata and remarks (Pym 2004) maintained at http://www.cs.bath.ac.uk/ pym/reductive-logic-errata.html.Google ScholarGoogle Scholar
  28. David Pym, Peter O'Hearn, and Hongseok Yang. Possible worlds and resources: The semantics of BI. Theoretical Computer Science, 315(1):257--305, 2004. Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. S. Read. Relevant Logic: A Philosophical Examination. Basil Blackwell, 1987.Google ScholarGoogle Scholar
  30. John C. Reynolds. Separation logic: A logic for shared mutable data structures. In Proceedings of 17th LICS, 2002. Google ScholarGoogle ScholarDigital LibraryDigital Library
  31. Harold Schellinx. Some syntactical observations on linear logic. Journal of Logic and Computation, 1(4):537--559, 1991.Google ScholarGoogle ScholarCross RefCross Ref

Index Terms

  1. Classical BI: a logic for reasoning about dualising resources

        Recommendations

        Comments

        Login options

        Check if you have access through your login credentials or your institution to get full access on this article.

        Sign in

        PDF Format

        View or Download as a PDF file.

        PDF

        eReader

        View online with eReader.

        eReader
        About Cookies On This Site

        We use cookies to ensure that we give you the best experience on our website.

        Learn more

        Got it!