Skip to main content
SpringerLink
Log in
Book cover

International Conference on Automated Reasoning with Analytic Tableaux and Related Methods

TABLEAUX 2019: Automated Reasoning with Analytic Tableaux and Related Methods pp 130–146Cite as

Relating Labelled and Label-Free Bunched Calculi in BI Logic

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11714))

Abstract

In this paper we study proof translations between labelled and label-free calculi for the logic of Bunched Implications (\(\mathrm {BI}\)). We first consider the bunched sequent calculus \(\mathrm {LBI}\) and define a labelled sequent calculus, called \(\mathrm {GBI}\), in which labels and constraints reflect the properties of a specifically tailored Kripke resource semantics of \(\mathrm {BI}\) with two total resource composition operators and explicit internalization of inconsistency. After showing the soundness of \(\mathrm {GBI}\) w.r.t. our specific Kripke frames, we show how to translate any \(\mathrm {LBI}\)-proof into a \(\mathrm {GBI}\)-proof. Building on the properties of that translation we devise a tree property that every \(\mathrm {LBI}\)-translated \(\mathrm {GBI}\)-proof enjoys. We finally show that any \(\mathrm {GBI}\)-proof enjoying this tree property (and not only \(\mathrm {LBI}\)-translated ones) can systematically be translated to an \(\mathrm {LBI}\)-proof.

Work supported by the TICAMORE project (ANR grant 16-CE91-0002).

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Abrusci, M., Ruet, P.: Non-commutative logic I: the multiplicative fragment. Ann. Pure Appl. Log. 101, 29–64 (2000)

    Article  MathSciNet  Google Scholar 

  2. Basin, D., D’Agostino, M., Gabbay, D.M., Matthews, S., Viganó, L. (eds.): Labelled Deduction, Volume 17 of Applied Logic Series. Kluwer Academic Publishers, Dordrecht (2000)

    Google Scholar 

  3. Brotherston, J., Villard, J.: Sub-classical Boolean bunched logics and the meaning of par. In: International Conference on Computer Science Logic, CSL 2014, LIPIcs, Dagsthul, Vienna, Austria, pp. 325–342 (2014)

    Google Scholar 

  4. Collinson, M., Pym, D.: Algebra and logic for resource-based systems modelling. Math. Struct. Comput. Sci. 19(5), 959–1027 (2009)

    Article  MathSciNet  Google Scholar 

  5. Fitting, M.: Proof Methods for Modal and Intuitionistic Logics. Reidel Publishing Company, Dordrecht (1983)

    Book  Google Scholar 

  6. Gabbay, D.M.: Labelled Deductive Systems, Volume I - Foundations. Oxford University Press, Oxford (1996)

    Google Scholar 

  7. Galatos, N., Jipsen, P.: Distributive residuated frames and generalized bunched implication algebras. Algebra Universalis 78(3), 303–336 (2017)

    Article  MathSciNet  Google Scholar 

  8. Galmiche, D., Méry, D.: Semantic labelled tableaux for propositional BI without bottom. J. Log. Comput. 13(5), 707–753 (2003)

    Article  Google Scholar 

  9. Galmiche, D., Méry, D., Pym, D.: The semantics of BI and Resource Tableaux. Math. Struct. Comput. Science 15(6), 1033–1088 (2005)

    Article  MathSciNet  Google Scholar 

  10. Girard, J.Y.: Linear logic. Theor. Comput. Sci. 50(1), 1–102 (1987)

    Article  MathSciNet  Google Scholar 

  11. Hóu, Z., Tiu, A., Goré, R.: A labelled sequent calculus for BBI: proof theory and proof search. In: Galmiche, D., Larchey-Wendling, D. (eds.) TABLEAUX 2013. LNCS (LNAI), vol. 8123, pp. 172–187. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40537-2_16

    Chapter  MATH  Google Scholar 

  12. Ishtiaq, S., O’Hearn, P.: BI as an assertion language for mutable data structures. In: 28th ACM Symposium on Principles of Programming Languages, POPL 2001, London, UK, pp. 14–26 (2001)

    Google Scholar 

  13. Kaminski, M., Francez, N.: The Lambek calculus extended with intuitionistic propositional logic. Studia Logica 104(5), 1051–1082 (2016)

    Article  MathSciNet  Google Scholar 

  14. Larchey-Wendling, D.: The formal strong completeness of partial monoidal Boolean BI. J. Log. Comput. 26(2), 605–640 (2014)

    Article  MathSciNet  Google Scholar 

  15. O’Hearn, P.W., Pym, D.: The logic of bunched implications. Bull. Symb. Log. 5(2), 215–244 (1999)

    Article  MathSciNet  Google Scholar 

  16. Pinto, L., Uustalu, T.: Proof search and counter-model construction for bi-intuitionistic propositional logic with labelled sequents. In: Giese, M., Waaler, A. (eds.) TABLEAUX 2009. LNCS (LNAI), vol. 5607, pp. 295–309. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02716-1_22

    Chapter  MATH  Google Scholar 

  17. Pym, D.: On bunched predicate logic. In: 14th Symposium on Logic in Computer Science, Trento, Italy, pp. 183–192. IEEE Computer Society Press, July 1999

    Google Scholar 

  18. Pym, D.: The Semantics and Proof Theory of the Logic of Bunched Implications, volume 26 of Applied Logic Series. Kluwer Academic Publishers, Dordrecht (2002)

    MATH  Google Scholar 

  19. Reynolds, J.: Separation logic: a logic for shared mutable data structures. In: IEEE Symposium on Logic in Computer Science, Copenhagen, Danemark, pp. 55–74, July 2002

    Google Scholar 

  20. Schmitt, S., Kreitz, C.: Converting non-classical matrix proofs into sequent-style systems. In: McRobbie, M.A., Slaney, J.K. (eds.) CADE 1996. LNCS, vol. 1104, pp. 418–432. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61511-3_104

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Lorraine, CNRS, LORIA, 54506, Vandoeuvre-lès-Nancy, France

    Didier Galmiche, Michel Marti & Daniel Méry

Authors
  1. Didier Galmiche
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michel Marti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel Méry
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Méry .

Editor information

Editors and Affiliations

  1. IBISC, Univ. Evry, Université Paris-Saclay, Evry, France

    Serenella Cerrito

  2. Middlesex University London, London, UK

    Andrei Popescu

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Galmiche, D., Marti, M., Méry, D. (2019). Relating Labelled and Label-Free Bunched Calculi in BI Logic. In: Cerrito, S., Popescu, A. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2019. Lecture Notes in Computer Science(), vol 11714. Springer, Cham. https://doi.org/10.1007/978-3-030-29026-9_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-29026-9_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-29025-2

  • Online ISBN: 978-3-030-29026-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation