Skip to main content
SpringerLink
Log in

Analytic Tableau Systems for Propositional Bimodal Logics of Knowledge and Belief

  • Conference paper
  • First Online:
Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2002)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2381))

Abstract

We give sound and complete analytic tableau systems for the propositional bimodal logics KB, KB −C , KB −5, and KB −5C . These logics have two universal modal operators K and B, where K stands for knowing and B stands for believing. The logic KB is a combination of the modal logic S5 (for K) and KD45 (for B) with the interaction axioms I: K φ → B φ and C: B φ → KB φ. The logics KB −C , KB −5, KB −5C are obtained from KB respectively by deleting the axiom C (for KB −C ), the axioms 5 (for KB −5), and both of the axioms C and 5 (for KB −5C ). As analytic sequent-like tableau systems, our calculi give simple decision procedures for reasoning about both knowledge and belief in the mentioned logics.

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

Access this chapter

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 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. F. Baader and U. Sattler. An overview of tableau algorithms for description logics. Studia Logica, 69:5–40, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Baldoni, L. Giordano, and A. Martelli. A tableau calculus for multimodal logics and some (un)decidability results. In H. de Swart, editor, Proceeding of TABLEAUX’98, LNCS 1397, pages 44–59. Springer-Verlag, 1998.

    Google Scholar 

  3. B. Beckert and D. Gabbay. Fibring semantic tableaux. In H. de Swart, editor, Proceeding of TABLEAUX’98, LNCS 1397, pages 77–92. Springer-Verlag, 1998.

    Google Scholar 

  4. S. Demri. Complexity of simple dependent bimodal logics. In Roy Dyckhoff (Ed.): TABLEAUX 2000, LNAI 1847, pages 190–204. Springer, 2000.

    Google Scholar 

  5. K. Fine and G. Schurz. Transfer theorems for stratified multimodal logics. In J. Copeland, editor, Logic and Reality, Essays in Pure and Applied Logic. In Memory of Arthur Prior, pages 169–213. Oxford University Press, 1996.

    Google Scholar 

  6. M. Fitting. Proof Methods for Modal and Intuitionistic Logics. Volume 169 of Synthese Library. D. Reidel, Dordrecht, Holland, 1983.

    MATH  Google Scholar 

  7. M. Fitting, L. Thalmann, and A. Voronkov. Term-modal logics. Studia Logica, 69:133–169, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  8. D. Gabbay. Fibred semantics and the weaving of logics part 1: Modal and intuitionistic logics. The Journal of Symbolic Logic, 61(4):1057–1120, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  9. D. Gabbay and V. Shehtman. Products of modal logics, part 1. Logic Journal of the IGPL, 6(1):73–146, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. Goré. Tableau methods for modal and temporal logics. In D’Agostino Gabbay Hähnle Posegga, editor, Handbook of Tableau Methods, pages 297–396. Kluwer Academic Publishers, 1999.

    Google Scholar 

  11. K.J.J. Hintikka. Form and content in quantification theory. Acta Philosophica Fennica, 8:3–55, 1955.

    Google Scholar 

  12. M. Kracht and F. Wolter. Properties of independently axiomatisable bimodal logic. Journal of Symbolic Logic, 56:1469–1485, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  13. M. Kracht and F. Wolter. Simulation and transfer results in modal logic-a survey. Studia Logica, 59:149–177, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  14. F. Massacci. Single step tableaux for modal logics. Journal of Automated Reasoning, 24(3):319–364, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  15. L.A. Nguyen. Analytic tableau systems and interpolation for the modal logics KB, KDB, K5, KD5. Studia Logica, 69(1):41–57, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  16. W. Rautenberg. Modal tableau calculi and interpolation. Journal of Philosophical Logic, 12:403–423, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Rosati. A sound and complete tableau calculus for reasoning about only knowing and knowing at most. Studia Logica, 69:171–191, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  18. W. van der Hoek and J.-J. Meyer. Modalities for reasoning about knowledge and uncertainties. In Patrick Doherty, editor, Partiality, Modality, and Nonmonotonicity. CSLI Publications, 1996.

    Google Scholar 

  19. A. Visser. A course on bimodal provability logic. Annals of Pure and Applied Logic, 73:109–142, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. Wooldridge, C. Dixon, and M. Fisher. A tableau-based proof method for temporal logics of knowledge and belief. Journal of Applied Non-Classical Logics, 8(3):225–258, 1998.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Informatics, Warsaw University, ul. Banacha 2, 02-097, Warsaw, Poland

    Linh Anh Nguyen

Authors
  1. Linh Anh Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Abteilung Wisssenbasierte Systeme 184/3, Technische Univesität Wien, Favoritenstr. 9-11, 1040, Vienna, Austria

    Uwe Egly

  2. Institut für Computersprachen AG Theoretische Informatik und Logik 185.2, Technische Universität Wien, Favoritenstr. 9-11, 1040, Vienna, Austria

    Chritian G. Fermüller

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Nguyen, L.A. (2002). Analytic Tableau Systems for Propositional Bimodal Logics of Knowledge and Belief. In: Egly, U., Fermüller, C.G. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2002. Lecture Notes in Computer Science(), vol 2381. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45616-3_15

Download citation

  • DOI: https://doi.org/10.1007/3-540-45616-3_15

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43929-5

  • Online ISBN: 978-3-540-45616-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation