Skip to main content
SpringerLink
Log in

Efficient loop-check for backward proof search in some non-classical propositional logics

  • Contributed Papers
  • Conference paper
  • First Online:
Theorem Proving with Analytic Tableaux and Related Methods (TABLEAUX 1996)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1071))

Abstract

We consider the modal logics KT and S4, the tense logic Kt, and the fragment IPC(∧,→) of intuitionistic logic.

For these logics backward proof search in the standard sequent or tableau calculi does not terminate in general. In terms of the respective Kripke semantics, there are several kinds of non-termination: loops inside a world (KT), infinite resp. looping branches (S4, IPC(∧,→)), and infinite branching degree (Kt).

We give uniform sequent-based calculi that contain specifically tailored loop-checks such that the efficiency of proof search is not deteriorated. Moreover all these loop-checks are easy to implement and can be combined with optimizations.

Note that our calculus for S4 is not a pure contraction-free sequent calculus, but this (theoretical) defect does not hinder its application in practice.

Work supported by the Swiss National Science Foundation, SPP 5003-34279.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Brian F. Chellas. Modal logic: an introduction. Cambridge University Press, 1980.

    Google Scholar 

  2. Roy Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. The Journal of Symbolic Logic, 57(3):795–807, 1992.

    Google Scholar 

  3. Melvin Fitting. Proof Methods for Modal and Intuitionistic Logics. Reidel, Dordrecht, 1983.

    Google Scholar 

  4. Melvin Fitting. Basic modal logic. In Dov M. Gabbay, C. J. Hogger, and J. A. Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming, volume 1, pages 368–448. Clarendon Press, Oxford, 1993.

    Google Scholar 

  5. D M Gabbay. Algorithmic Proof with Diminishing Resources Part 1. In E. Börger, editor, CSL 90, LNCS 533, pages 156–173, 1991.

    Google Scholar 

  6. Rajeev Goré. Tableau methods for modal and temporal logics. Technical report, TR-15-95, Automated Reasoning Project, Australian National University, Canberra, Australia, 1995. To appear in Handbook of Tableau Methods, Kluwer, 199?.

    Google Scholar 

  7. Rajeev Goré, Wolfgang Heinle, and Alain Heuerding. Relations between propositional normal modal logics: an overview. Technical report, TR-16-95, Automated Reasoning Project, Australian National University, Canberra, Australia, 1995.

    Google Scholar 

  8. Alain Heuerding, Gerhard Jäger, Stefan Schwendimann, and Michael Seyfried. Prepositional logics on the computer. In Peter Baumgartner, Reiner Hähnle, and Joachim Posegga, editors, Theorem Proving with Analytic Tableaux and Related Methods, LNCS 918, pages 310–323, 1995.

    Google Scholar 

  9. Jörg Hudelmaier. An O(n log n)-space decision procedure for intuitionistic propositional logic. Journal of Logic and Computation, 3(1):63–75, 1993.

    Google Scholar 

  10. Jörg Hudelmaier. On a contraction free sequent calculus for the modal logic S4. In TABLEAUX 94. 3rd Workshop on Theorem Proving with Analytic Tableaux and Related Methods, 1994.

    Google Scholar 

  11. S.C. Kleene. Introduction to Metamathematics. North-Holland, Amsterdam, 1952.

    Google Scholar 

  12. Richard E. Ladner. The computational complexity of provability in systems of modal prepositional logic. SIAM Journal on Computing, 6(3):467–480, 1977.

    Google Scholar 

  13. LWBtheory. http://1wbwww.unibe.ch:8080/LWBtheory.html, 1995.

    Google Scholar 

  14. Nicholas Rescher and Alasdair Urquhart. Temporal Logic. Springer, 1971.

    Google Scholar 

  15. Michael Seyfried. A sequent calculus for proof search in the (→, ∧)-fragment of intuitionistic logic. Unpublished, 1994.

    Google Scholar 

  16. Heinrich Zimmermann. A directed tree calculus for minimal tense logic. Master's thesis, IAM, University of Berne, Switzerland, 1995.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Informatik und angewandte Mathematik, Universität Bern, Switzerland

    Alain Heuerding, Michael Seyfried & Heinrich Zimmermann

Authors
  1. Alain Heuerding
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Seyfried
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Heinrich Zimmermann
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

P. Miglioli U. Moscato D. Mundici M. Ornaghi

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Heuerding, A., Seyfried, M., Zimmermann, H. (1996). Efficient loop-check for backward proof search in some non-classical propositional logics. In: Miglioli, P., Moscato, U., Mundici, D., Ornaghi, M. (eds) Theorem Proving with Analytic Tableaux and Related Methods. TABLEAUX 1996. Lecture Notes in Computer Science, vol 1071. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61208-4_14

Download citation

  • DOI: https://doi.org/10.1007/3-540-61208-4_14

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61208-7

  • Online ISBN: 978-3-540-68368-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation