Skip to main content
SpringerLink
Log in

From Display to Labelled Proofs for Tense Logics

  • Conference paper
  • First Online:
Logical Foundations of Computer Science (LFCS 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10703))

Included in the following conference series:

Abstract

We introduce an effective translation from proofs in the display calculus to proofs in the labelled calculus in the context of tense logics. We identify the labelled calculus proofs in the image of this translation as those built from labelled sequents whose underlying directed graph possesses certain properties. For the basic normal tense logic \(\textsf {Kt}\), the image is shown to be the set of all proofs in the labelled calculus \(\textsf {G3Kt}\).

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 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

Institutional subscriptions

Notes

  1. 1.

    More specifically, this is true of a display calculus for a logic such that every structural connective can be interpreted as a connective of the logic.

  2. 2.

    Extending to primitive tense axioms [14] is straightforward though more syntactically involved.

References

  1. Belnap Jr., N.D.: Display logic. J. Philos. Logic 11(4), 375–417 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  3. Brünnler. K.: Deep sequent systems for modal logic. In: Advances in Modal Logic, vol. 6, pp. 107–119. College Publications, London (2006)

    Google Scholar 

  4. Chagrov, A., Zakharyashchev, M.: Modal companions of intermediate propositional logics. Stud. Logica. 51(1), 49–82 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ciabattoni, A., Ramanayake, R.: Power and limits of structural display rules. ACM Trans. Comput. Logic 17(3), 1–39 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dyckhoff, R., Negri, S.: Proof analysis in intermediate logics. Arch. Math. Log. 51(1–2), 71–92 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dyckhoff, R., Negri, S.: Geometrization of first-order logic. Bull. Symbolic Logic 21, 123–163 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fitting, M.: Proof Methods for Modal and Intuitionistic Logics. Synthese Library, vol. 169. D. Reidel Publishing Co., Dordrecht (1983)

    Book  MATH  Google Scholar 

  9. Fitting, M.: Prefixed tableaus and nested sequents. Ann. Pure Appl. Logic 163(3), 291–313 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Goré, R., Postniece, L., Tiu, A.: On the correspondence between display postulates and deep inference in nested sequent calculi for tense logics. Log. Methods Comput. Sci. 7(2), 1–38 (2011). (2:8)

    Article  MathSciNet  MATH  Google Scholar 

  11. Goré, R., Ramanayake, R.: Labelled tree sequents, tree hypersequents and nested (deep) sequents. In: Advances in Modal Logic, vol. 9. College Publications, London (2012)

    Google Scholar 

  12. Greco, G., Ma, M., Palmigiano, A., Tzimoulis, A., Zhao, Z.: Unified correspondence as a proof-theoretic tool. J. Logic Comput. (2016, to appear). https://doi.org/10.1093/logcom/exw022

  13. Kashima, R.: Cut-free sequent calculi for some tense logics. Stud. Logica. 53(1), 119–135 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kracht, M.: Power and weakness of the modal display calculus. In: Proof Theory of Modal Logic (Hamburg, 1993) Applied Logic Series, vol. 2, pp. 93–121. Kluwer Academic Publishers, Dordrecht (1996)

    Google Scholar 

  15. Lemmon, E.J., Scott, D.S.: The ‘Lemmon Notes’: An Introduction to Modal Logic. Blackwell, Oxford (1977)

    MATH  Google Scholar 

  16. Mints, G.: Indexed systems of sequents and cut-elimination. J. Philos. Logic 26(6), 671–696 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Negri, S.: Proof analysis in modal logic. J. Philos. Logic 34(5–6), 507–544 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ramanayake, R.: Inducing syntactic cut-elimination for indexed nested sequents. In: Proceedings of IJCAR, pp. 416–432 (2016)

    Google Scholar 

  19. Restall, G.: Comparing modal sequent systems. http://consequently.org/papers/comparingmodal.pdf

  20. Restall, G., Poggiolesi, F.: Interpreting and applying proof theory for modal logic. In: Restall, G., Russell, G. (eds.) New Waves in Philosophical Logic, pp. 39–62 (2012)

    Google Scholar 

  21. Viganò, L.: Labelled Non-Classical Logics. Kluwer Academic Publishers, Dordrecht (2000). With a foreword by Dov M. Gabbay

    Book  MATH  Google Scholar 

  22. Wansing, H.: Displaying Modal Logic. Trends in Logic-Studia Logica Library, vol. 3. Kluwer Academic Publishers, Dordrecht (1998)

    MATH  Google Scholar 

Download references

Acknowledgments

Work supported by the FWF projects: START Y544-N23 and I 2982.

Author information

Authors and Affiliations

  1. Institut für Computersprachen, Technische Universität Wien, 1040, Wien, Austria

    Agata Ciabattoni, Tim Lyon & Revantha Ramanayake

Authors
  1. Agata Ciabattoni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tim Lyon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Revantha Ramanayake
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tim Lyon .

Editor information

Editors and Affiliations

  1. City University of New York, New York, New York, USA

    Sergei Artemov

  2. Cornell University, Ithaca, New York, USA

    Anil Nerode

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Ciabattoni, A., Lyon, T., Ramanayake, R. (2018). From Display to Labelled Proofs for Tense Logics. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2018. Lecture Notes in Computer Science(), vol 10703. Springer, Cham. https://doi.org/10.1007/978-3-319-72056-2_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-72056-2_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-72055-5

  • Online ISBN: 978-3-319-72056-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation