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International Symposium on Logical Foundations of Computer Science

LFCS 2018: Logical Foundations of Computer Science pp 120–139Cite as

From Display to Labelled Proofs for Tense Logics

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

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 \(\textsf {G3Kt}\).

Keywords

  • Display calculus
  • Labelled calculus
  • Structural proof theory
  • Tense logic
  • Modal logic

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  • DOI: 10.1007/978-3-319-72056-2_8
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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)

    MathSciNet  CrossRef  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)

    CrossRef  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)

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    CrossRef  MATH  Google Scholar 

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

    MathSciNet  CrossRef  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)

    MathSciNet  CrossRef  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)

    MathSciNet  CrossRef  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)

    MathSciNet  CrossRef  MATH  Google Scholar 

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

    MathSciNet  CrossRef  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

    CrossRef  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 

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Acknowledgments

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

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Authors and Affiliations

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

    Agata Ciabattoni, Tim Lyon & Revantha Ramanayake

Authors
  1. Agata Ciabattoni
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  2. Tim Lyon
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  3. Revantha Ramanayake
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Corresponding author

Correspondence to Tim Lyon .

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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

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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

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

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