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International Workshop on Theorem Proving with Analytic Tableaux and Related Methods

TABLEAUX 1996: Theorem Proving with Analytic Tableaux and Related Methods pp 48–64Cite as

Almost duplication-free tableau calculi for prepositional lax logics

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

Abstract

In this paper we provide tableau calculi for the intuitionistic modal logics PLL and PLL 1, where the calculus for PLL 1 is duplication-free while among the rules for PLL there is just one rule that allows duplication of formulas. These logics have been investigated by Fairtlough and Mendler in relation to the problem of Formal Hardware Verification. In order to develop these calculi we extend to the modal case some ideas presented by Miglioli, Moscato and Ornaghi for intuitionistic logic. Namely, we enlarge the language containing the usual sings T and F with the new sign F c. PLL and PLL 1 logics are characterized by a Kripke-semantics which is a “weak” version of the semantics for ordinary intuitionistic modal logics. In this paper we establish the soundness and completeness theorems for these calculi.

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References

  1. G. Amati and F. Pirri. A uniform tableau method for intuitionistic modal logics I*. Studia Logica, 53(1):29–60, 1994.

    Google Scholar 

  2. M. Bozic and K. Dosen. Models for normal intuitionistic modal logics. Studia Logica, 43:217–245, 1984.

    Google Scholar 

  3. M.J. Cresswell and G.E. Hughes. A Companion to Modal Logic. Methuen, 1984.

    Google Scholar 

  4. H.B. Curry. The elimination theorem when modality is present. The Journal of Symbolic Logic, 17:249–265, 1952.

    Google Scholar 

  5. M. Ferrari. Cut-free tableau calculi for some intuitionistic modal logics. Technical Report 133-95, Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, 1995.

    Google Scholar 

  6. M. Fitting. Proof Methods for Modal and Intuitionistic Logics. Reidel, Dordrecht, 1983.

    Google Scholar 

  7. M. Fairtlough and M. Mendler. An intuitionistic modal logic with applications to the formal verification of hardware. In LNCS Springer-Verlag, editor, Proceedings of Computer Science Logic '94, Kazimierz, Poland, pages 354–368, 1994.

    Google Scholar 

  8. G. Fischer Servi. On modal logics with an intuitionistic base. Studia Logica, 36:141–149, 1977.

    Google Scholar 

  9. G. Fischer Servi. Axiomatizations for some intuitionistic modal logics. Rend. Sem. Mat. Univers. Polit. Torino, 42:179–194, 1984.

    Google Scholar 

  10. D.M. Gabbay. Semantical Investigations in Heyting's Intuitionistic Logic. Reidel, Dordrecht, 1981.

    Google Scholar 

  11. M. Mendler. Constrained proofs: A logic for dealing with behavioural constraints in formal hardware verification. In G.Jones and M.Sheeran, editors, Designing Correct Circuits, pages 1–28. Springer, 1990.

    Google Scholar 

  12. M. Mendler. A Modal Logic for Handling Behavioural Constraints in Formal Hardware Verification. PhD thesis, Edinburgh University, Department of Computer Science, ECS-LFCS-93-255, 1993.

    Google Scholar 

  13. P. Miglioli, U. Moscato, and M. Ornaghi. How to avoid duplications in refutation systems for intuitionistic logic and Kuroda logic. In K. Broda, M. D' Agostino, R. Goré, R. Johnson, and S. Reeves, editors, Proceedings of the 3rd Workshop on Theorem Proving with Analytic Tableaux and Related Methods, pages 169–187. Abingdon, U.K., May 4–6 1994.

    Google Scholar 

  14. P. Miglioli, U. Moscato, and M. Ornaghi. An improved refutation system for intuitionistic predicate logic. Journal of Automated Reasoning, 12:361–373, 1994.

    Google Scholar 

  15. G. Plotkin and C. Stirling. A framework for intuitionistic modal logic. In J.Y. Halpern, editor, Theoretical Aspects of Reasoning about Knowledge, pages 399–406. Morgan-Kaufmann, 1986.

    Google Scholar 

  16. A.S. Troelstra and D. van Dalen. Constructivism in mathematics: an introduction. Volume 2. In Studies in logic and the foundations of mathematics, volume 122. North-Holland, 1988.

    Google Scholar 

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Author information

Author notes

  1. Mauro Ferrari

    Present address: Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, Via Comelico 39, 20135, Milano, Italy

Authors and Affiliations

  1. Dipartimento di Scienze dell'Informazione, Università degli Studi di Milano, Via Comelico 39, 20135, Milano, Italy

    Alessandro Avellone

Authors
  1. Alessandro Avellone
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  2. Mauro Ferrari
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Corresponding author

Correspondence to Mauro Ferrari .

Editor information

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

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© 1996 Springer-Verlag Berlin Heidelberg

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Avellone, A., Ferrari, M. (1996). Almost duplication-free tableau calculi for prepositional lax 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_4

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  • DOI: https://doi.org/10.1007/3-540-61208-4_4

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61208-7

  • Online ISBN: 978-3-540-68368-1

  • eBook Packages: Springer Book Archive

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