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A Labelled Natural Deduction System for Linear Temporal Logic

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Abstract

The paper is devoted to the concise description of some Natural Deduction System (ND for short) for Linear Temporal Logic. The system's distinctive feature is that it is labelled and analytical. Labels convey necessary semantic information connected with the rules for temporal functors while the analytical character of the rules lets the system work as a decision procedure. It makes it more similar to Labelled Tableau Systems than to standard Natural Deduction. In fact, our solution of linearity representation is rather independent of the underlying proof method, provided that some form of (analytic) cut is admissible. We will also discuss some generalisations of the system and compare it with other formalizations of linearity.

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References

  1. D'Agostino, M., ‘Tableau Methods for Classical Propositional Logic’ in: M. D'Agostino et al. (eds.), Handbook of Tableau Methods, pp. 45–123, Kluwer Academic Publishers, Dordrecht 1999.

    Google Scholar 

  2. Basin, D., S. Mathews, and L. Vigano, ‘Labelled Propositional Modal Logics: Theory and Practice’, Journal of Logic and Computation 7(6):685–717, 1997.

    Google Scholar 

  3. Basin, D., S. Mathews, and L. Vigano, ‘Natural Deduction for Non-Classical Logics’, Studia Logica 60:119–160, 1998.

    Google Scholar 

  4. Blackburn, P., ‘Representation, Reasoning, and Relational Structures: a Hybrid Logic Manifesto’, Logic Journal of the IGPL, 8(3):339–365, 2000.

    Google Scholar 

  5. Blackburn, P., ‘Internalizing Labelled Deduction’, Journal of Logic and Computation, 10(1):137–168, 2000.

    Google Scholar 

  6. Burgess, J. P., ‘Basic Tense Logic’, in: D. Gabbay, F. Guenthner, (eds.), Handbook of Philosophical Logic, vol II, pp. 89–133, Reidel Publishing Company, Dordrecht 1984.

    Google Scholar 

  7. Bonette, N., and R. GorÉ, ‘A Labelled Sequent System for Tense Logic Kt’, in: AI98: Proceedings of the Australian Joint Conference on Artificial Inteligence, LNAI 1502:71–82, Springer 1998.

  8. Fisher, M., C. Dixon, and M. Peim, ‘Clausal Temporal Resolution’, ACM Transactions on Computational Logic 1(4), 2001.

  9. Fitch, F., Symbolic Logic, Ronald Press Co, New York 1952.

    Google Scholar 

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

    Google Scholar 

  11. Gabbay, D., LDS — Labelled Deductive Systems, Clarendon Press, Oxford 1996.

    Google Scholar 

  12. Gentzen, G., ‘Untersuchungen über das Logische Schliessen’, Mathematische Zeitschrift 39:176–210 and 39:405–431, 1934.

    Google Scholar 

  13. Goldblatt, R. I. Logics of Time and Computation, CSLI Lecture Notes, Stanford 1987.

  14. GorÉ, R. ‘Tableau Methods for Modal and Temporal Logics’, in: M. D'Agostino et al., (eds.), Handbook of Tableau Methods, pp. 297–396, Kluwer Academic Publishers, Dordrecht 1999.

    Google Scholar 

  15. Horrocks, I. Optimising Tableaux Decision Procedures for Description Logics, PhD Thesis, University of Manchester 1997.

  16. Horrocks, I., U. Satler, and S. Tobies, ‘Practical Reasoning for Very Expressive Description Logics’ Logic Journal of the IGPL 8(3):239–263, 2000.

    Google Scholar 

  17. Indrzejczak, A., ‘Natural Deduction System for Tense Logics’, Bulletin of the Section of Logic 23(4):173–179, 1994.

    Google Scholar 

  18. Indrzejczak, A., ‘A Survey of Natural Deduction Systems for Modal Logics’, Logica Trianguli 3:55–84, 1999.

    Google Scholar 

  19. Indrzejczak, A., ‘Multiple Sequent Calculus for Tense Logics’, Abstracts of AiML and ICTL 2000:93–104, Leipzig 2000.

    Google Scholar 

  20. Indrzejczak, A., ‘Decision Procedure for Modal Logics in Natural Deduction setting’, unpublished draft, Lodz 2001.

  21. Indrzejczak, A., ‘Labelled Analytic Tableaux for S4.3.’, Bulletin of the Section of Logic 31(1):15–26, 2002.

    Google Scholar 

  22. Indrzejczak, A., ‘Resolution based Natural Deduction’, Bulletin of the Section of Logic 31(3):159–170, 2002.

    Google Scholar 

  23. Indrzejczak, A., ‘Hybrid system for (not only) Hybrid Logic’, Abstracts of AiML 2002:137–156, Toulouse 2002.

    Google Scholar 

  24. JaŚkowski, S., ‘On the Rules of Suppositions in Formal Logic’ Studia Logica 1:5–32, 1934.

    Google Scholar 

  25. Kalish, D., and R. Montague, Logic, Techniques of Formal Reasoning, Harcourt, Brace and World, New York 1964.

    Google Scholar 

  26. Kanger, S., Provability in Logic, Almqvist & Wiksell, Stockholm 1957.

    Google Scholar 

  27. Kashima, R., ‘Cut-free sequent calculi for some tense logics’, Studia Logica, 53:119–135, 1994.

    Google Scholar 

  28. Lichtenstein, O., and A. Pnueli, ‘Propositional Temporal Logics: Decidability and Completeness’, Logic Journal of the IGPL 8(1):55–85, 2000.

    Google Scholar 

  29. Marx, M., S. Mikulas, and M. Reynolds, ‘The Mosaic Method for Temporal Logics’, in: R. Dyckhoff, (ed.), Automated Reasoning with Analytic Tableaux and Related Methods, Proc. of International Conference TABLEAUX 2000, Saint Andrews, Scotland, LNAI 1847:324–340, Springer 2000.

  30. Massacci, F., ‘Strongly Analytic Tableaux for Normal Modal Logics’, in: A. Bundy, (ed.), Proc. CADE-12, LNAI 814:723–737, Springer 1994.

  31. Ono, H., and A. Nakamura, ‘On the Size of Refutation Kripke Models for some Linear Modal and Tense Logics’, Studia Logica 39:325–333, 1980.

    Google Scholar 

  32. Pelletier, F.J. ‘Automated Natural Deduction in THINKER’, Studia Logica 60:3–43, 1998.

    Google Scholar 

  33. Prawitz, D. Natural Deduction, Almqvist and Wiksell, Stockholm 1965.

    Google Scholar 

  34. Rescher, N., and A. Urquhart, Temporal Logic, Springer-Verlag, New York 1971.

    Google Scholar 

  35. Russo, A., ‘Modal Labelled Deductive Systems’, Dept. of Computing, Imperial College, London, Technical Report 95/7, 1995.

    Google Scholar 

  36. Seligman, J., ‘The Logic of Correct Description ’ in: M. de Rijke, (ed.), Advances in Intensional Logic, pp. 107–135. Kluwer 1997

  37. Shimura, T., ‘Cut-free systems for the modal logic S4.3 and S4.3GRZ’, Reports on Mathematical Logic, 25:57–73, 1991.

    Google Scholar 

  38. Sieg, W., Mechanisms and Search: Aspects of Proof Theory, Associazione Italiana di Logica e sue Applicazioni, Padova 1992.

    Google Scholar 

  39. Smullyan, R., First-Order Logic, Springer, Berlin 1968.

    Google Scholar 

  40. Takano, M., ‘Subformula Property as a substitute for Cut-Elimination in Modal Propositional Logics’, Mathematica Japonica 37(6):1129–1145, 1992.

    Google Scholar 

  41. Tzakova, M., ‘Tableaux Calculi for Hybrid Logics’, to appear in: N. Murray, (ed.), Conference on Tableaux Calculi and Related Methods (TABLEAUX), Saratoga Springs, Springer 2002

  42. Wansing, H., Displaying Modal Logics, Kluwer Academic Publishers, Dordrecht 1999.

    Google Scholar 

  43. Wolper, P., ‘The Tableau Method for Temporal Logic: An overview’, Logique et Analyse 28(110/111):119–136, 1985.

    Google Scholar 

  44. Zeman, J. J., Modal Logic, Oxford University Press, Oxford 1973.

    Google Scholar 

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Authors and Affiliations

  1. Department of Logic, University of Lódź, Kopcinskiego 16/18, 90-232, Lódź, Poland

    Andrzej Indrzejczak

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  1. Andrzej Indrzejczak
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Indrzejczak, A. A Labelled Natural Deduction System for Linear Temporal Logic. Studia Logica 75, 345–376 (2003). https://doi.org/10.1023/B:STUD.0000009565.98020.9b

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  • DOI: https://doi.org/10.1023/B:STUD.0000009565.98020.9b

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