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Analytic Tableaux for Non-deterministic Semantics

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12842))

Abstract

Analytic tableau systems for the family of non-deterministic semantics are introduced. These are based on tableaux for many-valued logics using sets-as-signs DNF representations. Karnaugh maps illustrate the construction of tableau rules. In contrast to classical many-valued tableaux, we add a rule called sign intersection. Soundness and completeness are shown. As an example demonstrates, some tableau systems would be incomplete without sign intersection. There is a correspondence to well-studied canonical calculi based on sequent systems: Tableau systems can be translated into canonical calculi, but not vice-versa (structural rules are missing on the tableau side).

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Notes

  1. 1.

    Also note that a tableau system for a particular non-deterministic semantics is already listed in [19, p. 211]. Unfortunately, we could not find out more about it; an approach to contact the author was unsuccessful.

  2. 2.

    A usage of this nmatrix in a meaningful logic would be nice but is not intended.

References

  1. Avron, A., Arieli, O., Zamansky, A.: Theory of Effective Propositional Paraconsistent Logics. College Publications (2018)

    Google Scholar 

  2. Avron, A., Konikowska, B.: Multi-valued calculi for logics based on non-determinism. Log. J. IGPL 13(4), 365–387 (2005). https://doi.org/10.1093/jigpal/jzi030

    Article  MathSciNet  MATH  Google Scholar 

  3. Avron, A., Lev, I.: Canonical propositional Gentzen-type systems. In: Goré, R., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS, vol. 2083, pp. 529–544. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45744-5_45

    Chapter  Google Scholar 

  4. Avron, A., Lev, I.: Non-deterministic multiple-valued structures. J. Log. Comput. 15(3), 241–261 (2005). https://doi.org/10.1093/logcom/exi001

    Article  MathSciNet  MATH  Google Scholar 

  5. Avron, A., Zamansky, A.: Canonical signed calculi, non-deterministic matrices and cut-elimination. In: Artemov, S., Nerode, A. (eds.) LFCS 2009. LNCS, vol. 5407, pp. 31–45. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-92687-0_3

    Chapter  Google Scholar 

  6. Avron, A., Zamansky, A.: Non-deterministic semantics for logical systems. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 16, 2nd edn, pp. 227–304. Springer, Heidelberg (2011). https://doi.org/10.1007/978-94-007-0479-4_4

    Chapter  Google Scholar 

  7. Baaz, M., Fermüller, C.G., Salzer, G.: Automated deduction for many-valued logics. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. 2, chap. 19, pp. 1355–1402. Elsevier and MIT Press (2001)

    Google Scholar 

  8. Baaz, M., Fermüller, C.G., Zach, R.: Dual systems of sequents and tableaux for many-valued logics. Bull. EATCS 51, 192–197 (1993)

    MATH  Google Scholar 

  9. Baaz, M., Lahav, O., Zamansky, A.: Finite-valued semantics for canonical labelled calculi. J. Autom. Reason. 51(4), 401–430 (2013). https://doi.org/10.1007/s10817-013-9273-x

    Article  MathSciNet  MATH  Google Scholar 

  10. Beckert, B., Hähnle, R., Manyà, F.: The SAT problem of signed CNF formulas. In: Basin, D., D’Agostino, M., Gabbay, D.M., Matthews, S., Viganò, L. (eds.) Labelled Deduction. APLS, vol. 17, pp. 59–80. Springer, Dordrecht (2000). https://doi.org/10.1007/978-94-011-4040-9_3

    Chapter  Google Scholar 

  11. Carnielli, W.A.: Systematization of finite many-valued logics through the method of tableaux. J. Symb. Log. 52(2), 473–493 (1987). https://doi.org/10.2307/2274395

    Article  MathSciNet  MATH  Google Scholar 

  12. Coniglio, M.E., del Cerro, L.F., Newton, M.P.: Modal logic with non-deterministic semantics: part I–propositional case. Log. J. IGPL 28(3), 281–315 (2020). https://doi.org/10.1093/jigpal/jzz027

    Article  MathSciNet  Google Scholar 

  13. D’Agostino, M., Mondadori, M.: The taming of the cut. Classical refutations with analytic cut. J. Log. Comput. 4(3), 285–319 (1994). https://doi.org/10.1093/logcom/4.3.285

  14. Fermüller, C.G.: On matrices, Nmatrices and games. J. Log. Comput. 26(1), 189–211 (2016). https://doi.org/10.1093/logcom/ext024

    Article  MathSciNet  MATH  Google Scholar 

  15. Hähnle, R.: Advanced many-valued logics. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 2, 2nd edn, pp. 297–395. Springer, Dordrecht (2001). https://doi.org/10.1007/978-94-017-0452-6_5

    Chapter  Google Scholar 

  16. Hähnle, R.: Towards an efficient tableau proof procedure for multiple-valued logics. In: Börger, E., Kleine Büning, H., Richter, M.M., Schönfeld, W. (eds.) CSL 1990. LNCS, vol. 533, pp. 248–260. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-54487-9_62

    Chapter  Google Scholar 

  17. Hähnle, R.: Automated Deduction in Multiple-valued Logics. Oxford University Press (1993)

    Google Scholar 

  18. Hähnle, R.: Tableaux for many-valued logics. In: D’Agostino, M., Gabbay, D.M., Hähnle, R., Posegga, J. (eds.) Handbook of Tableau Methods, pp. 529–580. Kluwer (1999)

    Google Scholar 

  19. Ivlev, J.V.: Modal’naja logika. Moskovskogo Univ, Izdat (1991). (in Russian)

    Google Scholar 

  20. Konikowska, B.: Two over three: a two-valued logic for software specification and validation over a three-valued predicate calculus. J. Appl. Non-Classical Log. 3(1), 39–71 (1993). https://doi.org/10.1080/11663081.1993.10510795

    Article  MathSciNet  MATH  Google Scholar 

  21. Ohnishi, M., Matsumoto, K.: A system for strict implication. Ann. Jpn. Assoc. Philos. Sci. 2(4), 183–188 (1964). https://doi.org/10.4288/jafpos1956.2.183

    Article  MathSciNet  MATH  Google Scholar 

  22. Omori, H., Skurt, D.: More modal semantics without possible worlds. IfCoLog J. Log. Appl. 3(5), 815–846 (2016)

    Google Scholar 

  23. Paoli, F.: Substructural Logics: A Primer. Trends in Logic. Kluwer Academic Publishers (2002)

    Google Scholar 

  24. Pawlowski, P.: Tree-like proof systems for finitely-many valued non-deterministic consequence relations. Log. Univers. 14(4), 407–420 (2020). https://doi.org/10.1007/s11787-020-00263-0

    Article  MathSciNet  MATH  Google Scholar 

  25. Rosser, J.B., Turquette, A.R.: Many-Valued Logics. North-Holland (1952)

    Google Scholar 

  26. Rouseau, G.: Sequents in many valued logic I. Fundam. Math. 60(1), 23–33 (1967). https://doi.org/10.4064/fm-60-1-23-33

    Article  MathSciNet  Google Scholar 

  27. Smullyan, R.M.: First-Order Logic. Dover, 2 edn. (1995)

    Google Scholar 

  28. Surma, S.J.: An algorithm for axiomatizing every finite logic. In: Rine, D.C. (ed.) Computer Science and Multiple-Valued Logic, pp. 137–143. North-Holland Publishing Company (1977)

    Google Scholar 

  29. Zamansky, A., Avron, A.: Canonical signed calculi with multi-ary quantifiers. Ann. Pure Appl. Logic 163(7), 951–960 (2012). https://doi.org/10.1016/j.apal.2011.09.006

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

I am indebted to Peter Steinacker for the supervision of my master’s thesis and to Andreas Maletti, second reviewer, for spotting the mistake in my thesis which initiated the work on the present paper. I would like to thank Daniel Skurt, Hitoshi Omori, Reiner Hähnle, Richard Bubel, Elio La Rosa, Pawel Pawlowski for useful suggestions.

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Authors and Affiliations

  1. Technische Universität Darmstadt, Darmstadt, Germany

    Lukas Grätz

  2. Universität Leipzig, Leipzig, Germany

    Lukas Grätz

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  1. Lukas Grätz
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Correspondence to Lukas Grätz .

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Editors and Affiliations

  1. University of Birmingham, Birmingham, UK

    Anupam Das

  2. University of Genoa, Genoa, Italy

    Sara Negri

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Grätz, L. (2021). Analytic Tableaux for Non-deterministic Semantics. In: Das, A., Negri, S. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2021. Lecture Notes in Computer Science(), vol 12842. Springer, Cham. https://doi.org/10.1007/978-3-030-86059-2_3

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  • DOI: https://doi.org/10.1007/978-3-030-86059-2_3

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