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Infinite-Valued First-Order Łukasiewicz Logic: Hypersequent Calculi Without Structural Rules and Proof Search for Sentences in the Prenex Form

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Abstract

The rational first-order Pavelka logic is an expansion of the infinite-valued first-order Łukasiewicz logic Ł∀ by truth constants. For this logic, we introduce a cumulative hypersequent calculus G1Ł∀ and a noncumulative hypersequent calculus G2Ł∀ without structural inference rules. We compare these calculi with the Baaz–Metcalfe hypersequent calculus GŁ∀ with structural rules. In particular, we show that every GŁ∀-provable sentence is G1Ł∀-provable and a Ł∀-sentence in the prenex form is GŁ∀-provable if and only if it is G2Ł∀-provable. For a tableau version of the calculus G2Ł∀, we describe a family of proof search algorithms that allow us to construct a proof of each G2Ł∀-provable sentence in the prenex form.

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References

  1. M. Baaz and G. Metcalfe, “Herbrand’s theorem, Skolemization and proof systems for first-order Lukasiewicz logic,” J. Logic Comput. 20, 35 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Berend and T. Tassa, “Improved bounds on Bell numbers and on moments of sums of random variables,” Probab.Math. Statist. 30, 185 (2010).

    MathSciNet  MATH  Google Scholar 

  3. M. Bongini, A. Ciabattoni, and F. Montagna, “Proof search and Co-NP completeness for many-valued logics,” Fuzzy Sets and Systems 292, 130 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Ciabattoni, C. G. Fermüller, and G. Metcalfe, “Uniform rules and dialogue games for fuzzy logics,” in Logic for Programming, Artificial Intelligence, and Reasoning, 496 (Springer-Verlag, Berlin, 2005).

    Google Scholar 

  5. P. Cintula, P. Hájek, and C. Noguera (Eds.), Handbook of Mathematical Fuzzy Logic, Vol. 2 (College Publications, London, 2011).

    MATH  Google Scholar 

  6. M. Fitting, First-Order Logic and Automated Theorem Proving (Springer-Verlag, New York, 1996).

    Book  MATH  Google Scholar 

  7. G. Gentzen, “Untersuchungen über das logische Schlieβen. I, II,” Math. Z. 39, 176, 405 (1934) [in German].

    MathSciNet  MATH  Google Scholar 

  8. A. S. Gerasimov, “A sequent calculus-based predicate logic for modeling continuous scales,” in Proc. Tenth National Conf. Artificial Intelligence CAI-06 (Obninsk, 2006), 339 (Fizmatlit, Moscow, 2006).

    Google Scholar 

  9. A. S. Gerasimov, Design and Implementation of a Proof Search Algorithm for an Extension of the Infinite-Valued Predicate Lukasiewicz Logic (Ph. D. Thesis, St. Petersburg Univ., St. Petersburg, 2007) [in Russian].

    Google Scholar 

  10. A. S. Gerasimov, “Free-variable semantic tableaux for the logic of fuzzy inequalities,” Algebra i logika 55, 156 (2016) [Algebra and Logic 55, 103 (2016)].

    MATH  Google Scholar 

  11. R. Hähnle and P. H. Schmitt, “The liberalized δ-rule in free variable semantic tableaux,” J. Autom. Reasoning 13, 211 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Hájek, Metamathematics of Fuzzy Logic (Kluwer Academic Publishers, Dordrecht, 1998).

    Book  MATH  Google Scholar 

  13. R. Kaye, “A Gray code for set partitions,” Inf. Process. Lett. 5, 171 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  14. S. C. Kleene, Mathematical Logic (Dover Publications, Mineola, NY, 2002).

    MATH  Google Scholar 

  15. N. K. Kosovskiĭ, “Level logics,” Zap. Nauchn. Seminar. POMI 220, 72 (1995) [J. Math. Sci. 87, 3221 (1997)].

    Google Scholar 

  16. S. Yu. Maslov, Theory of Deductive Systems and its Applications (Radio i Svyaz’, Moscow, 1986) [Theory of Deductive Systems and its Applications (TheMIT Press, Cambridge,MA, 1987)].

    MATH  Google Scholar 

  17. G. Metcalfe, N. Olivetti, and D. Gabbay, Proof Theory for Fuzzy Logics (Springer, Dordrecht, 2009).

    MATH  Google Scholar 

  18. G. E. Mints, “The Skolem method in intuitionistic calculi,” Trudy Mat. Inst. Steklova 121, 67 (1972) [Proc. Steklov Inst.Math. 121, 73 (1974)].

    MathSciNet  Google Scholar 

  19. E. Orlowska and J. Golińska-Pilarek, Dual Tableaux. Foundations, Methodology, Case Studies (Springer, Dordrecht, 2011).

    Book  MATH  Google Scholar 

  20. M. E. Ragaz, Arithmetische Klassifikation von Formelmengen der Unendlichwertigen Logik (Ph. D. Thesis, ETH Zürich, Zürich, 1981).

    MATH  Google Scholar 

  21. A. S. Troelstra and H. Schwichtenberg, Basic Proof Theory (Cambridge Univ. Press, Cambridge, 2000).

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. 39-35 Ozerkovaya St., Saint Petersburg, 198516, Russia

    A. S. Gerasimov

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  1. A. S. Gerasimov
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Correspondence to A. S. Gerasimov.

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Original Russian Text © A.S. Gerasimov, 2017, published in Matematicheskie Trudy, 2017, Vol. 20, No. 2, pp. 3–34.

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Gerasimov, A.S. Infinite-Valued First-Order Łukasiewicz Logic: Hypersequent Calculi Without Structural Rules and Proof Search for Sentences in the Prenex Form. Sib. Adv. Math. 28, 79–100 (2018). https://doi.org/10.3103/S1055134418020013

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