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Goedel logics: Prenex fragments (2407.16683v2)

Published 23 Jul 2024 in cs.LO

Abstract: In this paper, we provide a complete classification for the first-order Goedel logics concerning the property that the formulas admit logically equivalent prenex normal forms. We show that the only first-order Goedel logics that admit such prenex forms are those with finite truth value sets since they allow all quantifier-shift rules and the logic $G_\uparrow$ with only one accumulation point at 1 in the infinite truth value set. In all the other cases, there are generally no logically equivalent prenex normal forms. We will also see that $G_\uparrow$ is the intersection of all finite first-order Goedel logics. The second part of this paper investigates the existence of effective equivalence between the validity of a formula and the validity of some prenex normal form. The existence of such a normal form is obvious for finite valued Goedel logic and $G_\uparrow$. Goedel logics with an uncountable truth value set admit the prenex normal forms if and only if every surrounding of 0 is uncountable or 0 is an isolated point. Otherwise, uncountable Goedel logics are not recursively enumerable, however, the prenex fragment is always recursively enumerable. Therefore, there is no effective translation between the valid formula and the valid prenex normal form. However, the existence of effectively constructible validity equivalent prenex forms for the countable case is still up for debate.

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References (56)
  1. Juan P Aguilera and Matthias Baaz. Unsound inferences make proofs shorter. arXiv preprint arXiv:1608.07703, 2016.
  2. A nonhyperarithmetical Gödel logic. In International Symposium on Logical Foundations of Computer Science, pages 1–8. Springer, 2022.
  3. Matthias Baaz. Infinite-valued Gödel logic with 0-1-projections and relativisations. In Petr Hájek, editor, Gödel’96: Logical Foundations of Mathematics, Computer Science, and Physics, volume 6 of Lecture Notes in Logic, pages 23–33. Springer-Verlag, Brno, 1996.
  4. A Schütte-Tait style cut-elimination proof for first-order Gödel logic. In Uwe Egly and Christian G. Fermüller, editors, Automated Reasoning with Analytic Tableaux and Related Methods, International Conference, TABLEAUX 2002, volume 2381 of LNCS, pages 24–38, Berlin, 2002. Springer.
  5. Hypersequent calculi for Gödel logics: A survey. Journal of Logic and Computation, 13(6):835–861, 2003.
  6. Monadic fragments of Gödel logics: Decidability and undecidability results. In Nachum Dershowitz and Andrei Voronkov, editors, Logic for Programming, Artificial Intelligence, and Reasoning, volume 4790/2007 of Lecture Notes in Computer Science, pages 77–91, 2007.
  7. SAT in monadic Gödel logics: A borderline between decidability and undecidability. In Hiroakira Ono, Makoto Kanazawa, and Ruy de Queiroz, editors, Logic, Language, Information and Computation, 16th Workshop, WoLLIC 2009, number 5514 in LNAI, pages 113–123, 2009.
  8. First-order satisfiability in Gödel logics: An NP-complete fragment. Theoretical Computer Science, 412:6612–6623, 2011.
  9. Quantified propositional Gödel logic. In Andrei Voronkov and Michel Parigot, editors, Logic for Programming and Automated Reasoning. 7th International Conference, LPAR 2000, volume 1995 of LNAI, pages 240–256, Berlin, 2000. Springer.
  10. A resolution mechanism for prenex Gödel logic. In Anuj Dawar and Helmut Veith, editors, Computer Science Logic, 24th International Workshop, CSL 2010, 19th Annual Conference of the EACSL, volume 6247 of LNCS, pages 67–79, Berlin, 2010. Springer.
  11. Completeness of a first-order temporal logic with time-gaps. Theoretical Computer Science, 160(1–2):241–270, 1996.
  12. Incompleteness of a first-order Gödel logic and some temporal logics of programs. In Computer Science Logic, volume 1092/1996 of Lecture Notes in Computer Science, pages 1–15, 1996.
  13. Quantifier elimination for quantified propositional logics on Kripke frames of type omega. Journal of Logic and Computation, 18:649–668, 2008.
  14. Matthias Baaz, Norbert Preining, Gödel–Dummett logics, in: Petr Cintula, Petr Hájek, Carles Noguera (Eds.) Handbook of Mathematical Fuzzy Logic vol.2, College Publications, 2011, pp.585–626, chapterVII.
  15. On the classification of first order Goedel logics. Ann. Pure Appl. Log 170:36–57, 2019.
  16. Characterization of the axiomatizable prenex fragments of first-order Gödel logics. In 33rd IEEE International Symposium on Multiple-Valued Logic (ISMVL 2003), pages 175–180, Los Alamitos, 2003. IEEE Computer Society.
  17. Completeness of a hypersequent calculus for some first-order Göde logics with delta. In 36th International Symposium on Multiple-valued Logic (ISMVL 2006). IEEE Computer Society, 2006.
  18. First-order Gödel logics. Annals of Pure and Applied Logic, 147(1–2):23–47, 2007.
  19. An axiomatization of quantified propositional Gödel logic using the Takeuti-Titani rule. In Logic Colloquium 1999, volume 13 of Lecture Notes in Logic, pages 91–104, 2000.
  20. Matthias and Helmut Veith. Interpolation in Fuzzy Logic Archive for Mathematical Logic, accepted for publication.
  21. Matthias and Helmut Veith. Quantifier Elimination in Fuzzy Logic CSL 1998.
  22. Compact propositional Gödel logics. In Proceedings of 28th International Symposium on Multiple-Valued Logic, pages 108–113, Los Alamitos, CA, 1998. IEEE Computer Society Press.
  23. Hypersequents and the proof theory of intuitionistic fuzzy logic. In Peter G. Clote and Helmut Schwichtenberg, editors, Proceedings of 14th CSL Workshop, volume 1862 of Lecture Notes in Computer Science, pages 187–201, Berlin, 2000. Springer-Verlag.
  24. J.R Buechi. On a decision method in restricted second order arithmetic. In Logic, Methodology, and Philosophy of Science Proc. of the 1960 Congress, Stanford University Press, pp. 1-11.
  25. Continuous Fraïssé conjecture. Order, 25(4):281–298, 2008.
  26. Linear Kripke frames and Gödel logics. Journal of Symbolic Logic, 72(1):26–44, 2007.
  27. Arnold Beckmann, Norbert Preining, Deciding logics of linear Kripke frames with scattered end pieces, Soft Comput. 21(1) (2017) 191–197.
  28. Arnold Beckmann, Norbert Preining, Separating intermediate predicate logics of well-founded and dually well-founded structures by monadic sentences, J. Logic Comput. 25(3) (2015) 527–547.
  29. Agata Ciabattoni. A proof-theoretical investigation of global intuitionistic (fuzzy) logic. Archive for Mathematical Logic, 44(4):435–457, 2005.
  30. Brian A. Davey. On the lattice of subvarieties. Houston Journal of Mathematics, 5(2):183–192, 1979.
  31. Michael Dummett. A propositional calculus with denumerable matrix. Journal of Symbolic Logic, 24(2): 97–106, 1959.
  32. Melvin Chris Fitting. Intuitionistic Logic, Model Theory and Forcing. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam/London, 1969.
  33. Melvin Chris Fitting. Proof Methods for Modal and Intuitionistic Logics. Kluwer, 1983.
  34. Roland Fraïssé. Sur la comparaison des types d’ordres. Comptes Rendus Hebdomadaire des Séances de l’Académie des Sciences, Paris, 226:1330–1331, 1948.
  35. Kurt Gödel. Zum intuitionistischen Aussagenkalkül. Anzeiger Akademie der Wissenschaften Wien, 69: 65–66, 1932.
  36. S. Gottwald. Mehrwertige Logik. Berlin 1989.
  37. Petr Hájek. Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic. Kluwer, Dordrecht, 1998.
  38. Petr Hájek. A non-arithmetical Gödel logic. Logic Journal of the IGPL, 13(4):435–441, 2005.
  39. Alfred Horn. Logic with truth values in a linearly ordered Heyting algebras. Journal of Symbolic Logic, 34(3):395–408, 1969.
  40. Alexander S. Kechris. Classical Descriptive Set Theory, volume 159 of Graduate Texts in Mathematics. Springer, 1995.
  41. Nichtklassische Logik. Akademie-Verlag, Berlin 1988.
  42. L. Maksimova. Craig’s interpolation theorem and amalgamable varieties. Doklady Akademii Nauk SSSR, 237/6, 1281-1284, 1977.
  43. Richard Mansfield. Perfect subsets of definable sets of real numbers. Pacific Journal of Mathematics, 35:451–457, 1970.
  44. Yiannis N. Moschovakis. Descriptive Set Theory, volume 100 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1980.
  45. Hiroakira Ono. Kripke models and intermediate logics. Publications of the Research Institute for Mathematical Sciences Kyoto University, 6:461–476, 1971.
  46. Hiroakira Ono. A study of intermediate predicate logics. Publications of the Research Institute for Mathematical Sciences Kyoto University, 8:619–649, 1972/73.
  47. Norbert Preining. Complete Recursive Axiomatizability of Gödel Logics. PhD thesis, Vienna University of Technology, Austria, 2003.
  48. Norbert Preining, Gödel logics and Cantor–Bendixon analysis, in: M. Baaz, A. Voronkov (Eds.), Logic for Programming and Automated Reasoning, LPAR 2002. Proceedings, in: Lecture Notes in Artificial Intelligence, vol.2514, Springer, 2002, pp.327–336.
  49. George N. Raney. Completely distributive complete lattices. Proceedings of the American Mathematical Society, 3:677–680, 1952.
  50. Joseph G. Rosenstein. Linear Orderings. Academic Press, New York, 1982.
  51. Mitio Takano. Another proof of the strong completeness of the intuitionistic fuzzy logic. Tsukuba Journal of Mathematics, 11(1):101–105, 1987.
  52. Mitio Takano. Ordered sets R and Q as bases of Kripke models. Studia Logica, 46:137–148, 1987.
  53. Gaisi Takeuti Proof Theory, 2nd ed., North-Holland , 1987.
  54. Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. Journal of Symbolic Logic, 49(3):851–866, 1984.
  55. Anne S. Troelstra. Aspects of constructive mathematics. In Jon Barwise, editor, Handbook of Mathematical Logic, pages 973–1052. North-Holland, 1977.
  56. Reinhard Winkler. How much must an order theorist forget to become a topologist? In Contributions of General Algebra 12, Proc. of the Vienna Conference, pages 420–433, Klagenfurt, Austria, 1999. Verlag Johannes Heyn.

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