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Satisfiability in Łukasiewicz Logic and Its Unbounded Relative

Authors Zuzana Haniková , Filip Jankovec

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Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
Keywords
  • unbounded Łukasiewicz Logic
  • Łukasiewicz Logic
  • Abelian Logic
  • existential theory
  • computational complexity
  • NP-completeness

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Abstract

Unbounded Łukasiewicz logic is a substructural logic that combines features of infinite-valued Łukasiewicz logic with those of abelian logic. The logic is finitely strongly complete w.r.t. the additive 𝓁-group on the reals expanded with a distinguished element -1. We show that the existential theory of this structure is NP-complete. This provides a complexity upper bound for the set of theorems and the finite consequence relation of unbounded Łukasiewicz logic. The result is obtained by reducing the problem to the existential theory of the MV-algebra on the reals, the standard semantics of Łukasiewicz logic. This provides a new connection between both logics. The result entails a translation of the existential theory of the standard MV-algebra into itself.

Cite As Get BibTex

Zuzana Haniková and Filip Jankovec. Satisfiability in Łukasiewicz Logic and Its Unbounded Relative. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CSL.2026.14

Author Details

Zuzana Haniková
  • Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic
Filip Jankovec
  • Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic

Funding

Both authors were supported by ERDF-Project Knowledge in the Age of Distrust, No. CZ.02.01.01/00/23_025/0008711.
  • Jankovec, Filip: SVV-2025-260837.

Acknowledgements

Three anonymous reviews helped to improve the text.

References

  1. M. E. Adams and W. Dziobiak. Q-universal Quasivarieties of Algebras. Proceedings of the American Mathematical Society, 120(4):1053-1059, 1994. Google Scholar
  2. Stefano Aguzzoli. Geometric and Proof-Theoretic Issues in Łukasiewicz Propositional Logics. PhD thesis, University of Siena, Siena, 1998. Google Scholar
  3. Stefano Aguzzoli. An asymptotically tight bound on countermodels for Łukasiewicz logic. International Journal of Approximate Reasoning, 43(1):76-89, 2006. URL: https://doi.org/10.1016/J.IJAR.2006.02.003.
  4. Stefano Aguzzoli and Agata Ciabattoni. Finiteness in infinite-valued Łukasiewicz logic. Journal of Logic, Language and Information, 9(1):5-29, 2000. URL: https://doi.org/10.1023/A:1008311022292.
  5. Stefano Aguzzoli and Brunella Gerla. Finite-valued reductions of infinite-valued logics. Archive for Mathematical Logic, 41(4):361-399, 2002. URL: https://doi.org/10.1007/S001530100118.
  6. Chen Chung Chang. Algebraic analysis of many-valued logics. Transactions of the American Mathematical Society, 88(2):467-490, 1958. Google Scholar
  7. Chen Chung Chang. A new proof of the completeness of the Łukasiewicz axioms. Transactions of the American Mathematical Society, 93(1):74-80, 1959. Google Scholar
  8. Roberto Cignoli, Itala M.L. D'Ottaviano, and Daniele Mundici. Algebraic Foundations of Many-Valued Reasoning, volume 7 of Trends in Logic. Kluwer, Dordrecht, 1999. Google Scholar
  9. Petr Cintula, Berta Grimau, Carles Noguera, and Nicholas J. J. Smith. These degrees go to eleven: fuzzy logics and gradable predicates. Synthese, 200(6):Paper No. 445, 38, 2022. URL: https://doi.org/10.1007/s11229-022-03909-2.
  10. Petr Cintula and Petr Hájek. Complexity issues in axiomatic extensions of Łukasiewicz logic. Journal of Logic and Computation, 19(2):245-260, 2009. URL: https://doi.org/10.1093/LOGCOM/EXN052.
  11. Petr Cintula, Petr Hájek, and Carles Noguera, editors. Handbook of Mathematical Fuzzy Logic, volume 1&2 of Studies in Logic, vol. 37-38. College Publications, London, 2011. Google Scholar
  12. Petr Cintula, Filip Jankovec, and Carles Noguera. Superabelian logics, 2024. submitted. URL: https://arxiv.org/abs/2409.20170.
  13. A. H. Clifford. Partially ordered abelian groups. Ann. of Math. (2), 41:465-473, 1940. URL: https://doi.org/10.2307/1968728.
  14. Wojciech Dzik. Unification in some substructural logics of BL-algebras and hoops. Reports on Mathematical Logic, 43:73-83, 2008. URL: https://rml.tcs.uj.edu.pl/rml-43/a-dzi-43.htm.
  15. Ronald Fagin, Joseph Y. Halpern, and Nimrod Megiddo. A logic for reasoning about probabilities. Information and Computation, 87(1-2):78-128, 1990. URL: https://doi.org/10.1016/0890-5401(90)90060-U.
  16. Melvin C. Fitting. Many-valued modal logics. Fundamenta Informaticae, 15(3-4):235-254, 1991. Google Scholar
  17. Melvin C. Fitting. Many-valued modal logics II. Fundamenta Informaticae, 17(1-2):55-73, 1992. Google Scholar
  18. Tommaso Flaminio and Franco Montagna. A logical and algebraic treatment of conditional probability. Archive for Mathematical Logic, 44(2):245-262, 2005. URL: https://doi.org/10.1007/S00153-004-0253-Z.
  19. Nikolaos Galatos, Peter Jipsen, Tomasz Kowalski, and Hiroakira Ono. Residuated Lattices: An Algebraic Glimpse at Substructural Logics, volume 151 of Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam, 2007. Google Scholar
  20. Yu. Sh. Gurevich and A. I. Kokorin. Universal equivalence of ordered abelian groups. Algebra i Logika. Sem., 2(1):37-39, 1963. Google Scholar
  21. Reiner Hähnle. Many-valued logic and mixed integer programming. Annals of Mathematics and Artificial Intelligence, 12(3-4):231-264, December 1994. URL: https://doi.org/10.1007/BF01530787.
  22. Petr Hájek. Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic. Kluwer, Dordrecht, 1998. URL: https://doi.org/10.1007/978-94-011-5300-3.
  23. Petr Hájek and Dagmar Harmancová. A many-valued modal logic. In Proceedings IPMU’96. Information Processing and Management of Uncertainty in Knowledge-Based Systems, pages 1021-1024, 1996. Google Scholar
  24. Zuzana Haniková. Computational complexity of propositional fuzzy logics. In Petr Cintula, Petr Hájek, and Carles Noguera, editors, Handbook of Mathematical Fuzzy Logic, volume 2, pages 793-851. College Publications, 2011. Google Scholar
  25. Zuzana Haniková and Petr Savický. Term satisfiability in FL_ew-algebras. Theoretical Computer Science, 631:1-15, 2016. URL: https://doi.org/10.1016/J.TCS.2016.03.009.
  26. Emil Jeřábek. The complexity of admissible rules of Łukasiewicz logic. Journal of Logic and Computation, 23(3):693-705, 2013. URL: https://doi.org/10.1093/LOGCOM/EXS007.
  27. Yuichi Komori. Super-Łukasiewicz implicational logics. Nagoya Mathematical Journal, 72:127-133, 1978. Google Scholar
  28. Yuichi Komori. Super-Łukasiewicz propositional logics. Nagoya Mathematical Journal, 84:119-133, 1981. Google Scholar
  29. Jan Łukasiewicz. On determinism. The Polish Review, 13(3):47-61, 1968. Google Scholar
  30. Jan Łukasiewicz and Alfred Tarski. Untersuchungen über den Aussagenkalkül. Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, cl. III, 23(iii):30-50, 1930. Google Scholar
  31. Enrico Marchioni and Franco Montagna. Complexity and definability issues in ŁΠ1/2. Journal of Logic and Computation, 17(2):311-331, 2007. Google Scholar
  32. Robert McNaughton. A theorem about infinite-valued sentential logic. Journal of Symbolic Logic, 16(1):1-13, 1951. URL: https://doi.org/10.2307/2268660.
  33. George Metcalfe, Nicola Olivetti, and Dov M. Gabbay. Proof Theory for Fuzzy Logics, volume 36 of Applied Logic Series. Springer, 2008. Google Scholar
  34. Robert K. Meyer and John K. Slaney. Abelian logic from A to Z. In Graham Priest, Richard Routley, and Jean Norman, editors, Paraconsistent Logic: Essays on the Inconsistent, pages 245-288. Philosophia Verlag, 1989. Google Scholar
  35. Daniele Mundici. Satisfiability in many-valued sentential logic is NP-complete. Theoretical Computer Science, 52(1-2):145-153, 1987. URL: https://doi.org/10.1016/0304-3975(87)90083-1.
  36. Daniele Mundici. Advanced Łukasiewicz Calculus and MV-Algebras, volume 35 of Trends in Logics. Springer, 2011. Google Scholar
  37. Jan Pavelka. On fuzzy logic I, II, III. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 25:45-52, 119-134, 447-464, 1979. Google Scholar
  38. Matthias Emil Ragaz. Arithmetische Klassifikation von Formelmengen der unendlichwertigen Logik. PhD thesis, Swiss Federal Institute of Technology, Zürich, 1981. Google Scholar
  39. Greg Restall. An Introduction to Substructural Logics. Routledge, New York, 2000. Google Scholar
  40. Alan Rose and J.Barkley Rosser. Fragments of many-valued statement calculi. Transactions of the American Mathematical Society, 87(1):1-53, 1958. Google Scholar
  41. Bruno Scarpellini. Die Nichtaxiomatisierbarkeit des unendlichwertigen Prädikatenkalküls von Łukasiewicz. Journal of Symbolic Logic, 27(2):159-170, 1962. Google Scholar
  42. Amanda Vidal. Undecidability and non-axiomatizability of modal many-valued logics. The Journal of Symbolic Logic, 87(4):1576-1605, 2022. URL: https://doi.org/10.1017/JSL.2022.32.
  43. Volker Weispfenning. The complexity of the word problem for Abelian 𝓁-groups. Theoretical Computer Science, 48(1):127-132, 1986. URL: https://doi.org/10.1016/0304-3975(86)90089-7.
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