Studia Logica

, Volume 100, Issue 6, pp 1271–1290 | Cite as

Lewis Dichotomies in Many-Valued Logics



In 1979, H. Lewis shows that the computational complexity of the Boolean satisfiability problem dichotomizes, depending on the Boolean operations available to formulate instances: intractable (NP-complete) if negation of implication is definable, and tractable (in P) otherwise [21]. Recently, an investigation in the same spirit has been extended to nonclassical propositional logics, modal logics in particular [2, 3]. In this note, we pursue this line in the realm of many-valued propositional logics, and obtain complexity classifications for the parameterized satisfiability problem of two pertinent samples, Kleene and Gödel logics.


Parameterized satisfiability Complexity dichotomy Many-valued logics 


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  1. 1.
    Aguzzoli S., Gerla B.: Normal Forms and Free Algebras for Some Extensions of MTL. Fuzzy Sets and Systems 159, 1131–1152 (2008)CrossRefGoogle Scholar
  2. 2.
    Bauland, M., H. Schnoor, E. Hemaspaandra, and I. Schnoor, Generalized Modal Satisfiability, Proceedings of the 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS’06), 2006, pp. 500–511.Google Scholar
  3. 3.
    Bauland, M., H. Schnoor I. Schnoor, T. Schneider, and H. Vollmer, The Complexity of Generalized Satisfiability for Linear Temporal Logic, Logical Methods in Computer Science 5:1, 2008.Google Scholar
  4. 4.
    Berman J., Blok W. J.: Stipulations, Multivalued Logic, and De Morgan Algebras. Journal of Multiple-Valued Logic and Soft Computing 7(5-6), 391–416 (2001)Google Scholar
  5. 5.
    Berman J., Mukaidono M.: Enumerating fuzzy switching functions and free Kleene algebra. Computers and Mathematics with Applications 10(1), 25–35 (1984)CrossRefGoogle Scholar
  6. 6.
    Beyersdorff , Thomas M., Meier A., Vollmer H.: The Complexity of Propositional Implication. Information Processing Letters 109, 1071–1077 (2009)CrossRefGoogle Scholar
  7. 7.
    Bodnarchuk, V. G., V. N. Kotov, L. A. Kalužnin, and B. A. Romov, Galois Theory for Post Algebras, I, II, Cybernetics 5:243–252, 531–539, 1969.Google Scholar
  8. 8.
    Böhler E., Schnoor H., Reith S., Vollmer H.: Bases for Boolean Co-Clones. Information Processing Letters 96(2), 59–66 (2005)CrossRefGoogle Scholar
  9. 9.
    Böhler E., Reith S., Creignou N., Vollmer H.: Playing with Boolean Blocks. Part I: Post’s Lattice with Applications to Complexity Theory.. SIGACT News 34(4), 38–52 (2003)CrossRefGoogle Scholar
  10. 10.
    Bulatov A.: Identities in Lattices of Closed Classes. Discrete Applied Mathematics 3(6), 601–609 (1993)Google Scholar
  11. 11.
    Cook, S. A., The Complexity of Theorem Proving Procedures, Proceedings of the 3rd Symposium on Theory of Computation (STOC’71), 1971, pp. 151–158.Google Scholar
  12. 12.
    Geiger D.: Closed Systems of Functions and Predicates. Pacific Journal of Mathematics 27(2), 228–250 (1968)Google Scholar
  13. 13.
    Gerla B.: A Note on Functions Associated with Gödel Formulas. Soft Computing 4(4), 206–209 (2000)CrossRefGoogle Scholar
  14. 14.
    Hájek, P., Metamathematics of Fuzzy Logic, Kluwer, 1998.Google Scholar
  15. 15.
    Horn A.: Free L-Algebras. Journal of Symbolic Logic 34, 475–480 (1969)CrossRefGoogle Scholar
  16. 16.
    Kalman J. A.: Lattices With Involution. Transactions of the American Mathematical Society 87(2), 485–491 (1958)CrossRefGoogle Scholar
  17. 17.
    Ladner R. E.: On the Structure of Polynomial-Time Reducibility. Journal of the ACM 22, 155–171 (1975)CrossRefGoogle Scholar
  18. 18.
    Ladner R. E.: The Circuit Value Problem is Logspace Complete for P. SIGACT News 9(2), 18–20 (1977)Google Scholar
  19. 19.
    Lau, D., Function Algebras on Finite Sets: Basic Course on Many-Valued Logic and Clone Theory, Springer, 2006.Google Scholar
  20. 20.
    Levin L. A.: Universal Sequential Search Problems. Problems of Information Transmission 9(3), 265–266 (1975)Google Scholar
  21. 21.
    Lewis H. R.: Satisfiability Problems for Propositional Calculi. Mathematical Systems Theory 13, 45–53 (1979)CrossRefGoogle Scholar
  22. 22.
    Papadimitriou, C. H., Computational Complexity, Addison Wesley Longman, 1995.Google Scholar
  23. 23.
    Post E. L.: The Two-Valued Iterative Systems of Mathematical Logic. Annals of Mathematical Studies 5, 1–122 (1941)Google Scholar
  24. 24.
    Reith, S., Generalized Satisfiability Problems, Ph.D. thesis, Institut fär Informatik, Universität Wärzburg, 2001.Google Scholar
  25. 25.
    Schechter, E., Classical and Nonclassical Logics, Princeton University Press, 2005.Google Scholar
  26. 26.
    Turner, R., Logics for Artificial Intelligence, Ellis Horwood, 1984.Google Scholar

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© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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