Modeling Decisions for Artificial Intelligence

MDAI 2015: Modeling Decisions for Artificial Intelligence pp 221-229 | Cite as

The Complexity of 3-Valued Łukasiewicz Rules

  • Miquel Bofill
  • Felip Manyà
  • Amanda Vidal
  • Mateu Villaret
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9321)

Abstract

It is known that determining the satisfiability of n-valued Łukasiewicz rules is NP-complete for \(n \ge 4\), as well as that it can be solved in time linear in the length of the formula in the Boolean case (when \(n=2\)). However, the complexity for \(n=3\) is an open problem. In this paper we formally prove that the satisfiability problem for 3-valued Łukasiewicz rules is NP-complete. Moreover, we also prove that when the consequent of the rule has at most one element, the problem is polynomially solvable.

References

  1. 1.
    Aguzzoli, S., Gerla, B., Haniková, Z.: Complexity issues in basic logic. Soft Comput. 9(12), 919–934 (2005)CrossRefMATHGoogle Scholar
  2. 2.
    Ansótegui, C., Bofill, M., Manyà, F., Villaret, M.: Building automated theorem provers for infinitely-valued logics with satisfiability modulo theory solvers. In: Proceedings of 42nd International Symposium on Multiple-Valued Logics (ISMVL), pp. 25–30. IEEE CS Press, Victoria (2012)Google Scholar
  3. 3.
    Ansótegui, C., Bofill, M., Manyà, F., Villaret, M.: Automated theorem provers for multiple-valued logics with satisfiability modulo theory solvers. Fuzzy Sets Syst. (2015). doi:10.1016/j.fss.2015.04.011
  4. 4.
    Ansótegui, C., Bofill, M., Manyà, F., Villaret, M.: SAT and SMT technology for many-valued logics. Multiple-Valued Logic Soft Comput. 24(1–4), 151–172 (2015)Google Scholar
  5. 5.
    Bofill, M., Manyà, F., Vidal, A., Villaret, M.: Finding hard instances of satisfiability in Łukasiewicz logics. In: Proceedings of 45th International Symposium on Multiple-Valued Logics (ISMVL), page In press. IEEE CS Press, Waterloo (2015)Google Scholar
  6. 6.
    Borgwardt, S., Cerami, M., Peñaloza, R.: Many-valued Horn logic is hard. In: Proceedings of the First Workshop on Logics for Reasoning about Preferences, Uncertainty, and Vagueness, PRUV 2014, co-located with IJCAR 2014, Vienna, Austria, pp. 52–58 (2014)Google Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, San Francisco (1979)MATHGoogle Scholar
  8. 8.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)CrossRefMATHGoogle Scholar
  9. 9.
    Metcalfe, G., Olivetti, N., Gabbay, D.M.: Proof Theory of Fuzzy Logics. Applied Logic Series, vol. 36. Springer, The Netherlands (2009)Google Scholar
  10. 10.
    Vidal, A.: NiBLoS: a nice BL-logics solver. Master’s thesis, Universitat de Barcelona, Barcelona, Spain (2012)Google Scholar
  11. 11.
    Vidal, A., Bou, F., Godo, L.: An SMT-based solver for continuous t-norm based logics. In: Hüllermeier, E., Link, S., Fober, T., Seeger, B. (eds.) SUM 2012. LNCS, vol. 7520, pp. 633–640. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  12. 12.
    Zhang, H., Stickel, M.E.: An efficient algorithm for unit propagation. In: Proceedings of the Fourth International Symposium on Artificial Intelligence and Mathematics (AI-MATH’96), Fort Lauderdale, Florida pp. 166–169 (1996)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Miquel Bofill
    • 1
  • Felip Manyà
    • 2
  • Amanda Vidal
    • 2
  • Mateu Villaret
    • 1
  1. 1.Universitat de GironaGironaSpain
  2. 2.Artificial Intelligence Research Institute (IIIA, CSIC)BellaterraSpain

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