Herbrand Theorems and Skolemization for Prenex Fuzzy Logics

  • Matthias Baaz
  • George Metcalfe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5028)

Abstract

Approximate Herbrand theorems are established for first-order fuzzy logics based on continuous t-norms, and used to provide proof-theoretic proofs of Skolemization for their Prenex fragments. Decidability and complexity results for particular fragments are obtained as consequences.

Keywords

Herbrand Theorem Skolemization Fuzzy Logics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baaz, M., Ciabattoni, A., Fermüller, C.G.: Herbrand’s theorem for prenex Gödel logic and its consequences for theorem proving. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 201–216. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. 2.
    Baaz, M., Ciabattoni, A., Montagna, F.: Analytic calculi for monoidal t-norm based logic. Fundamenta Informaticae 59(4), 315–332 (2004)MathSciNetMATHGoogle Scholar
  3. 3.
    Baaz, M., Iemhoff, R.: The skolemization of existential quantifiers in intuitionistic logic. Annals of Pure and Applied Logic 142(1-3), 269–295 (2006)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Baaz, M., Metcalfe, G.: Herbrand’s theorem, skolemization, and proof systems for Łukasiewicz logic (submitted), http://www.math.vanderbilt.edu/people/metcalfe/publications
  5. 5.
    Baaz, M., Zach, R.: Hypersequents and the proof theory of intuitionistic fuzzy logic. In: Clote, P.G., Schwichtenberg, H. (eds.) CSL 2000. LNCS, vol. 1862, pp. 187–201. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  6. 6.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)MATHGoogle Scholar
  7. 7.
    Hájek, P.: Arithmetical complexity of fuzzy logic – a survey. Soft Computing 9, 935–941 (2005)CrossRefMATHGoogle Scholar
  8. 8.
    Hájek, P.: Making fuzzy description logic more general. Fuzzy Sets and Systems 154(1), 1–15 (2005)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Hanikova, Z.: A note on the complexity of tautologies of individual t-algebras. Neural Network World 12(5), 453–460 (2002)Google Scholar
  10. 10.
    Montagna, F.: Three complexity problems in quantified fuzzy logic. Studia Logica 68(1), 143–152 (2001)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Mostert, P.S., Shields, A.L.: On the structure of semigroups on a compact manifold with boundary. Annals of Mathematics 65, 117–143 (1957)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Novák, V.: On the Hilbert-Ackermann theorem in fuzzy logic. Acta Mathematica et Informatica Universitatis Ostraviensis 4, 57–74 (1996)MathSciNetMATHGoogle Scholar
  13. 13.
    Ragaz, M.E.: Arithmetische Klassifikation von Formelmengen der unendlichwertigen Logik. PhD thesis, ETH Zürich (1981)Google Scholar
  14. 14.
    Straccia, U.: Reasoning within fuzzy description logics. Journal of Artificial Intelligence Research 14, 137–166 (2001)MathSciNetMATHGoogle Scholar
  15. 15.
    Vojtás, P.: Fuzzy logic programming. Fuzzy Sets and Systems 124, 361–370 (2001)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Willard, S.: General Topology. Dover (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Matthias Baaz
    • 1
  • George Metcalfe
    • 2
  1. 1.Institute of Discrete Mathematics and GeometryTechnical University ViennaWienAustria
  2. 2.Department of MathematicsVanderbilt UniversityNashvilleUSA

Personalised recommendations