Advertisement

Adding Real Coefficients to Łukasiewicz Logic: An Application to Neural Networks

  • Antonio Di Nola
  • Brunella Gerla
  • Ioana Leustean
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8256)

Abstract

In this paper we shall deal with an extension of Łukasiewicz propositional logic obtained by considering scalar multiplication with real numbers, and we focus on the description of its Lindenbaum algebra, i.e., the algebra of truth functions. We show the correspondence between truth tables of such logic and multilayer perceptrons in which the activation function is the truncated identity.

Keywords

Many-valued logic Łukasiewicz logic McNaughton functions Neural Networks MV-algebras Riesz MV-algebras 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aguzzoli, S., Bova, S., Gerla, B.: Free Algebras and Functional Representation for Fuzzy Logics. In: Cintula, P., Hájek, P., Noguera, C. (eds.) Handbook of Mathematical Fuzzy Logic. Studies in Logic, vol. 38, pp. 713–791. College Publications, London (2011)Google Scholar
  2. 2.
    Amato, P., Di Nola, A., Gerla, B.: Neural networks and rational McNaughton functions. Journal of Multiple-Valued Logic and Soft Computing 11, 95–110 (2005)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et anneaux réticulés. Lectures Notes in Mathematics, vol. 608. Springer (1977)Google Scholar
  4. 4.
    Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer (1982)Google Scholar
  5. 5.
    Castro, J.L., Trillas, E.: The logic of neural networks. Mathware and Soft Computing 5, 23–27 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of many-valued Reasoning. Kluwer, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  7. 7.
    Dhompongsa, S., Kreinovich, V., Nguyen, H.T.: How to interpret Neural Networks in terms of fuzzy logic? In: Proceedings of VJFUZZY 2001, pp. 184–190 (2001)Google Scholar
  8. 8.
    Di Nola, A., Leus̨tean, I.: Riesz MV-algebras and their logic. In: Proceedings of EUSFLAT-LFA 2011, pp. 140–145 (2011)Google Scholar
  9. 9.
    Di Nola, A., Leus̨tean, I.: Łukasiewicz logic and Riesz spaces. Soft Computing (accepter for publication)Google Scholar
  10. 10.
    Gerla, B.: Rational Łukasiewicz logic and Divisible MV-algebras. Neural Networks World 11, 159 (2001)Google Scholar
  11. 11.
    Haykin, S.: Neural Neworks – A Comprehensive Foundation. Prentice-Hall, Englewood Cliffs (1999)Google Scholar
  12. 12.
    Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland, Amsterdam (1971)zbMATHGoogle Scholar
  13. 13.
    McNaughton, R.: A theorem about infinite-valued sentential logic. The Journal of Symbolic Logic 16, 1–13 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mundici, D.: A constructive proof of McNaughton’s theorem in infinite-valued logics. Journal of Symbolic Logic 59, 596–602 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mundici, D.: Averaging the truth value Łukasiewicz logic. Studia Logica 55, 113–127 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ovchinnikov, S.: Max-Min Representation of Piecewise Linear Functions. Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry 43, 297–302 (2002)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Antonio Di Nola
    • 1
  • Brunella Gerla
    • 2
  • Ioana Leustean
    • 3
  1. 1.Università di SalernoSalernoItaly
  2. 2.Università dell’InsubriaVareseItaly
  3. 3.University of BucharestBucharestRomania

Personalised recommendations