Skip to main content
SpringerLink
Log in

Interval Valued QL-Implications

  • Conference paper
Logic, Language, Information and Computation (WoLLIC 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4576))

Abstract

The aim of this work is to analyze the interval canonical representation for fuzzy QL-implications and automorphisms. Intervals have been used to model the uncertainty of a specialist’s information related to truth values in the fuzzy propositional calculus: the basic systems are based on interval fuzzy connectives. Thus, using subsets of the real unit interval as the standard sets of truth degrees and applying continuous t-norms, t-conorms and negation as standard truth interval functions, the standard truth interval function of an QL-implication can be obtained. Interesting results on the analysis of interval canonical representation for fuzzy QL-implications and automorphisms are presented. In addition, commutative diagrams are used in order to understand how an interval automorphism acts on interval QL-implications, generating other interval fuzzy QL-implications.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Hickey, T., Ju, Q., Emdem, M.: Interval arithmetic: from principles to implementation. Journal of the ACM 48(5), 1038–1068 (2001)

    Article  MathSciNet  Google Scholar 

  2. Moore, R.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)

    Google Scholar 

  3. Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, New York (1983)

    MATH  Google Scholar 

  4. Moore, R.E.: Interval Arithmetic and Automatic Error Analysis in Digital Computing. PhD thesis, Stanford University, Stanford (1962)

    Google Scholar 

  5. Kearfort, R.B., Kreinovich, V. (eds.): Applications of Interval Computations. Kluwer Academic Publishers, Boston (1996)

    Google Scholar 

  6. Nguyen, H., Kreinovich, V., Zuo, Q.: Interval-valued degrees of belief: applications of interval computations to expert systems and intelligent control. International Journal of Uncertainty, Fuzziness, and Knowledge-Based Systems 5(3), 317–358 (1997)

    Article  MathSciNet  Google Scholar 

  7. Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman & Hall, London (1991)

    MATH  Google Scholar 

  8. Walley, P.: Measures of uncertainty. Artificial Intelligence 83, 1–58 (1996)

    Article  MathSciNet  Google Scholar 

  9. Dubois, D., Prade, H.: Interval-valued fuzzy sets, possibility theory and imprecise probability. In: Proc. Intl. Conf. Fuzzy Logic and Technology, Barcelona, pp. 314–319 (2005)

    Google Scholar 

  10. Lodwick, W.: Preface. Reliable Computing 10(4), 247–248 (2004)

    Article  MathSciNet  Google Scholar 

  11. Turksen, I.: Fuzzy normal forms. Fuzzy Sets and Systems 69, 319–346 (1995)

    Article  MathSciNet  Google Scholar 

  12. Moore, R., Lodwick, W.: Interval analysis and fuzzy set theory. Fuzzy Sets and Systems 135(1), 5–9 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gasse, B.V., Cornelis, G., Deschrijver, G., Kerre, E.: On the properties of a generalized class of t-norms in interval-valued fuzzy logics. New Mathematics and Natural Computation 2, 29–42 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning - I. Information Sciences 6, 199–249 (1975)

    Article  MathSciNet  Google Scholar 

  15. Bedregal, B., Takahashi, A.: The best interval representation of t-norms and automorphisms. Fuzzy Sets and Systems 157(24), 3220–3230 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Bedregal, B., Takahashi, A.: Interval valued versions of t-conorms, fuzzy negations and fuzzy implications. In: IEEE Proc. Intl. Conf. on Fuzzy Systems, Vancouver, Vancouver, pp. 9553–9559. IEEE Computer Society Press, Los Alamitos (2006)

    Google Scholar 

  17. Santiago, R., Bedregal, B., Acióly, B.: Formal aspects of correctness and optimality in interval computations. Formal Aspects of Computing 18(2), 231–243 (2006)

    Article  MATH  Google Scholar 

  18. Callejas-Bedregal, R., Bedregal, B.: Intervals as a domain constructor. TEMA 2(1), 43–52 (2001), http://www.sbmac.org.br/tema

    Google Scholar 

  19. Kearfott, R.B.: Rigorous Global Search: Continuous problems. Kluwer Academic Publishers, Dordrecht (1996)

    MATH  Google Scholar 

  20. Caprani, O., Madsen, K., Stauning, O.: Existence test for asynchronous interval iteration. Reliable Computing 3(3), 269–275 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  21. Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analisys: With examples in parameter and state estimation, robust control and robotic. Springer, Heidelberg (2001)

    Google Scholar 

  22. Klement, E., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)

    MATH  Google Scholar 

  23. Bustince, H., Burilo, P., Soria, F.: Automorphism, negations and implication operators. Fuzzy Sets and Systems 134, 209–229 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Baczynski, M.: Residual implications revisited. Notes on the Smets-Magrez. Fuzzy Sets and Systems 145(2), 267–277 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Balasubramaniam, J.: Yager’s new class of implications J f and some classical tautologies. Information Sciences (2006)

    Google Scholar 

  26. Fodor, J.C.: On fuzzy implication operators. Fuzzy Sets and Systems 42, 293–300 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  27. Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht (1994)

    MATH  Google Scholar 

  28. Horcik, R., Navara, M.: Validation sets in fuzzy logics. Kybernetika 38(2), 319–326 (2002)

    MathSciNet  Google Scholar 

  29. Leski, J.: ε-insensitive learning techniques for approximate reasoning system. International Jour. of Computational Cognition 1(1), 21–77 (2003)

    Google Scholar 

  30. Ruan, D., Kerre, E.: Fuzzy implication operators and generalized fuzzy methods of cases. Fuzzy Sets and Systems 54, 23–37 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  31. Yager, R.: On the implication operator in fuzzy logic. Information Sciences 31, 141–164 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  32. Yager, R.: On global requirements for implication operators in fuzzy modus ponens. Fuzzy Sets and Systems 106, 3–10 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  33. Yager, R.: On some new classes of implication operators and their role in approximate reasoning. Information Sciences 167, 193–216 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  34. Shi, Y., Ruan, D., Kerre, E.E.: On the characterizations of fuzzy implications satisfying I(x,y) = I(x,I(x,y)). Information Sciences 177(14), 2954–2970 (2007)

    Article  Google Scholar 

  35. Klement, E., Navara, M.: A survey on different triangular norm-based fuzzy logics. Fuzzy Sets and Systems 101, 241–251 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  36. Navara, M.: Characterization of measures based on strict triangular norms. Mathematical Analysis and Applications 236, 370–383 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  37. Gehrke, M., Walker, C., Walker, E.: Some comments on interval valued fuzzy sets. International Journal of Intelligent Systems 11, 751–759 (1996)

    Article  MATH  Google Scholar 

  38. Gehrke, M., Walker, C., Walker, E.: Algebraic aspects of fuzzy sets and fuzzy logics. In: Proc. of the Work. on Current Trends and Development in Fuzzy Logic, Thessaloniki, pp. 101–170 (1999)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Programa de Pós-Graduaçäo em Informática, Universidade Católica de Pelotas, Rua Felix da Cunha 412, 96010-000 Pelotas, Brazil

    R. H. S. Reiser & G. P. Dimuro

  2. Depto de Informáticae Matemática Aplicada, Universidade Federal do Rio Grande do Norte, Campus Universitário s/n, 59072-970 Natal, Brazil

    B. C. Bedregal & R. H. N. Santiago

Authors
  1. R. H. S. Reiser
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. P. Dimuro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. C. Bedregal
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. H. N. Santiago
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Daniel Leivant Ruy de Queiroz

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer Berlin Heidelberg

About this paper

Cite this paper

Reiser, R.H.S., Dimuro, G.P., Bedregal, B.C., Santiago, R.H.N. (2007). Interval Valued QL-Implications. In: Leivant, D., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2007. Lecture Notes in Computer Science, vol 4576. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73445-1_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-73445-1_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73443-7

  • Online ISBN: 978-3-540-73445-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation