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International Conference on Artificial Intelligence and Soft Computing

ICAISC 2010: Artificial Intelligence and Soft Computing pp 3–10Cite as

On the Distributivity of Fuzzy Implications over Continuous Archimedean Triangular Norms

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6113))

Abstract

Recently, we have examined solutions of the following distributive functional equation I(x,S 1(y,z)) = S 2(I(x,y),I(x,z)), when S 1, S 2 are continuous Archimedean t-conorms and I is an unknown function [5,3]. Earlier, in [1,2], we have also discussed solutions of the following distributive equation I(x,T 1(y,z)) = T 2(I(x,y),I(x,z)), when T 1, T 2 are strict t-norms. In particular, in both cases, we have presented solutions which are fuzzy implications in the sense of Fodor and Roubens. In this paper we continue these investigations for the situation when T 1, T 2 are continuous Archimedean t-norms, thus we give a partial answer for one open problem postulated in [2]. Obtained results are not only theoretical – they can be also useful for the practical problems, since such distributive equations have an important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems.

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References

  1. Baczyński, M.: On a class of distributive fuzzy implications, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 9, 229–238 (2001)

    MATH  Google Scholar 

  2. Baczyński, M.: Contrapositive symmetry of distributive fuzzy implications, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 10, 135–147 (2002)

    Article  MATH  Google Scholar 

  3. Baczyński, M.: On the distributivity of fuzzy implications over continuous and Archimedean triangular conorms. Fuzzy Sets and Systems 161, 1406–1419 (2010)

    Article  MATH  Google Scholar 

  4. Baczyński, M., Jayaram, B.: On the characterizations of (S,N)-implications. Fuzzy Sets and Systems 158, 1713–1727 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Baczyński, M., Jayaram, B.: On the distributivity of fuzzy implications over nilpotent or strict triangular conorms. IEEE Trans. Fuzzy Syst. 17, 590–603 (2009)

    Article  Google Scholar 

  6. Baczyński, M., Drewniak, J.: Monotonic fuzzy implication. In: Szczepaniak, P.S., Lisboa, P.J.G., Kacprzyk, J. (eds.) Fuzzy Systems in Medicine. Studies in Fuzzines and Soft Computing, vol. 41, pp. 90–111. Physica-Verlag, Heidelberg (2000)

    Google Scholar 

  7. Balasubramaniam, J., Rao, C.J.M.: R-implication operators and rule reduction in Mamdani-type fuzzy systems. In: Proc. 6th Joint Conf. Information Sciences, Fuzzy Theory, Technology, Durham, USA, pp. 82–84 (2002)

    Google Scholar 

  8. Balasubramaniam, J., Rao, C.J.M.: On the distributivity of implication operators over T- and S-norms. IEEE Trans. Fuzzy Syst. 12, 194–198 (2004)

    Article  Google Scholar 

  9. Bustince, H., Pagola, M., Barrenechea, E.: Construction of fuzzy indices from fuzzy DI-subsethood measures: Application to the global comparison of images. Inform. Sci. 177, 906–929 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Combs, W.E., Andrews, J.E.: Combinatorial rule explosion eliminated by a fuzzy rule configuration. IEEE Trans. Fuzzy Syst. 6, 1–11 (1998)

    Article  Google Scholar 

  11. Combs, W.E.: Author’s reply. IEEE Trans. Fuzzy Syst. 7, 371, 478–479 (1999)

    Google Scholar 

  12. De Baets, B., Kerre, E., Gupta, M.: The fundamentals of fuzzy mathematical morphology, Part 1: Basic concepts. Int. J. Gen. Syst. 23, 155–171 (1994)

    Article  Google Scholar 

  13. Dick, S., Kandel, A.: Comments on Combinatorial rule explosion eliminated by a fuzzy rule configuration. IEEE Trans. Fuzzy Syst. 7, 475–477 (1999)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  15. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht (2000)

    MATH  Google Scholar 

  16. Mendel, J.M., Liang, Q.: Comments on Combinatorial rule explosion eliminated by a fuzzy rule configuration. IEEE Trans. Fuzzy Syst. 7, 369–371 (1999)

    Article  Google Scholar 

  17. Ruiz-Aguilera, D., Torrens, J.: Distributivity of strong implications over conjunctive and disjunctive uninorms. Kybernetika 42, 319–336 (2005)

    MathSciNet  Google Scholar 

  18. Ruiz-Aguilera, D., Torrens, J.: Distributivity of residual implications over conjunctive and disjunctive uninorms. Fuzzy Sets and Systems 158, 23–37 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Sokhansanj, B.A., Rodrigue, G.H., Fitch, J.P.: Applying URC fuzzy logic to model complex biological systems in the language of biologists. In: 2nd Int. Conf. Systems Biology (ICSB 2001), Pasadena, USA, p. 102 (2001)

    Google Scholar 

  20. Trillas E., Alsina C.: On the law [p ∧ q→r] = [(p →r) ∨ (q→r)] in fuzzy logic. IEEE Trans. Fuzzy Syst. 10, 84–88 (2002)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Silesia, 40-007, Katowice, ul. Bankowa 14, Poland

    Michał Baczyński

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  1. Michał Baczyński
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Editor information

Editors and Affiliations

  1. Department of Artificial Intelligence, Academy of Humanities and Economics, Poland

    Leszek Rutkowski

  2. Academy of Humanities and Economics in Łódź,, ul. Rewolucji 1905 nr 64, Łódź, Poland

    Rafał Scherer

  3. Institute of Automatics,, AGH University of Science and Technology, Al. Mickiewicza 30, PL-30-059, Kraków, Poland

    Ryszard Tadeusiewicz

  4. Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Initiative in Soft Computing (BISC), 94720-1776, Berkeley, CA,  

    Lotfi A. Zadeh

  5. Computational Intelligence Laboratory Department of Electrical and Computer Engineering, University of Louisville, 40292, Louisville, KY

    Jacek M. Zurada

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Baczyński, M. (2010). On the Distributivity of Fuzzy Implications over Continuous Archimedean Triangular Norms. In: Rutkowski, L., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds) Artificial Intelligence and Soft Computing. ICAISC 2010. Lecture Notes in Computer Science(), vol 6113. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13208-7_1

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  • DOI: https://doi.org/10.1007/978-3-642-13208-7_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13207-0

  • Online ISBN: 978-3-642-13208-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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