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On Linear and Quadratic Constructions of Fuzzy Implication Functions

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Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations (IPMU 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 853))

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Abstract

In this paper a new construction method of fuzzy implication functions from a given one, based on ternary polynomial functions is presented. It is proved that the case of linear polynomial functions leads only to trivial solutions and thus the quadratic case is studied in depth. It is shown that the quadratic method allows many different possibilities depending on the usual properties of fuzzy implications functions that we want to preserve. Specifically, there are infinitely many quadratic functions that transform fuzzy implication functions satisfying properties like the neutrality principle, the identity principle, or the law of contraposition with respect to the classical negation, into new fuzzy implication functions satisfying them.

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References

  1. Aguiló, I., Suñer, J., Torrens, J.: New types of contrapositivisation of fuzzy implications with respect to fuzzy negations. Inf. Sci. 322, 223–236 (2015)

    Article  MathSciNet  Google Scholar 

  2. Aguiló, I., Suñer, J., Torrens, J.: A new look on fuzzy implication functions: FNI-implications. In: Carvalho, J.P., Lesot, M.-J., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R.R. (eds.) IPMU 2016. CCIS, vol. 610, pp. 375–386. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40596-4_32

    Chapter  Google Scholar 

  3. Baczyński, M., Beliakov, G., Bustince, H., Pradera, A.: Advances in Fuzzy Implication Functions. Studies in Fuzziness and Soft Computing Series, vol. 300. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-35677-3

    Book  MATH  Google Scholar 

  4. Baczyński, M., Jayaram, B.: Fuzzy Implications. Studies in Fuzziness and Soft Computing, vol. 231. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-69082-5

    Book  MATH  Google Scholar 

  5. Baczyński, M., Jayaram, B., Massanet, S., Torrens, J.: Fuzzy implications: past, present, and future. In: Kacprzyk, J., Pedrycz, W. (eds.) Springer Handbook of Computational Intelligence, pp. 183–202. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-43505-2_12

    Chapter  Google Scholar 

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

    Book  Google Scholar 

  7. Kolesárová, A., Mesiar, R.: On linear and quadratic constructions of aggregations functions. Fuzzy Sets Syst. 268, 1–14 (2015)

    Article  MathSciNet  Google Scholar 

  8. Mas, M., Monserrat, M., Torrens, J., Trillas, E.: A survey on fuzzy implication functions. IEEE Trans. Fuzzy Syst. 15(6), 1107–1121 (2007)

    Article  Google Scholar 

  9. Massanet, S., Torrens, J.: On some properties of threshold generated implications. Fuzzy Sets Syst. 205, 30–49 (2012)

    Article  MathSciNet  Google Scholar 

  10. Massanet, S., Torrens, J.: Threshold generation method of construction of a new implication from two given ones. Fuzzy Sets Syst. 205, 50–75 (2012)

    Article  MathSciNet  Google Scholar 

  11. Massanet, S., Torrens, J.: On the vertical threshold generation method of fuzzy implication and its properties. Fuzzy Sets Syst. 226, 32–52 (2013)

    Article  MathSciNet  Google Scholar 

  12. Massanet, S., Torrens, J.: An overview of construction methods of fuzzy implications. In: Baczyński, M., Beliakov, G., Bustince, H., Pradera, A. (eds.) Advances in Fuzzy Implication Functions. STUDFUZZ, vol. 300, pp. 1–30. Springer, Berlin Heidelberg (2013). https://doi.org/10.1007/978-3-642-35677-3_1

    Chapter  MATH  Google Scholar 

  13. Shi, Y., Van Gasse, B., Ruan, D., Kerre, E.: On a new class of implications in fuzzy logic. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. CCIS, vol. 80, pp. 525–534. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14055-6_55

    Chapter  MATH  Google Scholar 

  14. Trillas, E., Mas, M., Monserrat, M., Torrens, J.: On the representation of fuzzy rules. Int. J. Approx. Reason. 48(2), 583–597 (2008)

    Article  MathSciNet  Google Scholar 

  15. Vemuri, N.R., Jayaram, B.: The -composition of fuzzy implications: closureswith respect to properties, powers and families. Fuzzy Sets Syst. 275, 58–87 (2015)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

This paper has been partially supported by the Spanish Grant TIN2016-75404-P, AEI/FEDER, UE.

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Authors and Affiliations

  1. Soft Computing, Image Processing and Aggregation (SCOPIA) Research Group, Department of Mathematics and Computer Science, University of the Balearic Islands, 07122, Palma, Spain

    Sebastia Massanet, Juan Vicente Riera & Joan Torrens

  2. Balearic Islands Health Research Institute (IdISBa), 07010, Palma, Spain

    Sebastia Massanet, Juan Vicente Riera & Joan Torrens

Authors
  1. Sebastia Massanet
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  2. Juan Vicente Riera
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  3. Joan Torrens
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Corresponding author

Correspondence to Sebastia Massanet .

Editor information

Editors and Affiliations

  1. Universidad de Cádiz, Cádiz, Cadiz, Spain

    Jesús Medina

  2. Universidad de Málaga, Málaga, Málaga, Spain

    Manuel Ojeda-Aciego

  3. Universidad de Granada, Granada, Spain

    José Luis Verdegay

  4. Universidad de Granada, Granada, Spain

    David A. Pelta

  5. Universidad de Málaga, Málaga, Málaga, Spain

    Inma P. Cabrera

  6. LIP6, Université Pierre et Marie Curie, CNRS, Paris, France

    Bernadette Bouchon-Meunier

  7. Iona College, New Rochelle, New York, USA

    Ronald R. Yager

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Massanet, S., Riera, J.V., Torrens, J. (2018). On Linear and Quadratic Constructions of Fuzzy Implication Functions. In: Medina, J., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations. IPMU 2018. Communications in Computer and Information Science, vol 853. Springer, Cham. https://doi.org/10.1007/978-3-319-91473-2_53

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  • DOI: https://doi.org/10.1007/978-3-319-91473-2_53

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-91472-5

  • Online ISBN: 978-3-319-91473-2

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