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Activating Generalized Fuzzy Implications from Galois Connections

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Enric Trillas: A Passion for Fuzzy Sets

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 322))

Abstract

This paper deals with the relation between fuzzy implications and Galois connections, trying to raise the awareness that the fuzzy implications are indispensable to generalise Formal Concept Analysis. The concrete goal of the paper is to make evident that Galois connections, which are at the heart of some of the generalizations of Formal Concept Analysis, can be interpreted as fuzzy incidents. Thus knowledge processing, discovery, exploration and visualization as well as data mining are new research areas for fuzzy implications as they are areas where Formal Concept Analysis has a niche.

F.J. Valverde-Albacete—was partially supported by EU FP7 project LiMoSINe, (contract 288024).

C. Peláez-Moreno—was partially supported by the Spanish Government-CICYT project 2011-268007/TEC.

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References

  1. Dubois, D., Prade, H.: Fuzzy sets in approximate reasoning, part 1: inference with possibility distributions. Fuzzy Sets Syst. 100(Supplement 1), 73–132 (1999)

    Article  Google Scholar 

  2. Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice-Hall Inc., Upper Saddle River (1995)

    MATH  Google Scholar 

  3. Trillas, E., del Campo, C., Cubillo, S.: When QM-operators are implication functions and conditional fuzzy relations. Int. J. Intell. Syst. 15(7), 647–655 (2000)

    Article  MATH  Google Scholar 

  4. Yager, R.R.: On some new classes of implication operators and their role in approximate reasoning. Inf. Sci. 167(1–4), 193–216 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Alsina, C., Trillas, E., Valverde, L.: On some logical connectives for fuzzy sets theory. J. Math. Anal. Appl. 93, 11–26 (1983)

    Article  MathSciNet  Google Scholar 

  6. Trillas, E., Valverde, L.: On inference in fuzzy logic. In: Proceedings of the 2nd International Fuzzy Systems Association (IFSA) Congress, pp. 294–297 (1987)

    Google Scholar 

  7. Valverde, L., Trillas, E.: On modus ponens in fuzzy logic. In: Proceedings of 15th International Symposium on Multiple-Valued Logic, pp. 294–301 (1985)

    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. Barbut, M., Monjardet, B.: Ordre et classification. Algèbre et combinatoire, tome I. Méthodes mathématiques des sciences de l’Homme. Hachette (1970)

    Google Scholar 

  10. Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  11. Erné, M., Koslowski, J., Melton, A., Strecker, G.: A primer on Galois connections. In: Todd, A. (ed.) Annals of the New York Academy of Sciences, vol. 704, pp. 103–125. Madison (1993)

    Google Scholar 

  12. Carpineto, C., Romano, G.: Concept Data Analysis. Theory and Applications. Wiley, Chichester (2005)

    Book  Google Scholar 

  13. Poelmans, J., Kuznetsov, S.O., Ignatov, D.I., Dedene, G.: Formal concept analysis in knowledge processing: a survey on models and techniques. Expert Syst. Appl. 40(16), 6601–6623 (2013)

    Article  Google Scholar 

  14. Valverde-Albacete, F., Peláez-Moreno: Systems vs. methods: an analysis of the affordances of formal concept analysis for information retrieval. In: Formal Concept Analysis Meets Information Retrieval Workshop, Moscow, Russia, co-located with the 35th European Conference on Information Retrieval (ECIR 2013), pp. 113–126 (2013)

    Google Scholar 

  15. Burusco, A., Fuentes-González, R.: The study of the L-fuzzy concept lattice. Mathw. Soft Comput. 1(3), 209–218 (1994)

    MATH  Google Scholar 

  16. Bělohlávek, R.: Fuzzy Relational Systems. Foundations and Principles. IFSR International Series on Systems Science and Engineering, vol. 20. Kluwer Academic, New York (2002)

    Book  MATH  Google Scholar 

  17. Golan, J.S.: Semirings and Their Applications. Kluwer Academic, Dordrecht (1999)

    Book  MATH  Google Scholar 

  18. Valverde-Albacete, F.J., Peláez-Moreno, C.: Extending conceptualisation modes for generalised formal concept analysis. Inf. Sci. 181, 1888–1909 (2011)

    Article  MATH  Google Scholar 

  19. Belohlavek, R., Vychodil, V.: Attribute implications in a fuzzy setting. Formal Concept Analysis, pp. 45–60. Springer, Berlin (2006)

    Chapter  Google Scholar 

  20. Goguen, J.A.: L-fuzzy sets. J. Math. Anal. Appl 18(1), 145–174 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  21. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gondran, M., Minoux, M.: Dioids and semirings: links to fuzzy set and other applications. Fuzzy Sets Syst. 158, 1273–1294 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Golan, J.S.: Semirings and Affine Equations Over Them: Theory and Applications. Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht (2003)

    Google Scholar 

  24. Gondran, M., Minoux, M.: Graphs, Dioids and Semirings. New Models and Algorithms. Springer, New York (2008)

    MATH  Google Scholar 

  25. Golan, J.S.: Power algebras over Semirings. With Applications in Mathematics and Computer Science. Mathematics and its Applications, vol. 488. Kluwer Academic, Dordrecht (1999)

    Google Scholar 

  26. Cuninghame-Green, R.A., Cechlárová, K.: Residuation in fuzzy algebra and some applications. Fuzzy Sets Syst. 71(2), 227–239 (1995)

    Article  MATH  Google Scholar 

  27. Cuninghame-Green, R.A.: Minimax algebra and applications. Fuzzy Sets Syst. 41(3), 251–257 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  28. Davey, B., Priestley, H.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  29. Moreau, J.J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures et Appl. 49, 109–154 (1970)

    MATH  MathSciNet  Google Scholar 

  30. Cuninghame-Green, R.: Minimax Algebra. Lecture Notes in Economics and Mathematical Systems, vol. 166. Springer, Berlin (1979)

    MATH  Google Scholar 

  31. Erné, M.: Adjunctions and Galois Connections: Origins, History and Development. Mathematics and Its Applications, vol. 565. Kluwer Academic, Dordrecht (2004)

    Google Scholar 

  32. Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society, Providence (1967)

    MATH  Google Scholar 

  33. Gratzer, G.: General Lattice Theory. Birkhauser, Berlin (2003)

    Google Scholar 

  34. Peláez-Moreno, C., García-Moral, A.I., Valverde-Albacete, F.J.: Analyzing phonetic confusions using formal concept analysis. J. Acoust. Soc. Am. 128(3), 1377–1390 (2010)

    Google Scholar 

  35. Peláez-Moreno, C., García-Moral, A.I., Valverde-Albacete, F.J.: Eliciting a hierarchical structure of human consonant perception task errors using formal concept analysis. In: Proceedings of INTERSPEECH, pp. 828–831 (2009)

    Google Scholar 

  36. Peláez-Moreno, C., Valverde-Albacete, F.J.: Detecting features from confusion matrices using generalized formal concept analysis. In: Hybrid Artificial Intelligence Systems, pp. 375–382. Springer, Berlin (2010)

    Google Scholar 

  37. González Calabozo, J., Peláez-Moreno, C., Valverde-Albacete, F.: Gene expression array exploration using \({\cal {K}}\)-formal concept analysis. In: Valtchev, P., Jäschke, R. (eds.) Formal Concept Analysis. Lecture Notes in Computer Science, vol. 6628, pp. 119–134. Springer, Berlin (2011)

    Google Scholar 

  38. González-Calabozo, J.M., Peláez-Moreno, C., Valverde-Albacete, F.J.: WebGeneKFCA: an on-line conceptual analysis tool for genomic expression data. In: International Conference on Concept Lattices and Applications, pp. 345–350 (2012)

    Google Scholar 

  39. Valverde-Albacete, F.J., Peláez-Moreno, C.: Towards a generalisation of formal concept analysis for data mining purposes. In: Missaoui, R., Schmid, J. (eds.) Proceedings of the International Conference on Formal Concept Analisys. LNCS, vol. 3874, pp. 161–176. Springer (2006)

    Google Scholar 

  40. Chou, C.Y., Mei, H.: Analyzing tag-based Mashups with fuzzy FCA. In: IEEE International Symposium on Service-Oriented System Engineering, pp. 257–262 (2008)

    Google Scholar 

  41. Fenza, G., Loia, V., Senatore, S.: Concept mining of semantic web services by means of extended fuzzy formal concept analysis (FFCA). In: IEEE International Conference on Systems, Man and Cybernetics, pp. 240–245 (2008)

    Google Scholar 

  42. Zhou, B., Hui, S.C., Fong, A.C.M.: An effective approach for periodic web personalization. In: IEEE/WIC/ACM International Conference, pp. 284–292 (2006)

    Google Scholar 

  43. Fenza, G., Senatore, S.: Friendly web services selection exploiting fuzzy formal concept analysis. Soft. Comput. 14(8), 811–819 (2010)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Depto. Lenguajes y Sistemas Informáticos, UNED, Madrid, Spain

    Francisco J. Valverde-Albacete

  2. Depto. Teoría de Señal y Comunicaciones, UC3M, Madrid, Spain

    Carmen Peláez-Moreno

  3. Depto. Estadística e Investigación Operativa II, UCM, Madrid, Spain

    Cristina del Campo

Authors
  1. Francisco J. Valverde-Albacete
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  2. Carmen Peláez-Moreno
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  3. Cristina del Campo
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Correspondence to Cristina del Campo .

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Editors and Affiliations

  1. European Centre for Soft Computing, Asturias, Spain

    Luis Magdalena

  2. Depto. de Ciencias de la Computación e Inteligencia Artificial, University of Granada, Granada, Spain

    Jose Luis Verdegay

  3. Spanish Council for Scientific Research (CSIC), Artificial Intelligence Research Institute (IIIA-CSIC), Catalonia, Spain

    Francesc Esteva

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Valverde-Albacete, F.J., Peláez-Moreno, C., del Campo, C. (2015). Activating Generalized Fuzzy Implications from Galois Connections. In: Magdalena, L., Verdegay, J., Esteva, F. (eds) Enric Trillas: A Passion for Fuzzy Sets. Studies in Fuzziness and Soft Computing, vol 322. Springer, Cham. https://doi.org/10.1007/978-3-319-16235-5_15

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  • DOI: https://doi.org/10.1007/978-3-319-16235-5_15

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  • Online ISBN: 978-3-319-16235-5

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