Abstract
There exist several proposals for extending formal concept analysis (FCA) to fuzzy settings. They focus mainly on mathematical aspects and assume generally a residuated algebra in order to maintain the required algebraic properties for the definition of formal concepts. However, less efforts have been devoted for discussing what are the possible reasons for introducing degrees in the relation linking objects and properties (which defines a formal context in the FCA sense), and thus what are the possible meanings of the degrees and how to handle them in agreement with their intended semantics. The paper investigates three different semantics, namely i) the graduality of the link associating properties to objects, pointing out various interpretations of a fuzzy formal context; ii) the uncertainty pervading this link (in case of binary properties) when only imperfect information is available and represented in the framework of possibility theory; and lastly, iii) the typicality of objects and the importance of definitional properties within a class. Remarkably enough, the uncertainty semantics has been hardly considered in the FCA setting, and the third semantics apparently not. Moreover, we provide an algorithm for building the whole fuzzy concept lattice based on Gödel implication for handling gradual properties in a qualitative manner.
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References
Barbut, M., Monjardet, B.: Ordre et Classification. Algèbre et Combinatoire. Tome 2. Hachette, Paris (1970)
Bělohlávek, R.: Fuzzy Galois connections. Math. Logic Quart 45, 497–504 (1999)
Bělohlávek, R., Vychodil, V.: What is a fuzzy concept lattice. In: Proc. CLA 2005, Olomounc. Czech Republic, pp. 34–45 (2005)
Bělohlávek, R., Vychodil, V.: Graded LinClosure and its role in relational data analysis. In: Yahia, S.B., Nguifo, E.M., Belohlavek, R. (eds.) CLA 2006. LNCS (LNAI), vol. 4923, pp. 139–154. Springer, Heidelberg (2008)
Birkhoff, G.: Théorie et applications des treillis. Annales de l’IHP 11(5), 227–240 (1949)
Bosc, P., Pivert, O.: About yes/no queries against possibilistic databases. Int. J. Intell. Syst. 22(7), 691–721 (2007)
Burmeister, P., Holzer, R.: Treating incomplete knowledge in formal concepts analysis. In: Ganter, B., Stumme, G., Wille, R. (eds.) Formal Concept Analysis. LNCS (LNAI), vol. 3626, pp. 114–126. Springer, Heidelberg (2005)
Burusco, A., Fuentes-González, R.: The study of the L-fuzzy concept lattice. Mathware & Soft Computing 3, 209–218 (1994)
Burusco, A., Fuentes-González, R.: Construction of the L-fuzzy concept lattice. Fuzzy Sets and Systems 97(1), 109–114 (1998)
De Baets, B., Kerre, E.: Fuzzy relational compositions. Fuzzy Sets Syst. 60, 109–120 (1993)
Djouadi, Y., Dubois, D., Prade, H.: On the possible meanings of degrees when making formal concept analysis fuzzy. In: Proc. EUROFUSE 2009, pp. 253–258 (2009)
Djouadi, Y., Prade, H.: Interval-valued fuzzy formal concept analysis. In: Rauch, J., Raś, Z.W., Berka, P., Elomaa, T. (eds.) Foundations of Intelligent Systems. LNCS(LNAI), vol. 5722, pp. 592–601. Springer, Heidelberg (2009)
Djouadi, Y., Prade, H.: Interval-valued fuzzy Galois connections: Algebraic requirements and concept lattice construction. Fundamenta Informaticae 99(2), 169–186 (2010)
Dubois, D., Prade, H.: A theorem on implication functions defined from triangular norms. Stochastica 8, 267–279 (1984)
Dubois, D., Prade, H.: Possibility Theory. Plenum Press, New York (1988)
Dubois, D., Prade, H.: Gradual inference rules in approximate reasoning. Information Sciences 61, 103–122 (1992)
Dubois, D., Prade, H.: Upper and lower images of a fuzzy set induced by a fuzzy relation - Applications to fuzzy inference and diagnosis. Information Sciences 64, 203–232 (1992)
Fan, S.Q., Zhang, W.X., Ma, J.M.: Fuzzy inference based on fuzzy concept lattice. Fuzzy Sets and Systems 157(24), 3177–3187 (2006)
Fodor, J.C.: Nilpotent minimum and related connectives for fuzzy logic. In: Fourth IEEE Int. Conf. Conference on Fuzzy Systems, Yokohama, Japan, pp. 2077–2082 (1995)
Georgescu, G., Popescu, A.: Non-dual fuzzy connections. Arch. Math. Log. 43(8), 1009–1039 (2004)
Guénoche, A.: Construction du treillis de Galois d’une relation binaire. Mathématiques et Sciences Humaines 109, 41–53 (1990)
Kerre, E.E.: An overview of fuzzy relational calculus and its applications. In: Torra, V., Narukawa, Y., Yoshida, Y. (eds.) MDAI 2007. LNCS (LNAI), vol. 4617, pp. 1–13. Springer, Heidelberg (2007)
Kerre, E.E., Nachtegael, M.: Fuzzy relational calculus and its application to image processing. In: Gesù, V.D., Pal, S.K., Petrosino, A. (eds.) Proceedings of Fuzzy Logic and Applications, 8th International Workshop, WILF 2009. LNCS, vol. 5571, pp. 179–188. Springer, Heidelberg (2009)
Medina, J., Ojeda-Aciego, M., Ruiz-Calviño, J.: Formal concept analysis via multi-adjoint concept lattices. Fuzzy Sets and Systems 160(2), 130–144 (2009)
Messai, N., Devignes, M., Napoli, A., Tabbone, M.: Many-valued concept lattices for conceptual clustering and information retrieval. In: Proc. 18th Europ. Conf. on Artif. Intellig., Patras, pp. 722–727 (2008)
Pollandt, S.: Fuzzy Begriffe. Springer, Heidelberg (1997)
Ward, M., Dilworth, R.P.: Residuated lattices. Trans. AMS 45, 335–354 (1939)
Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered Sets, pp. 445–470. Reidel, Dordrecht (1982)
Wolff, K.E.: Concepts in fuzzy scaling theory: order and granularity. Fuzzy Sets and Systems 132, 63–75 (2002)
Xie, C., Yi, L., Du, Y.: An algorithm for fuzzy concept lattices building with application to social navigation. In: ISKE 2007, International Conference on Intelligent Systems and Knowledge Engineering, China (2007)
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Djouadi, Y., Dubois, D., Prade, H. (2010). Graduality, Uncertainty and Typicality in Formal Concept Analysis. In: Cornelis, C., Deschrijver, G., Nachtegael, M., Schockaert, S., Shi, Y. (eds) 35 Years of Fuzzy Set Theory. Studies in Fuzziness and Soft Computing, vol 261. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16629-7_7
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