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Putting Rough Sets and Fuzzy Sets Together

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Book cover Intelligent Decision Support

Part of the book series: Theory and Decision Library ((TDLD,volume 11))

Abstract

In this paper we argue that fuzzy sets and rough sets aim to different purposes and that it is more natural to try to combine the two models of uncertainty (vagueness for fuzzy sets and coarseness for rough sets) in order to get a more accurate account of imperfect information. First, the upper and lower approximations of a fuzzy set are defined, when the universe of discourse of a fuzzy sets is coarsened by means of an equivalence relation. We then come close to Caianiello’s C-calculus. Shafer’s concept of coarsened belief functions also belongs to the same line of thought and is reviewed here. Another idea is to turn the equivalence relation relation into a fuzzy similarity relation, for a more expressive modeling of coarseness. New results on the representation of similarity relations by means of a fuzzy partition of fuzzy clusters of more or less indiscernible points are surveyed. The properties of upper and lower approximations of fuzzy sets by similarity relations are thoroughly studied. Lastly the potential usefulness of the fuzzy rough set notions for logical inference in the presence of both fuzzy predicates and graded indiscernibility is indicated. Especially fuzzy rough sets may provide a nice semantic background for modal logic involving fuzzy modalities and/or fuzzy sentences.

This paper draws from and continues a previous article by the authors, entitled “Rough fuzzy sets and fuzzy rough sets”, that appeared in the Int.J.of General Systems in 1990.

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Authors and Affiliations

  1. Institut de Recherche en Informatique de Toulouse — C.N.R.S., Université Paul Sabatier, 31062, Toulouse, France

    Didier Dubois & Henri Prade

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  1. Didier Dubois
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  2. Henri Prade
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Editors and Affiliations

  1. Institute of Computing Science, Technical University of Poznań, Poland

    Roman Słowiński

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Dubois, D., Prade, H. (1992). Putting Rough Sets and Fuzzy Sets Together. In: Słowiński, R. (eds) Intelligent Decision Support. Theory and Decision Library, vol 11. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7975-9_14

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  • DOI: https://doi.org/10.1007/978-94-015-7975-9_14

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