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Development of Near Sets Within the Framework of Axiomatic Fuzzy Sets

Wang G., Ren Y., Liu X.

Fundamenta Informaticae

2012

Vol. 118, nr 3

291-304

Near sets result from a generalization of rough sets, which introduced by Peters in 2006, and later formally defined in 2007. Near set theory provides a new framework for representation of objects characterized by the features that describe them. AFS (Axiomatic Fuzzy Set) theory was proposed by Liu (1998), which is a semantic methodology relating to the fuzzy theory. In this paper, a new version of near sets based on AFS theory is established, in which every object has an AFS fuzzy description with definitely semantics. The proposed approach to assessing the nearness (closeness) of objects is not defined directly using a distance metric, but depend on similarity of their fuzzy descriptions. It is also a natural linguistic description that is similar to humans perception. Moreover, an approach to set approximation based on the union of families of objects with similar fuzzy descriptions is given. The near sets based on AFS theory can be viewed as a new development of near sets within the fuzzy context.

The transportation distance for fuzzy descriptions of measurements

Allevard T., Benoit E., Foulloy L.

Metrology and Measurement Systems

2007

Vol. 14, nr 1

25-37

Fuzzy nominal scales were introduced in order to propose a formalism to the representation of empirical quantities by fuzzy subsets of words. This scale proposes a similarity relation and an associated bounded distance that can be used to perform signal processing on fuzzy subsets of words. Due to the limits of this last distance, we studied distances associated to this formalism and proposed a new distance operator named transportation distance. This paper presents the results of these studies.