Fuzzy Sets, Near Sets, and Rough Sets. Sets in the Computational Intelligence Spectrum

This keynote talk considers how one might utilize fuzzy sets, near sets, and rough sets, taken separately or taken together in hybridizations in solving a variety of problems commonly faced in science and engineering. These technologies offer set theoretic approaches to solving problems such as classifying sensor output, image retrieval and image correspondence. Fuzzy sets result from the introduction of a membership function that generalizes the traditional characteristic function. The notion of a fuzzy set was introduced by L. Zadeh in 1965. Fifteen years later, rough sets were introduced by Z. Pawlak in 1981. A set is considered rough the boundary between its lower and upper approximation is non-empty. Of the three forms of sets, a near set is newest, introduced in 2007 by J.F. Peters in a perception-based approach to the study of perceptual objects. Perceptual systems provide stepping stones leading to nearness relations and properties of near sets. This work has been motivated by an interest in finding a solution to the problem of discovering perceptual granules that are, in some sense, near each other. Near set theory provides a description-based approach to observing, comparing and classifying perceptual granules. Near sets result from the introduction of a description-based approach to perceptual objects and a generalization of the traditional rough set approach to granulation that is independent of the notion of the boundary of a set approximation. Near set theory has strength by virtue of the strength it gains from rough set theory, starting with a L2 norm-based perceptual indiscernibility relation, a new extension of the traditional indiscernibility equivalence relation. This keynote talk highlights a context for three forms of sets that are now part of the computational intelligence spectrum of tools useful in pattern recognition. By way of introduction to near sets, we consider various perceptual nearness relations that define partitions of sets of perceptual objects that are near each other. Every perceptual granule is represented by a set of perceptual objects that have their origin in the physical world. Objects that have the same appearance are considered perceptually near each other, i.e., objects with matching descriptions. Pixels, pixel windows, and segmentations of digital images are given by way of illustration of sample near sets. The contribution of this paper is an overview of the links between fuzzy sets, near sets and rough sets as well as the relation between these sets and the original notion of a set introduced by Cantor in 1883.