Granular Computing: Fuzzy Logic and Rough Sets

Abstract

The primary goal of granular computing is to elevate the lower level data processing to a high level knowledge processing. Such an elevation is achieved by granulating the data space into a concept space. Each granule represents certain primitive concept, and the granulation as a whole represents a knowledge. In this paper, such an intuitive idea is formalized into a mathematical theory: Zadeh’s informal words are taken literally as a formal definition of granulation. Such a mathematical notion is a mild generalization of the “old” notion of crisp/fuzzy neighborhood systems of (pre-)topological spaces. A crisp/fuzzy neighborhood is a granule and is assigned a meaningful name to represent certain primitive concept or to summarize the information content. The set of all linear combinations of these names, called formal words, mathematically forms a vector space over real numbers. Each vector is intuitively an advanced concept represented by some “weighted averaged” of primitive concepts. In terms of these concepts, the universe can be represented by a formal word table; this is one form of Zadeh’s veristic constraints. Such a representation is useful; fuzzy logic designs can be formulated as series of table transformations. So table processing techniques of rough set theory may be used to simplify these tables and their transformations. Therefore the complexity of large scaled fuzzy systems may be reduced; details will be reported in future papers.