Granular Computing: Topological and Categorical Aspects of Near and Rough Set Approaches to Granulation of Knowledge

Abstract

Knowledge (information) granulation is one of the fundamental concepts of information processing leading to a new discipline called granular computing. One of the basic problems addressed by granular computing is the higher order granulation: how to collect basic information granules into a new granule. In the paper we address this problem by purely mathematical means using two well-established methodologies of information processing: near set theory and rough set theory. We start with the simple fact that the theory of near sets and the theory of rough sets share a common metric root. Since a probe function and an equivalence relation can be regarded as a pseudometric on U, in actual fact the underlying structure of both theories is a family of pseudometrics. The same starting point one can find in metric topology: an arbitrary family of pseudometrics is called a pregauge structure and when this family additionally separates all points, it is called a gauge structure. Pregauge structures characterise all completely regular spaces, whereas gauge structures correspond to all Hausdorff completely regular spaces (often called gauge spaces). In consequence, a perceptual system and an information system can be regarded as both pregauge structures and as topological completely regular spaces. A perceptual system or an approximation space does usually not separate all points and thus does not form a gauge space. Therefore in the paper we introduce the concept of a separating completion of a pregauge structure. This notion allows us to build a non-trivial topology on the set of perceptual elementary granules of a perceptual system (or an information system); in other words, a separating completion induces the higher order granulation. The completion requirement induces also a topology on a set of objects, which is locally homeomorphic to the the topology on basic information granules. Apart from topological results, we shall also discuss both topologies using category theory. A perceptual system may be actually enriched to an abelian group or a vector space, while unchanging the original granulation. It also gives rise to a quite rich sheaf of all real-valued functions preserving the basic granulation. Summing up, our aim is to build rich mathematical structures which do not change basic granulation and may be used to solve the problem of higher order granules.