Granular Structures and Approximations in Rough Sets and Knowledge Spaces

Granular computing is an emerging field of study focusing on structured thinking, structured problem solving and structured information processing with multiple levels of granularity [1, 2, 7, 8, 13, 17, 19, 20, 21, 22, 24, 25, 29]. Many theories may be interpreted in terms of granular computing. The main objective of this chapter is to examine the granular structures and approximations used in rough set analysis [10, 11] and knowledge spaces [3, 4]. A primitive notion of granular computing is that of granules. Granules may be considered as parts of a whole. A granule may be understood as a unit that we use for describing and representing a problem or a focal point of our attention at a specific point of time. Granules can be organized based on their inherent properties and interrelationships. The results are a multilevel granular structure. Each level is populated by granules of the similar size or the similar nature. Depending on a particular context, levels of granularity may be interpreted as levels of abstraction, levels of details, levels of processing, levels of understanding, levels of interpretation, levels of control, and many more. An ordering of levels based on granularity provides a hierarchical granular structure. The formation of granular structures is based on a vertical separation of levels and a horizontal separation of granules in each level. It explores the property of loose coupling and nearly-decomposability [14] and searches for a good approximation [12]. Typically, elements in the same granules interact more than elements in different granules. Granules in the same level are relatively independent and granules in two adjacent levels are closely related.