Artificial Intelligence Perspectives on Granular Computing

The theory of rough sets, proposed by Pawlak (1982, 1991, 1998; Pawlak and Skowron, 2007a, 2007b), offers systematic approaches for analyzing information tables in terms of indiscernibility, granulation and approximations. It has led to many useful and effective approaches for data analysis and machine learning. The notions of indiscernability and definability can be defined by a decision logic language in an information table (Yao, 2007a). A set of objects is definable if we can find a formula defining the set, namely, the set consists exactly of these objects satisfying the formula. Two objects are indiscernible if we cannot differentiate them by any formula, namely, they satisfy exactly the same set of formulas. This formulation is consistent with Leibniz Law that identify is defined by means of indiscernibility (Krause and Coelho, 2005). It is also in the same line of thought as the theory of granularity suggested by Hobbs (1985) and the theory of abstraction studied by Giunchglia and Walsh (1992). The indiscernibility of object is formally characterized by an equivalence relation, and equivalence classes are granules of the induced partition. Such a granulation of universe leads to approximations. Since not every subset of the universe can be expressed as a union of some equivalence classes, one needs to approximate it from below and above by a pair of sets that are unions of equivalence classes. One of the distinct features of rough set theory is that the iniscernibility is a relative notion defined with respect to a particular subset of attributes. When different subsets of attributes are used, one obtains different partitions. The relationships between these partitions are used to study the dependency of attribute sets. The philosophy of rough set analysis is general enough to be applicable to many problem-solving tasks. It, in fact, has a major influence on an emerging field of study known as granular computing (Inuiguchi, Hirano and Tsumoto,