The Study Of Mathematical Properties On Several Types Of Generalized Rough Set Models

Rough set theory is a mathematical tool to describe incompleteness and uncertainty, and it can effectively analyze imprecise, inconsistent and incomplete information. Pawlak rough set is based on the classification of the universe determined by the equivalence relation.However, in many practical problems, the data cannot be described by the equivalence relation, and so, the Pawlak rough set cannot be applied to deal with such data. In other word,the requirement of the equivalence relation seems to be a very restrictive condition that may limit the application domain of rough sets. To solve this issue, many extended rough set models have been proposed by relaxing the equivalence relation to more general binary relations. In this paper, we propose some new generalized rough set models, study their mathematical properties, and compare the properties of these models with those of Pawlak rough set model. The paper is organized as follows.In Chapter 1, the Pawlak rough set model and some generalized rough set models are introduced. The framework and innovations of this paper are also given in this chapter.In Chapter 2, some basic concepts and mathematical properties of Pawlak rough set model and some generalized rough set models are reviewed.In Chapter 3, one kind of knowledge granule,which is the intersection of successor neighborhoods of an object, is introduced,and the generalized rough set model based on this kind of knowledge granule and their properties are reviewed. Then three new kinds of generalized rough set model based on this kind of knowledge granule are proposed, and the mathematical properties of the upper and lower approximations are given.In Chapter 4, three types of generalized rough set models based on a family of binary relations are studied. Firstly, we define the lower/upper approximation with respect to the family of binary relations as the intersection/union of all the singleton lower/upper approximations determined by each binary relation. In the second definition, the lower approximation is defined by the intersection of all the subset lower approximations determined by each binary relation, and the upper approximation is given by the duality. In the third definition, the upper approximation is defined by the union of all the subset upper approximations determined by each binary relation, and the lower approximation is given bythe duaily. The properties of these three rough set models are also discussed.In Chapter 5, a new definition of lower and upper approximations is given in the neighborhood system, and the properties of the approximations are discussed.In Chapter 6, the work of this paper is summarized, and the questions need to study in the future is proposed.