The Coarsening And Refining Of Belief Structures

In this paper,a refining mapping on the collection of a frame of discernment's subsets and its inner and outer coarsening mappings on the collection of the refined frame of discernment's subsets are generalized to this two aspects,i.e. a fuzzy refining mapping on the collection of the frame of discernment's fuzzy subsets and its inner and outer fuzzy coarsening mappings on the collection of the refined frame of discernment's fuzzy subsets. A series of properties about them are discussed. Then the coarsened belief structure and the refined belief structure are defined,so the important results are obtained. The relationship between the coarsening or refining operations and the combining operations of a belief structure are investigated. It is proved that the refining operation and two kinds of combining operations are both commutative with each other;while the coarsening operation is only commutative with the disjunctive sum operation but not with the conjunctive sum operation. We pointed out the reason why the coarsening operation and the conjunctive sum operation cannot be commutative.In addition,weak inclusion and strong inclusion are introduced for discussing the relationship between strong inner or outer approximations and the coarsening or refining of belief structures. It is proved that the coarsening and refining operations can both preserve the strong inclusion of belief structures. As a result,the strong inner and outer approximations of the coarsened or refined belief structures can become the coarsening or refining of the strong inner and outer approximations of the belief structures.