On the belief structures and reductions of multigranulation spaces with decisions

Multigranulation spaces provide a unified framework for representing multigranulation knowledge. In this paper, we aim to construct belief structures and characterize knowledge reduction in terms of evidence theory for the multigranulation spaces where decision attributes are considered. First, we introduce a type-1 belief structure for a multigranulation space, and prove that a pair of lower and upper pessimistic multigranulation approximations correspond to a pair of belief and plausibility functions under this belief structure. Second, we point out that no belief structure can be induced when using the optimistic multigranulation approximations in general. By adding a special sufficient condition, a second type of belief structure is further explored to study those that cannot be induced. Third, we develop several measurements by using the belief and plausibility functions, and characterize knowledge reduction of multigranulation spaces based on these measurements. In the end, a numerical algorithm for reducing redundant granular structures in multigranulation spaces is designed. An example is used to examine the efficiency of the proposed algorithm.