Attribute reduction of a decision information system (DIS) using multi-granulation rough sets is one of the important applications of granular computing. Constructing discernibility matrices by rough sets to get attribute reducts of a DIS is an important reduction method. By analyzing the commonalities between the multi-granulation reduction structure of decision multi-granulation spaces and that of incomplete DISs based on discernibility tool, this paper explored a general model for the multi-granulation reduction of DISs by the discernibility technique. First, the definition of the generalized neighborhood decision information system (GNDIS) was presented. Second, knowledge reduction of GNDISs by multi-granulation rough sets was discussed, and discernibility matrices and discernibility functions were constructed to characterize multi-granulation reduction structures of GNDISs. Third, the multi-granulation reduction structures of decision multi-granulation spaces and incomplete DISs were characterized by the reduction theory of GNDISs based on discernibility. Then, the multi-granulation reduction of GNDISs by the discernibility tool provided a theoretical foundation for designing algorithms of multi-granulation reduction of DISs.

The multigranulation rough set model is an important rough set model that approximates the target concept using a multigranularity structure. The multigranulation reductions of the generalized neighborhood decision information systems (GNDISs) based on multigranularity rough sets are general models for the multigranulation reductions of decision information systems (DISs) with no missing decision attribute values. In practical applications, missing labels exist in many datasets. Unfortunately, the theory of multigranulation reduction of GNDISs is not suitable for attribute reduction of partially labeled data. For this reason, the concept of partially labeled generalized neighborhood decision information systems (p-GNDISs) is proposed in this paper, and pessimistic multigranulation reduction of p-GNDISs is discussed. Moreover, the related family-based approach is provided for getting all the partially labeled, pessimistic reducts (PLP-reducts) of a p-GNDIS. Meanwhile, the matrix operations of the generalized neighborhood pessimistic lower approximation and the pessimistic multigranulation positive region on a p-GNDIS are presented. Relationships between the Boolean matrix of the related family and the matrices for computing pessimistic lower approximations are explored. Then, a logic algorithm to get a PLP-reduct of a p-GNDIS by matrix operations is presented. The pessimistic multigranulation reduction of p-GNDISs by related families method and matrix operations provides a theoretical foundation for designing algorithms of multigranulation reduction for partially labeled data.