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The neighborhood system based rough set is a generalization of Pawlak's rough set model since the former uses the neighborhood system instead of the partition for constructing target approximation. In this paper, the neighborhood system based rough set approach is employed to deal with the incomplete information system. By the coverings induced by the maximal consistent blocks and the support sets of the descriptors, respectively, two neighborhood systems based rough sets are explored. By comparing with the original maximal consistent block and descriptor based rough sets, the neighborhood system based rough sets hold the same lower approximations and the smaller upper approximations. Furthermore, the concept of attribute reduction is introduced into the neighborhood systems and the corresponding rough sets. The judgement theorems and discernibility functions to compute reducts are also presented. Some numerical examples are employed to substantiate the conceptual arguments.

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A simple multigranulation rough set approach is to approximate the target through a family of binary relations. Optimistic and pessimistic multigranulation rough sets are two typical examples of such approach. However, these two multigranulation rough sets do not take frequencies of occurrences of containments or intersections into account. To solve such problem, by the motivation of the multiset, the model of the multiple multigranulation rough set is proposed, in which both lower and upper approximations are multisets. Such two multisets are useful when counting frequencies of occurrences such that objects belong to lower or upper approximations with a family of binary relations. Furthermore, not only the concept of approximate distribution reduct is introduced into multiple multigranulation rough set, but also a heuristic algorithm is presented for computing reduct. Finally, multiple multigranulation rough set approach is tested on eight UCI (University of California-Irvine) data sets. Experimental results show: 1. the approximate quality based on multiple multigranulation rough set is between approximate qualities based on optimistic and pessimistic multigranulation rough sets; 2. by comparing with optimistic and pessimistic multigranulation rough sets, multiple multigranulation rough set needs more attributes to form a reduct.

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