Decision-relative discernibility matrices in the sense of entropies

In rough set theory, attribute reduction is a basic issue, which aims to hold the discernibility of the attribute set. To obtain all of the reducts of an information system or a decision table, researchers have introduced many discernibility matrices based reduction methods. However, the reducts in the sense of positive region can only be obtained by using the existing discernibility matrices. In this paper, we introduce two discernibility matrices in the sense of entropies (Shannon's entropy and complement entropy). By means of the two discernibility matrices, we can achieve all of the reducts in the sense of Shannon's entropy and all of the reducts in the sense of complement entropy, respectively. Furthermore, we discover the relationships among the reducts in the sense of preserving positive region, Shannon's entropy and complement entorpy. The experimental studies show that by the proposed decision-relative discernibility matrices based reduction methods, all the reducts of a decision table in sense of entropies can be obtained.