Nullity-based matroid of rough sets and its application to attribute reduction

Rough sets were proposed to deal with vagueness and incompleteness of knowledge in information systems. In this field, there are many optimization issues such as attribute reduction. Matroids generalized from matrices have been widely used in many fields, particularly greedy algorithm design, which plays an important role in attribute reduction. Therefore, it is meaningful to combine matroids with rough sets to solve the optimization problems. In this paper, we construct a type of matroid of rough sets based on the concept of nullity and apply it to attribute reduction. First, we propose a nullity operator for rough sets to induce a matroid and then we present a specific type of matroid called a nullity-based matroid. Second, given the relationship between nullities and matrices, we present two types of matrices to characterize this type of matroid and its nullity operator. Third, the dual of this type of matroid is induced by the second type of matrix. Finally, we apply the obtained matroids to attribute reduction issues in information systems. In summary, this paper provides a new approach to studying rough sets.