Rough Sets And Matroids In Information Systems

Binary relation,an important structure in mathematics,has been the basis of some fields.As one of three basic theories of granular computing,classical rough set theory provides a useful tool for dealing with the uncertainty,vagueness and granularity in information systems.Matroid theory borrows extensively from liner algebra and graph theory,which has abundant and a perfect system.They have been widely use in many fields.In this paper,we constructed a matroidal structure induced by binary relation,discussed the attribute reduction in different-attribute information system and proposed a new covering-based rough set model in set-valued information system.Firstly,by imitating the upper approximation number,a relational approximation number is defined via binary relation.Then the relational approximation number is proved to satisfy submodular for all subsets of a universe.Meanwhile,a method is proposed for calculating relational approximation number of a set by counting the cards of predecessor neighborhoods of the elements in the set.In addition,by introducing a concept of a family of multisets,we give a sufficient and necessary condition under which the upper approximation number of a set is equal to its relational approximation number.Then this paper constructs a matroidal structure in terms of the equality and discusses some fundamental properties of the structure.In addition,there are both qualitative and quantitative attributes in many real-world problems.We discuss the attribute reduction of all attributes after obtaining the attribute reduction of all qualitative attributes by binary relation aggregation in different-attribute information system.As we all know,covering-based rough set theory is a generalization of classical rough set theory for handling covering data,which frequently appear in set-valued information systems.In this paper,we propose a covering in terms of attribute sets in a set-valued information system and study its responding three types of covering approximations.Moreover,we show that the covering approximation operators induced by indiscernible neighborhoods and neighborhoods are equal to the approximation operators induced by the tolerance andsimilarity relations,respectively.Meanwhile,the covering approximation operators induced by complementary neighborhoods are equal to the approximation operators induced by the inverse of the similarity relation.By introducing the concept of relational matrices,the relationships of these approximation operators are equivalently represented.Moreover,by introducing a concept of misclassification rate functions,an extended variable precision covering-based rough set model is proposed in this paper.In addition,we define the f-lower and f-upper approximations in terms of neighborhoods in the extended model and study their properties.Particularly,two coverings with the same reductions are proved to generate the same f-lower and f-upper approximations.Finally,we discuss the relationships between the new model and some other variable precision rough set models.