Characterization of attribute reduction of decision system through matroid theory

Attribute reduction is an important issue in rough sets. It has been widely used in pattern recognition, machine learning and data mining. Many mathematical theories, such as fuzzy sets, lattice and matrix, have been applied to study the issue for constructing more effective reduction algorithms. In this paper, the attribute reduction in the decision system is conducted under matroid theory. The matroid approximation space is proposed to build a bridge between the decision system and the matroid. In this way, a single valued decision system is represented by several matroids directly. By using the circuits of these matroids, the concept of equivalence class is formulated equivalently with the matroidal approach and then the lower and the upper matroidal approximations are established. Furthermore, some basic concepts of attribute reduction in rough sets are redescribed in the context of matroid approximation space. Finally, for any two subsets of condition attributes in a consistent decision system, a sufficient condition is proposed to judge whether the positive regions of them with respect to the decision attribute are equal.