Matroidal And Topological Approaches To Rough Sets

With the development of science and technology, in real life, there exists different degree of uncertainty, fuzzy data. Rough set theory, a new mathematical tool of dealing with imprecise, inconsistent, incomplete knowledge, uses its known knowledge base to characterize the object set through a pair of upper and lower approximation operators. Rough set theory provides theoretical guidance and support for the implementation of the data mining technology such as data processing, attribute value reduction and rule processing. It has been successfully applied to knowledge and data mining, pattern recognition and analysis,artificial intelligence, fault analysis, machine learning, environmental engineering and other fields.However, because rough set theory is in development, the limitation of its application in real life is obvious. For example, the classical rough sets, which are based on partition, have been restricted on application and theory. To solve these problems, scholars have made many extended research on rough sets from different viewpoints such as the investigation of rough sets combined with other theory, rough set model dilatation etc. As many problems of rough sets such as attribute reduction, rule extraction, are almost NP-hard problems which can not be solved by rough sets simply. The combinations of rough sets and other theories such as topology, fuzzy sets, lattices, matroids, have attracted widely attentions of scholars. Greedy algorithm is an important tool for solving NP-hard problems. In order to establish some mathematical structures for these problems of rough sets, matroids provide well-established platforms for these greedy algorithms, and topology and lattice can give full play their own advantages of theory to provide theoretical support for the construction of these mathematical structures. Therefore, in order to solve the optimal solution of attribute reduction in rough sets,the combinations of rough sets and matroids, topology, lattice, are significance.This paper established matroidal structures and topological structures of rough sets,respectively. Firstly, based on matroids, we establish matroidal structure of classical rough sets and matroidal structure of covering-based rough sets, respectively. Some characteristics of these matroidal structures of rough sets are depicted from different viewpoints such as matrix characterization. Moreover, we investigate the characteristics of these closed-set lattices induced by the matroidal structures of covering-based rough sets. Secondly, based on the topology, a topological structure of covering-based rough sets is established, and some properties of the topological structure and its application are investigated with dependence space. The main work of this paper is as follows:(1) Matroidal structure of classical rough sets. Through classical rough sets, we define a family of sets by the upper approximation operator and prove it to satisfy spanning set axiom.So a spanning matroid, namely the matroid structure of classical rough sets, can be obtained in this way. Moreover, some characteristics of the matroidal structure, such as bases,independent sets, hyperplanes are investigated with rough set and matrix approaches,respectively.(2) Conditions for six types of covering-based upper approximation operators to be closure operators of matroids. Through comparing the characterizations of six types of covering-based upper approximation operators with the characterizations of closure operators of matroids, we discuss the condition under which six types of covering-based upper approximation operators form closure operators of matroids, and get some important results.For example, the necessary and sufficient condition for the second type of covering-based upper approximation operator to satisfy idempotence is obtained. Moreover, by using the characterizations of unary coverings, close friends, neighborhoods, indiscernible neighborhoods and reduction, we present necessary and sufficient conditions for six types of covering-based upper approximation operators to be closure operators of matroids.(3) Matroidal structure of covering-based rough sets. Based on close friends,neighborhoods, indiscernible neighborhoods of covering-based rough sets, three families ofsets are constructed and proved to satisfy independent set axiom of matroids, and some characteristics of these matroidal structures, such as rank function, closure, close set, are investigated. Moreover, we investigate some characteristics of these types of closed-set lattices induced by these three types of matroids and the relationships among these closed-set lattices. Especially we prove that these three types of matroids induced by covering-based rough sets are all modular matroids.(4) Topological structure of covering-based rough sets and dependence space. By neighborhood systems of covering-based rough sets, a topological structure of covering-based rough sets is constructed, and some characteristics of the topological structure, such as interior operator, closure operator, close set, are presented. Based on these results, by using topological bases to construct a consistent relation, a dependence space of topology is induced.Some characteristics of the dependence space are investigated. Moreover, we apply the obtained results of the space to the attribute reduction in incomplete information systems. And a discernibility matrix is defined for the attribute reduction in incomplete information systems.