Theory And Method Of Numerical Characterizations For Covering Rough Set Theory

The rough set theory is a new mathematical tool to deal with uncertainty and vagueness of decision system and it has been applied successfully in many fields, such as artificial intelligence, data mining, pattern recognition and information processing fields so on. Rough set theory is used to identify the reduct of all condition attributes of the decision system. Covering rough sets are improvements of traditional rough sets by considering covers of universe instead of partitions. In this paper, numerical characterizations for covering rough sets are developed in terms of evidence theory. Firstly, belief and plausibility functions are proposed to characterize the lower and upper approximations in covering rough sets; then attribute reductions of covering information systems and decision systems are characterized by these functions. The concepts of significance and relative significance of coverings are also developed to design algorithms to find reducts. With these discussions, the connection between covering rough sets and evidence theory are set up and a basic framework of numerical characterizations of covering rough sets is presented. Using discernibility matrix theory can calculate all attribute reducts for covering rough sets, but computational complexity is NP-hard. Not all elements of the discemibility matrix are necessary, and we only need to find out its minimal elements. Further, we observe that every element in the discemibility matrix is corresponding to a sample pair at least. Therefore, we just need one of these samples which are corresponding to each minimal element. In this paper, taking consistent covering decision system for example, firstly we get the concept of relative discemability relation based on the covering rough sets, then we use this relative discernability relation to get the method of computing minimal elements of the discernibility matrix of coordinate covering decision system.