The Covering-based Rough Upper Approximation Operators And Their Applications

Pawlak proposed the rough set theory in 1982.Rough set theory is not only a powerful tool to deal with incomplete information systems,but also an effective tool to deal with uncertain knowledge.It has been used in pattern recognition,data mining,machine learning and other fields.Afterwards,they generalize the classical rough set to the covering rough set,so it has become one of hot research topics.Based on previous researches,this paper investigates characterizations of covering-based upper approximation operators being closure operators by adopting theories of General Topology.On covering approximation space,many researchers have defined many mean-ingful lower and upper approximation operators based on neighborhoods,and s-tudy properties of these reflexive upper approximation operators and their rela-tions.In addition,they investigate not only some basic properties of D1,D2 and D3 by using topological methods,but also some characterizations of these upper approximation operators being closure operators.Therefore,it is an open question that under what conditions a complex upper approximation operator becomes a closure operator.In this paper,we investigate characterizations of upper approxi-mation operators D6,D7 and D8.Firstly,we define the first symmetric condition,the second symmetric condition,and the third symmetric condition;Secondly,we obtain properties of the reflexive neighborhoods based on these conditions;Thirdly,we describe not only the topological characterizations,intuitive characterizations for these operators being closure operators,but also characterize these operators being closure operators in the information exchange system;Finally,we discuss relationship among D1,D2,…,D8 and provide some examples.We also study the properties of the upper approximation operators generat-ed by NS(U)and closure system S when neighborhoods are non-reflexive,and properties of NS(U)and closure system S,and get the necessary and sufficien-t conditions for NS(U)being weak-unary,and also investigate properties of the upper approximation operator aprNS based on NS(U).Beside this,we find out necessary and sufficient conditions for aprNSs being a closure operator,and discuss general characterizations,topological characterizations,intuitive characterizations of the upper approximation operators aprs and Aprn being closure operators,and relationship between aprS and R.Although some scholars have done some researches in the field of rough mem-bership functions,we define the covering-based rough membership function ?CX(y)10 and the correlation function gx(y)based on the upper approximation operator C10,and obtain some properties of the correlation function gx(y)and numerical char-acterizations,and obtain numerical characterizations of the covering-based rough membership function?CX(y)10 and discuss the relationship between ?CX(y)10 and C10.Finally,we show that this covering-based rough membership function is more realistic than Pawlak' s rough membership function in applications of real life.