Realization of Rough Set Approximation Toplogical Operations Based on Formal Concept Analysis

There is an intimate correlation between rough set theory and formal concept analysis theory, so rough set approximations can be realized by means of formal concept analysis. For any given multiple valued information system, the realization of rough set approximation operation has two major steps, firstly convert the information system from multiple valued one to single valued formal context, secondly realize rough set approximation operations aided by concept lattice, which is equivalent to a query operation under some necessary conditions.


Introduction
Rough set theory (RS), first described by a Polish computer scientist Zdzisław I. Pawlak [1], is a formal approximation of a crisp set (i.e., conventional set) in terms of a pair of sets which give the lower and the upper approximation of the original set.In the standard version of rough set theory, the lower-and upper-approximation sets are crisp sets, but in other variations, the approximating sets may be fuzzy sets.Formal concept analysis (FCA) is a principled way of automatically deriving ontology from a collection of objects and their properties [2].The term was introduced by Rudolf Wille in 1984, and built on applied lattice and order theory that was developed by Birkhoff and others in the 1930s.
Research about the relationship between formal concept analysis and rough set theory has gained some development.Kent [3] and Yao [4] bring upper approximation and lower approximation in rough set theory into formal concept analysis, and discuss many approximation operators based on formal concept analysis.Qu Kaishe [5] focuses on the theory research about the correlation between formal concept analysis and rough set.He first reveals limitations of data analysis and processing in the rough set theory, and then provides a way for the synthesis of formal concept analysis and rough set theory by using nominal scale.
In these studies, the link between rough set theory and concept lattice is pointed out, but in practice how to realize the operation of rough set by using concept lattice is still left as a problem.Therefore, in this article, we focus on the way to realize rough set approximation topological operations based concept lattice, which includes upper approximation and lower approximation.

Basic Concepts in Rough Set Theory
Indiscernibility Relation is a central concept in rough set theory, and is considered as a relation between two objects or more, where all the values are identical in relation to a subset of considered attributes.Indiscernibility relation is an equivalence relation, where all identical objects of set are considered as elementary [6].
Approximations is also other an important concept in Rough Sets Theory, being associated with the meaning of the approximation topological operations [7].The lower and the upper approximations of a set are interior and closure operations in a topology generated by the indiscernibility relation.Below is resented and described the types of approximations that are used in Rough Sets Theory.a) Lower Approximation ( ) B′ Lower Approximation is a description of the domain objects that are known with certainty to belong to the subset of interest.The Lower Approximation Set of a set X , with regard to R is the set of all of objects, which certainly can be classified with X regarding R , that is, set B′ .Formally, Upper Approximation is a description of the objects that possibly belong to the subset of interest.The Upper Approximation Set of a set X regarding R is the set of all of objects which can be possibly classified with , ∅ denote an empty set.

c) Boundary Region (BR)
Boundary Region is description of the objects that of a set X regarding R is the set of all the objects, which cannot be classified neither as X nor X − regarding R .If the boundary region is a set X = ∅ , then the set is considered "Crisp", that is, exact in relation to R ; otherwise, if the boundary region is a set X ≠ ∅ the set X "Rough" is considered.In that the boundary region is BR. Formally,

Basic Concepts in Formal Concept Analysis
Formal concept analysis, which is proposed by R. Wille, is founded on a basis of order theory and lattice theory, and is a mathematical structure which depicts relationship between objects and attributes according to basic information provided by data base.Formal concept analysis has been successfully used in many fields, and to some extend it has been treated as a means of external cognition [8]. , , X B , and denote as ( ) ( ) Given two concepts ( ) , X B of a formal context ( )

Realization of Approximation Operation Based on FCA
Realization of rough set approximation operation can be divided into two steps: first, convert multiple valued information system to multiple valued formal context, and then covert multiple value formal context to single value formal context; second, realize rough set approximation operation by using the technique of formal concept analysis based on concept lattice.

Convert Multiple Valued Is from to Single Valued Formal Context
In an information system ( ) is the set of attributes, for every attribute i a , if the value of i a be- longs to { } 0,1 in which 1 represents an object has this attribute and 0 represents an object doesn't has this attribute, then this information system corresponds to an single value formal context.Otherwise, if the value of i a belongs to a set i V , and the size of i V is greater than 2, then we have to convert this attribute to several attributes according to the size of i V .For example, if there is a attribute "shape", and the value of "shape" be- longs to {triangle, round, rectangle}, then attribute "shape" can convert to three attributes, namely "triangle", "round" and "rectangle", and the value of every attributes belongs to { } 0,1 .Definition 3: Let ( ) = be an information system, if the value of an attribute i a belongs to a set i V , and the size of i V is ( ) , , , n b b b  are derived attributes of i a .After carrying out this converting process, attributes set A convert to a new attri- butes set M , then M is called the derived attributes of A .
Definition 4: Let ( ) , IS U A = be an information system, assume G U = , M is the derived attributes of A , I is the relation between U and M , the we call ( )

, , K G M I
= is the single value formal context derived from information system IS .

Realization of Approximation Based on Concept Lattice
For the sake of narrative convenience, we agree several symbols in this paper.
Proof: According to the definition of concept in FCA, extent set ( ) L K denotes Concept lattice of formal context K and ( ) B K denotes all concepts of ( ) L K .Lower case letters represents concepts, for example, we use letter c denote a concept, and let value formal context derived from information system IS , assume N M ⊆ ， value formal context derived from information system IS , X U ⊆ , B A ⊆ , N M ⊆ , attribute B is derived from attribute A , context of K regarding N , ( ) L H is the concept lattice of H , Upper Approxima- tion:

∈ 3 R . Concept lattice ( ) 1 L K is shown in Figure 1 .
equals to [ ] Bx .According to definition of upper and lower approxima- tion, this theorem obviously holds.value formal context derived from information system IS , assume X U ⊆ , sub-context 1 K of formal context K , which includes the columns labeled by 2 R and

Table 1 .
Single valued formal context K.