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.
Rough Set Approximation Operation Formal Context Concept Lattice1. 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-approxima- tion 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 Kai- she [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.
2. About Rough Set Theory and FCA2.1. 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
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, with regard to is the set of all of objects, which certainly can be classified with regarding, that is, set.
Formally,
b) Upper Approximation
Upper Approximation is a description of the objects that possibly belong to the subset of interest. The Upper Approximation Set of a set regarding is the set of all of objects which can be possibly classified with regarding, that is, set.
Formally, , denote an empty set.
c) Boundary Region (BR)
Boundary Region is description of the objects that of a set regarding is the set of all the objects, which cannot be classified neither as nor regarding. If the boundary region is a set, then the set is considered “Crisp”, that is, exact in relation to; otherwise, if the boundary region is a set the set “Rough” is considered. In that the boundary region is BR.
Formally,.
2.2. 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] .
Definition 1: Given a formal context, , , if and satisfy, , and, , we call the pair is a concept of formal context. is called extent of concept, is called intent of concept. or is the set which contains of the concepts of [2] .
Definition 2: Given a formal context, , if or, then call is a son concept of, and denote as. Apparently, forms a lattice, which is called concept lattice [2] .
Given two concepts, of a formal context, we have: If, then, for; If, then, for.
3. 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.
3.1. Convert Multiple Valued Is from to Single Valued Formal Context
In an information system, is the set of attributes, for every attribute, if the value of belongs to 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 belongs to a set, and the size of is greater than 2, then we have to convert this attribute to several attributes according to the size of. For example, if there is a attribute “shape”, and the value of “shape” belongs 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.
Definition 3: Let be an information system, if the value of an attribute belongs to a set, and the size of is, then convert to a series of attributes, and call are derived attributes of. After carrying out this converting process, attributes set convert to a new attri- butes set, then is called the derived attributes of.
Definition 4: Let be an information system, assume, is the derived attributes of, is the relation between and, the we call is the single value formal context derived from information system.
3.2. Realization of Approximation Based on Concept Lattice
For the sake of narrative convenience, we agree several symbols in this paper. denotes Concept lattice of formal context and denotes all concepts of. Lower case letters represents concepts, for example, we use letter denote a concept, and let and represent extension and intension of concept respectively.
Definition 5: Let be an information system, is the single value formal context derived from information system, assume，, we call formal context is sub-context of regarding.
Theorem 1: Let be an information system, is the single value formal context derived from information system, , , , attribute is derived from attribute, is sub-context of regarding, is the concept lattice of, Upper Approximation:
Lower Approximation:
Proof: According to the definition of concept in FCA, extent set has attributes, and size of
equals to the size of, then forms the equivalence classes of the B-indiscernibility relation, which means equals to. According to definition of upper and lower approxima-
tion, this theorem obviously holds.
Theorem 2: Let be an information system, is the single value formal context derived from information system, assume, , contains all the concepts in concept lattice, Upper Approximation is:
Lower Approximation is:
Proof: According to the definition of concept in FCA, extent set has attributes, and
, then forms the equivalence classes of the B-indiscernibility relation, which means equals to. According to definition upper and lower approximation, this
theorem obviously holds.
3.3. An Example
In an information system, three are equivalent classes, namely
and
Let,. Now let’s compute and by using Theorem 1 and Theorem 2.
In the above information system, and divide into three equivalent classes, thus the size of the value range of and is 3, and then we must convert these two attributes form multiple value to single value. The derived single value context is shown in Table 1. By using Godin algorithm [9] , we can build the concept lattice of a given single valued formal context.
1) Compute and by using Theorem 1
Under the attributes constraint, we get sub-context offormal context, which includes the columns labeled by and. Concept lattice is shown in Figure 1.
Concept lattice L (K1)
Table 1
. Single valued formal context K
R1
R2
R3
a
b
c
d
e
f
g
h
1
*
*
*
2
*
*
*
3
*
*
*
4
*
*
*
5
*
*
*
6
*
*
*
7
*
*
*
8
*
*
*
In concept lattice, the concepts whose extent satisfy the constraint are
, and. According Theorem 1, we have, , and.
2) Compute and by using Theorem 2
First convert to single vale set and convert to single vale set. Besides, for the father concept of concept, the intent satisfy and, according to Theorem 2, objects 3 and 4 belongs to.
Finally, we get, and.
By using different theory, we get the same result, and to some extent this proves the correctness of our method.
4. Conclusions
There is an intimate link between rough set theory and concept lattice, so rough set approximation operation can be realized being aided by FCA. From this perspective, realizing rough set operation aided by FCA can be seen as a kind of generalization of the concept lattice queries that meet certain conditions.
Realization of rough set approximation operation on concept lattice can be divided into two steps: first, convert multiple value information system to multiple value formal context, and then covert multiple value formal context to single value formal context; second, realize rough set approximation operation aided by formal concept analysis on concept lattice.
The method is expected to be of further use for calculating approximation of rough set that with real values, interval numbers and other self-defined types.
Acknowledgements
The work presented in this paper is supported by National Science Foundation of China (60975033) and Doctorial Foundation of Henan Polytechnic University (B2011-102).
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