_{1}

^{*}

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 theory (RS), first described by a Polish computer scientist Zdzisław I. Pawlak [

Research about the relationship between formal concept analysis and rough set theory has gained some development. Kent [

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.

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 [

Approximations is also other an important concept in Rough Sets Theory, being associated with the meaning of the approximation topological operations [

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

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

Formally,

c) Boundary Region (BR)

Boundary Region is description of the objects that of a set

Formally,

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 [

Definition 1: Given a formal context

Definition 2: Given a formal context

Given two concepts

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.

In an information system_{ }is greater than 2, then we have to convert this attribute to several attributes according to the size of

Definition 3: Let

Definition 4: Let

For the sake of narrative convenience, we agree several symbols in this paper.

Definition 5: Let

Theorem 1: Let

Lower Approximation:

Proof: According to the definition of concept in FCA, extent set

_{ }forms the equivalence classes of the B-indiscernibility relation, which means

tion, this theorem obviously holds.

Theorem 2: Let

Lower Approximation is:

Proof: According to the definition of concept in FCA, extent set

_{ }forms the equivalence classes of the B-indiscernibility relation, which means _{ }equals to

theorem obviously holds.

In an information system

and

Let

In the above information system,

1) Compute

Under the attributes constraint_{ }of_{ }formal context

Concept lattice L (K_{1})

. Single valued formal context K

R_{1} | R_{2} | R_{3} | ||||||
---|---|---|---|---|---|---|---|---|

a | b | c | d | e | f | g | h | |

1 | * | * | * | |||||

2 | * | * | * | |||||

3 | * | * | * | |||||

4 | * | * | * | |||||

5 | * | * | * | |||||

6 | * | * | * | |||||

7 | * | * | * | |||||

8 | * | * | * |

In concept lattice

2) Compute

First _{ }convert to single vale set _{ }convert to single vale set

Finally, we get

By using different theory, we get the same result, and to some extent this proves the correctness of our method.

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.

The work presented in this paper is supported by National Science Foundation of China (60975033) and Doctorial Foundation of Henan Polytechnic University (B2011-102).