Dependent Space and Attribute Reduction on Fuzzy Information System

From equivalence relation B R on discourse domain U , we can derive equivalence relation R on the attribute set A . From equivalence relation R on discourse domain A , we can derive a congruence relation on the attribute power set A P and establish an object dependent space. And then,we discuss the reduction method of fuzzy information system on object dependent space. At last ,the example in this paper demonstrates the feasibility and effectiveness of the reduction method based on the congruence relation T providing an insight into the link between equivalence relation and congruence relation of dependent spaces in the rough set. In this way, the paper can provide powerful theoritical support to the combined using of reduction method, so it is of certain practical value. 1. Model Basis Definition 1 Suppose that F A U , , is a fuzzy information system,where U is a limited object discourse domain, A is a limited condition attribute set, F is a mapping set. represents 1 , 0 power set. R~ is a fuzzy equivalence relation on the discourse domain U , 1 , 0 : ~ 	U U R . Suppose that it has the following attributes: 1) 1 , ~ i i x x R ; 2) i j j i x x R x x R , ~ , ~ ; 3) j k k i j i x x R x x R x x R , ~ , ~ , ~ Definition 2 Let R~ be a fuzzy equivalence relation on the discourse domain U , the cut relation of the fuzzy equivalence relation is defined as: j i j i x x R x x R , ~ , 1 0  . Theorem1 Let R~ be a fuzzy equivalence relation on the discourse domain U , j i j i x x R x x R , ~ , 1 0  is an equivalence relation, and R U, is a approximation space. Proof . First ,it is obvious that R has reflexive and symmetric; The following lines show the transitivity of R : If R x x R x x j k k i  , , , , Then j k k i j i x x R x x R x x R , ~ , ~ , ~ , where means R x x j i , , Then R has transitivity and R is an equivalence relation, R U, is approximation space. Definition 3 Suppose that , S is a semi-lattice, S S C : is a closure operator, if the following conditions hold for S y x , : 1) ; x C x 2) ; y C x C y x 3) x C x C C . Theorem 2 Let R be an equivalence relation on, R U P ) ( is a division of U P (discourse power set ). Suppose that F A U , , is a fuzzy information system, B is the attribute subset of attribute set of A ,i.e. A B , the subset of the discourse domain U ,i.e. U X , denote R C on ) , ( U P as: Y X R U P Y Y X R C , ) ( then R C is a closure operator. Proof. It is obvious from the definition that DOI: 10.1051/ , 7120 12 ITA 2017 ITM Web of Conferences itmconf/201 04017 (2017) 4017 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). 1) X R C X holds; 2) Suppose 2 1 X X , we still need to prove that 2 1 X R C X R C . It means that Y X R U P Y X R C Y 1 1 ; because


Model Basis
Definition 1 Suppose that is a fuzzy information system,where U is a limited object discourse domain, A is a limited condition attribute set, F is a mapping set.
> @ > @ o u represents > @ 1 , 0 power set.R ~ is a fuzzy equivalence relation on the discourse domain U , > @ 1 , 0 : ~o uU U R .Suppose that it has the following attributes: 3) ~ be a fuzzy equivalence relation on the discourse domain U , the cut relation of the fuzzy equivalence relation is defined as: Theorem1 Let R ~be a fuzzy equivalence relation on the discourse domain is an equivalence relation, and is a approximation space.
Proof .First ,it is obvious that G R has reflexive and symmetric; The following lines show the transitivity of is a closure operator.
Proof.It is obvious from the definition that 1) From the conclusions of 1) and 2) it is obvious that be a fuzzy information system , B is the attribute subset of attribute set is a congruence relation on the semi-lattice Proof .First, it is obvious from the definition that it is an equivalence relation.

Next, let
be a fuzzy information system , B is the attribute subset of attribute set is an object dependent space on U ., define the relation:

Proof
, be a fuzzy information system , R ~ is a fuzzy equivalence relation on discourse 2) Symmetry: Transitivity: , , , be a fuzzy information system, R ~ is fuzzy equivalence relation on discourse Proof .From the definition, it is easy to know that G R is equivalence relation on A .

Theorem 7
Let F A U , , be a fuzzy information system, R ~ is a fuzzy equivalence relation on discourse 3) From 1) and 2) we know, for ; On the other hand: . Thus this law is proved.
be a fuzzy information system, R ~ is fuzzy equivalence relation on discourse ), then ^` is called the attribute dependent space on attribute set A .

Note:
The above deduction demonstrates that the closure operator on limited semi-lattice and congruence relation can determine each other.

Attribute Reduction Based On Dependent Space
It is feasible to discuss the issue of reducing attribute of fuzzy information system based on the attribute dependent space ^, Theorem 9 We define a binary operator on the non-empty set > @ ^, Law of commutation:

Definition 5
We define a binary relation d on the non-empty set > @ ^, is a join semi-lattice.
Proof .First, the binary relation d satisfies reflexivity, anti-symmetry and transitivity, so the binary relation d is a partial ordering relation on > @ ^, there exists a least upper bound about d in , the theorem is further proved using arbitrariness.T constitute a hierarchical congruence structure of the attribute power set

Definition 6 Let
To better prepare the attribute reduction for information system, we need to define an order relation on the semi-lattice , then for , then From the definition of congruence relation and idempotent law, T is a congruence relation on the semi-lattice , from the theorem 12, we can derive that is the biggest element in > @ G T x .

Note:
The corollary1 indicates that for different congruence structures, there is always a biggest element in each congruence class.This provides a theoretical basis for finding attribute reduction method using congruence relation.T is a congruence relation on the semi-lattice

Theorem 13 Let
Proof .The method is similar to the proof of theorem 7. , so

Definition.7 Let
, thus we prove that it is a congruence relation.

In the following discussion, D
V represents the smallest algebraic unit created by dividing D .

Theorem 15 Let
, be a fuzzy information system, there is an attribute dependent space , , the following conclusions can be drawn: 3)

1)
Because 2) From the definition of closure operator, it is obvious that the conclusion holds; 3) Prove There exists , suppose there exists , , , Therefore there exists , this is contradictory to the statement that X is the biggest.
So for is the smallest algebraic unit generated by G R A , thus the equation is proved.
Next, we discuss dependent relation which is unique in attribute dependent space.
Definition.8 Given a fuzzy information system F A U , , , for attribute dependent space , we say that the attribute set A is dependent on the attribute subset X , denoted as dr X A , (i.e.A is dependent on X ).

Definition 9 Let
G T A, be an attribute dependent space, we call X A is the reduction set of A .If the following conditions are satisfied: 1) .
be a fuzzy information system , for attribute dependent space , then the smallest set X (based on set inclusion relation) that satisfies dr X A , is the reduction set of the attribute set A .
, thus the theorem is proved.

Definition 10 Let
be a fuzzy information system , for attribute dependent space is the degree of dependence of the attribute set A on attribute subset X , where x represents the number of set elements.
The degree of dependence describes the degree to which the attribute set A depends on the attribute subset X .The degree of dependence has the following characteristics: 2) The bigger the dd is, the higher the degree to which the attribute set A depends on the attribute subset X , indicating that the attribute subset X is more important.The smaller the dd is, the lower the degree to which the attribute set A depends on the attribute subset X , indicating that the attribute subset is less important.
We can see from this that the reduction of the attribute set A can also be described as finding the attribute subset that makes the degree of dependence dd 1 , 0 reach the maximum value.
Next, given the congruence relation in the attribute dependent space, a system of attribute congruence classification (division) as well as a theory about attribute reduction and some reduction methods will be put .

Lemma 1 Let
be a fuzzy information system , attribute dependent space T is a congruence relation on the attribute power set A P , for , and , thus the lemma is proved.
. We call Y as a G T reduction of attribute set A .
The question now is to choose an attribute reduction that satisfies the conditions from all the subsets of the attribute set A .We know if there are N attributes in the attribute set A , then the number of its non-empty subsets is 1 2 N , because the attribute set is a limited set, so the reduction set can be found for certain using definition 12.However, usually there are multiple reductions.

Methods And Examples Of Attribute Reduction
The procedure of attribute reduction for fuzzy information system under the dependent space model is as follows: Step 1 Work out the congruence relation G T based on the given fuzzy information system F A U , , and choose the threshold G (the method of choosing G is the same as the method in example 1 where F distribution is used).
Step 2 Work out the congruence classification Step 5 Work out the intersection set of all reduction sets.If the intersection set is not empty, it is the core attribute set A CORE .If the intersection set is empty, there is no core attribute.  .According to the reduction algorithm in attribute dependent space, the smallest set congruent to the attribute set A is the attribute subsets ^àcd and ^àbd , therefore both of them are the reduction of the attribute set.The absolute necessary attribute set i.e. core attribute set ^àd can be found by calculating the intersection set.

Table 1 fuzzy information system
In the end, we wish that this article can provide beneficial theoretical and applied contributions finding better, faster, more effective method of attribute reduction for the fuzzy information system .

1 )
is an equivalence relation about attribute set B on the discourse domain U .Proof .Reflexivity: From the definitions of G B R and R ~, it is obvious that set A .The following is some basic knowledge of reduction theory.
of sets formed under the congruence relation G is a biggest element in each congruence class (equivalence class) > @ G Suppose congruence class (equivalence class) > @ G T x of the congruence relation G T related to A P is: biggest element in the congruence class G T related to A P

Step 3
Find out the smallest element set G T M (based on smallest elements with inclusion relation) of each congruence class in all congruence classes.Step 4 Write out the set made up of all G T reduction sets reduct -

G 3 GT
the cut relation matrix and the equivalence classes of the attribute set A : into a single class or treated as a separate class therefore there is no practical meaning.So, for threshold can be identified using the F statistics.With this threshold, the attribute set A is classified ^`^`^d c to find out the attribute power set ^`^`^`^`^`^`^`^`^`^` t is the attribute subset of attribute set of A ,i.e. B . From theorem 3 we know that B R is a congruence relation on U , from definition 3 we can easily know that