Attribute Reduction for Generalized Decision Systems*

—Attribute reduction of information system is one of the most important applications of rough set theory. This paper focuses on generalized decision system and aims at studying positive region reduction and distribution reduction based on generalized indiscernibility relation. The judgment theorems for attribute reductions and attribute reduction approaches are presented. Our approaches improved the existed discernibility matrix and discernibility conditions. Furthermore, the reduction algorithms based on discernible degree are proposed.


INTRODUCTION
The theory of rough sets, proposed by Pawlak [6], is an extension of the set theory.Rough set theory has been conceived as a tool to conceptualize, organize, and analyze various types of data, in particular, to deal with inexact, uncertain or vague knowledge in applications related to artificial intelligence.
Information systems (sometimes called data tables, attribute-value systems, decision system etc.) are used for representing knowledge.A basic problem related to many practical applications of information systems is whether the whole set of attributes is always necessary to define a given partition of a universe.This problem is referred to as knowledge reduction, i.e., removing superfluous attributes from the information systems in such a way that the remaining attributes are the most informative.A large variety of approaches have been proposed in the literatures for effective and efficient reduction of knowledge.Of all paradigms, rough set theory is perhaps the most recent one making significant contribution to the field.Based on this theory and discernibility functions, some approaches for attribute reduction in complete and discrete decision systems are proposed [5,9,11,14,16].
In many practical situations, it may happen that the precise values of some of the attributes in an information system are not known, i.e. are missing or known partially.Such a system is called an incomplete information system.In order to deal with incomplete information systems, classical rough sets have been extended to several general models by using other binary relations or covers on the universe [1,2,7,8,10,15,18,19].Based on these extended rough set models, the researchers have put forward several meaningful indiscernibility relations in incomplete information system to characterize the similarity of objects.For instance, Kryszkiewicz [3,4] introduced a kind of indiscernibility relation, called tolerance relation, to handle incomplete information tables.Stefanowski [12] introduced two generalizations of the rough sets theory to handle the missing value.The first generalization introduces the use of a non symmetric similarity relation in order to formalize the idea of absent value semantics.The second proposal is based on the use of valued tolerance relations.The tolerance relation has also been generalized to constrained similarity relation and constrained dissymmetrical similarity relation [2,13,17].Accordingly, some attribute reduction approached for incomplete decision systems have been proposed.In this paper, an approach to attribute reduction for incomplete decision systems based on generalized indiscernibility relation is presented.Specifically, this study is not limited to a particular indiscernibility relation, but focus on the indiscernibility relation that satisfies reflexivity and symmetry.A general theory frame of attribute reduction for incomplete decision system will be presented.The paper is organized as follows: In Section 2, we recall some notions and properties of rough sets and decision systems.In Section 3, we propose an approach for positive region reduction.The reduction algorithm based on discernible degree is also presented.Section 4 is devoted to distribution reduction.The paper is completed with some concluding remarks.

An information system is a triplet ( , , )
U A F , where U is a nonempty finite set of objects called the universe of discourse,  for all xU  , where j V is the domain of attribute j a .A decision system ( , is a special case of an information system, where d is a special attribute called decision.The elements of C are called conditional attributes.
In a generalized decision system, we do not care about the information function, but focus on the indiscernibility relations generated by attributes.Concretely, a generalized decision system is a triple , XU  , the lower approximation and upper approximation of X with respect to B R are defined as

III. ATTRIBUTE REDUCTION BASED ON POSITIVE REGION
The section is devoted to the discussion of positive region reduction of generalized decision systems.
The above definition shows that   .It is noted that A R is reflexive, therefore, D  need not to be symmetry in general.

Theorem 3.4 Let ( , , ) S U A d 
be a generalized decision system and   be the positive discernibility function of S .If is the reduced disjunctive form of   , then  Skowron [11] proposed the discernibility conditions for object pairs that need to discern with respect to positive region reduction.The discernibility conditions are ( , ) : ( ) ( ) According to above theorem, the object pair ( , ) xy that satisfies ( ) ( ) do not need to discern in the criterion of positive region reduction.To be specific, Skowrons' discernibility conditions can be simplified as following: In essence, based on Corollary 3.1, the discernibility condition is 1 ( , )   xy when the indiscernibility relation satisfies reflexivity and symmetry.Theorem 3.4 presents an approach to calculate the positive region reductions by discernibility function.Similarly as pointed out in [11], the approach is NP hard.In the following of this section, we present a heuristic algorithm based on discernibility matrix to calculate positive region reduction.

Let ( , , ) S U A d 
be a generalized decision system, BA  .By Theorem 3.3, B is a positive region consistent set of S if and only if ( , ) . It follows that D  is the set of element pairs that needs to be discerned with respect to positive region reduction.For an attribute aA  , {( Intuitively speaking, the bigger the () a  , the more important the attribute a .We propose the following algorithm., the neighborhoods are given by: . We note that A R is reflexive, but not symmetric and transitive.By routine computation, .
Consequently, B is a positive region consistent set.

V. CONCLUSIONS
Rough set under incomplete information has been extensively studied.Researchers have put forward several similarity relations on objects and some attribute reduction approaches for incomplete information systems.This paper is devoted to the study of positive region reduction and distribution reduction based on generalized indiscernibility relation.
the set of all elements of U that can be uniquely classified to blocks of the partition / Ud by means of B .If we take B as the set of conditional attributes, then () B x Pos d  means the decision rule with respect to x is definite.d  if and only if ( ) ( ) theorem shows that, with respect to positive region reduction, x and y need to be discerned if , xy satisfy ( , ) attribute is looked upon as a Boolean variable.In what follows, a R is reflexive and symmetric for any aA  , then the positive discernibility function of S is 1 decision system and a R an equivalence relation for any aA  .It is trivial that A R is an equivalence relation on U .We use [] A y to denote () ) S U A d  be a decision system and a R an equivalence relation for any aA  .
of object pairs that a can discern.Thus, the bigger the set {possible that a is an element of a reduction.Based on this observation, we propose the notion of discernible degree.

4 )(
Compute the positive region discernible degree () a If there are more than one attributes with this property, then any one of the attribute may be chosen), delete( , )

1 ).(.
Corollary 4.1 show the method of distribution reduction based on generalized indiscernibility relation, which only satisfies reflexivity.Obviously, the methods improve the conclusion of literature.Similarly, We propose the following algorithm to compute distribution reduction.Algorithm 2 Input the generalized decision system ( , , ) S U A d  www.ijarai.thesai.org4)Compute the distribution discernible degree () a If there are more than one attributes with this property, then any one of the attribute may be chosen), delete( , )   The following theorem shows the connection between the concepts of distribution reduction and positive region reduction.If B is a distribution consistent set, then B is a positive region consistent set.Proof:We suppose that B is a distribution consistent set.It follows that ( ) for each ir  .Thus ( ) ~(~) ~( ) ~( ) , then B is called a positive region consistent set of S .The minimal positive region consistent set of S (with respect to set inclusion relation) is called as positive region reduction of S .
AB x Pos d Pos d, we have( ) [ ] (1)For any ik  , i T is a positive region reduction of S .In fact, if there exist , for each ik  .www.ijarai.thesai.orgProof: .Hence we have the following corollary.