Dynamic Reduct and Its Properties In the Object-Oriented Rough Set Models M.Srivenkatesh

This Paper deals with a new type of Reduct in the object-oriented rough set model which is called dynamic reduct. In the object–oriented rough set models, objects are treated as instances of classes, and illustrate structural hierarchies among objects based on is-a relationship and has-a relationship[6]. In this paper, we propose dynamic reduct and the notation of core according to the dynamic reduct in the object-oriented rough set models. It describes various formal definitions of core and discusses some properties about dynamic core in the object-oriented rough set models.


I. INTRODUCTION
Rough set theory [1,2] deals with approximation and reasoning about data.In the aspect of approximation, the basic concepts are lower approximation, upper approximation by indiscernibilty relations which illustrate set-theoretic approximations of any given subset of data.we can find reducts and decision rules in traditional rough set theory.The object-oriented rough set model is an extension of the "traditional rough set theory" by introducing objectoriented paradigm [4] used in computer science and the Object-oriented rough set models [6] illustrates hierarchical structures between classes, names and objects based on is-a and has-a relationships.Kudo and Murai have extended the object-oriented rough set models to treat incomplete information [8].In the papers [6] [8], formulation of the object-oriented rough set model was concentrated to the aspect of approximation, and in the paper [7], reasoning about objects in the object-oriented rough set model by introducing decision rules in the object-oriented rough set model, and revised discernibility matrix for finding reducts in the object-oriented rough set model was discussed.However dynamic reducts and properties in the objectoriented rough set model have not been discussed.In paper [5], deals with a comparison of dynamic and non-dynamic rough set methods for extracting laws from decision tables.
In this paper, we propose dynamic reduct and its properties in the object-oriented rough set models.The Reducts are stable in decision rules are called dynamic reducts in the object-oriented rough set models.Dynamic Reducts define set of classes or set of classes with names are called dynamic core in the object-oriented rough set models.This is the classes or classes with names included in all dynamic reducts in the object-oriented rough set models.
The rest of the paper is organized as follows: In section II, we review the object-oriented rough set model.In section III, we introduce dynamic reduct in the object-oriented rough set models.In section IV, we introduce dynamic core in the object-oriented rough set models.In section V, we introduce (F- )-dynamic core in the object-oriented rough set model.
In section VI, we introduce generalized dynamic core in the object-oriented rough set model.In section VII, we draw conclusion.

II. THE OBJECT-ORIENTED ROUGH SET MODEL
We briefly review the object-oriented rough set model.First, we describe the concept of class, name and object.Next, we illustrate well-defined structures as a basic framework of the object-oriented rough set model.Moreover, we introduce equivalence relations based on "equivalence as instances".Note that the contents of this section are entirely based on the papers [6] [8].

A. Class, Name, Object
Formally, a class structure C, a name structure N and a object structure O are defined by the following triples, respectively: , where C, N and O are finite and disjoint non-empty sets  (1)   The class name and object structures have the following characteristics, respectively: -The class structure illustrates abstract data forms and those hierarchical structures based on part / whole relationship (has-a relation) and specialized/ generalized relationship (is-a relation).
-The name structure introduces numerical constraint of objects and that identification, which provide concrete design of objects.
-The object structure illustrates actual combination of objects.

Two relations
The relation X  is called a has-a relation, which illustrates part / whole relationship.
means that "the class j c has a class j c ", or " j c is a part of j c ". On the other hand, the relation X  is called an is-a a relation, which illustrates specialized / generalized relationship.
between super classes and subclasses, and  The naming function provides names to each class, which enable us to use plural instances of the same class simultaneously.On the other hand, the name assignment provides names to every object, which enable us to identify objects by names.

B. Well-Defined Structures
Formally, the naming function nf : N C is a surjective p-morphism between N and C, and satisfies the following name preservation constraint: is the set of objects that x has.na(x) = n means that the name of the object x is n.The uniqueness condition requires that all distinct parts y H O (x) have different names.
We say that C, N and O are well-defined if and only if there exist a naming function nf : N C and a name assignment na : O N such that We concentrate well-defined class, name and object structures.In well-defined structures, if a class i

C. Indiscernibility Relations in the Object -Oriented Rough Set Model
All equivalence relations in object-oriented rough set models are based on the concept of equivalence as instances.In [6], to evaluate equivalence of instances, an equivalence relation ~ on O are recursively defined as follows: x ~ y  x and y satisfy the following two conditions: , and, Val(x) is the "value" of the "value object" x.Because C is a finite non-empty set and C  is acyclic, there is at least one . We call such class c an attribute, and denote the set of attributes by AT.For any object x, if id C (x) = a and a AT, we call such object x a value object of the attribute a.
The value object x as an instance of the attribute a represents a "value" of the attribute.

D. Object-Oriented Rough Sets
x ~ y means that the object x is equivalent to the object y as an instance of the class

 
x id C .Using the equivalence relation ~, an equivalence relation B ~ with respect to a given subset B  N of names is defined as follows: x and y satisfy the following two conditions: Suppose OORS is Object-Oriented Rough Set Model, N is non empty subset of names , and ~ B be the equivalence relation defined by eq (5) .For any subset X⊆O of objects are called B-lower approximation and B-upper approximation respectively.The B-lower approximation is also called the position region denoted by POS B (X).
Note that the contents of this section are entirely based on the paper [7].

A. Decision Rule
Let OORS(C, N, O) be the object-oriented rough set model where be the well defined class, name, object structures, respectively.Similar to the decision table in rough set theory, we divide the set of names N into the following two parts: the set of names that may appear in antecedents of decision rules (called condition names) N CON , and the set of names that may appear in conclusions of decision rules   is equivalent to  are also equivalent to l m o. (l ≤ j), respectively.Thus, DR(c; o) describes a certain property about combination of objects as an instance of the class c.
As a special case, we allow rules that have no condition names, that is, the case of i = 0 in (7) as follows: This rule illustrates that all instances o of the class c have some parts , respectively.On the other hand, we require that there is at least one name .This means that any object that has no decision name are not the target of decision rule generation.

B. Discernibility Matrix
Definition 2: A discernibility matrix of the objectoriented rough set model is a k  k matrix whose elements ij  at the i-the row and j-the column is defined as follows.
Where k is the number of objects, that is, The intention of L( ij  ) is that, for example the case of Next, connecting all formulas L( ij  ) by the logical product, we get a formula ).This formula is the conjunctive normal form.Thus, finally, we transform this formula to the disjunctive normal form that is logically equivalent to where each conjunction This is because, for each element ij  of the revised discernibility matrix, R contains at least one expression c or

C. A Method of Generating Decision Rules
Let R be a reduct of the object-oriented rough set model.We consider decision rules from the reduct and each object in decision classes.However, for each object o in any decision class  Reducts generated from object-oriented rough set models are sensitive to changes in the models.This can be seen by removing a randomly chosen set of objects from the original object set.Those reducts frequently occurring in random sub-object-oriented models can be considered to be stable; it is this object-oriented reducts that are encompassed by dynamic reducts in the object-oriented rough set models.

Construct an expression
The reducts stable in decision rules are called objectoriented dynamic reducts.Dynamic reducts define set of classes or set of classes with names are called dynamic core in the object-oriented rough set models.These are the classes or classes with names included in all object-oriented dynamic reducts.
The rules calculated by means of dynamic reducts are better pre-disposed classify unseen cases, because these reducts in the object-oriented rough set models are in some sense the most stable reducts, and they are the most frequently appearing reducts in sub-object-oriented rough set models created by random samples of a given objectoriented rough set models.
Firstly, all reducts are calculated for the given objectoriented rough set model, OORS.Then, new sub-objectoriented rough set model S OOR  by randomly deleting one or more rows form OORS.All reducts found for each subobject-oriented rough set model, and the dynamic reducts are computed using   R P s F , which denotes the significance of reduct P with all reducts found, R.
be the sub-object-oriented rough set model, where  In many cases, a given object-oriented rough model may exist several reducts.Each reduct can produce a rule set, and it is difficult to justify which the best rule is set.Therefore it http://ijacsa.thesai.org/ is a important to search the most stable reduct in the objectoriented rough set model, and hence reduct in the objectoriented rough set model is proposed in this case., [5], and , is called an F - dynamic reduct, which describes the most stable reducts in object-oriented rough set models.From the definition of dynamic reduct in the object-oriented rough set model, it follows that, it is also reduct of all sub-object-oriented rough set models from a given family F by random sampling.
The concept of dynamic core in the object-oriented rough set model is introduced here.

Dynamic Core in the Object-Oriented Rough Set Model.
Reduct finding is the basic problem in object-oriented rough set models, and the computation of feature core in the object-oriented rough set model is especially important for resolving this problem.All classes with names or classes in the feature core will be presence in any reduct, otherwise revised discernible relation in object-oriented rough set models can be ensured.According to the feature core one can construct object-oriented reduct heuristically, and the efficiency of reduct can be improved greatly.For dynamic reduct in the object-oriented rough set model, the need for feature core is to be probed.
represents classes and classes with names belongs to reducts in the object-oriented rough set model.
represents classes and classes with names belongs to reducts in the sub-object-oriented rough set model.

VI. GENERALIZED DYNAMIC CORE IN THE OBJECT-ORIENTED ROUGH SET MODEL
According to the definition of dynamic core in the objectoriented rough set model, if some feature classes or classes with names of any sub-object-oriented rough set models in F family are comprised by dynamic core, then it is certainly a feature classes or classes with names of object-oriented rough set model.This notion can be sometimes not convenient because we are interested in useful sets of classes or classes with names which are not necessarily reducts of the object-oriented rough set model.Therefore we have to generalize the notion of a dynamic core in the object-oriented rough set model.
the class of o is identified by the class identifier function.The class identifier id C is a p-morphism between O and C [3], that is, the function C id : O  C satisfies the following conditions: 1.
that the object o is an instance of the class c.The object structure O and the class structure C are also connected through the name structure N by the naming function nf : N C and the name assignment na : O N.
good property enables us the following description for clear http://ijacsa.thesai.org/representation of objects.Suppose we have , , 2 1

B
x B ~ y means that x and y are equivalent as instances of the class defined.Note that, in the "traditional" rough set theory, all equivalence classes concern the same attributes.On the other hand, each equivalence class of the objectoriented rough set model may concern different classes.In particular, if B ~is the set of objects that are not concerned any class

(
called decision names) N DEC .Note that N = N CON  N DEC and N CON  N DEC = .The decision names provide decision classes as equivalence classes [x] DEC N ~ based on the equivalence relation N DEC by (5).Decision rules in the object-oriented rough set model are defined as follows.Definition 1: A decision rule in the object-oriented rough set model has the following form:

.
We call this http://ijacsa.thesai.org/rule a decision rule of the class c by the object o, and denote DR(C;O) .The decision rule DR(c; o) means that, for any object o O  , if o is an instance of c and each part k n o .
[x] ~NDEC , not all classes o.Thus, for each object o [x] ~NDEC such that   , c o id C  we construct a decision rule DR(c; o) in the object-oriented rough set model as follows: http://ijacsa.thesai.org/ 1. Select the class c such that the reduct R.

4 .
the object o, and connect the class c and these expressions by  follows: m  N DEC  H N (na(o)), and connect the class c and these expressions by  as follows: Construct the decision rule DR(c : o) by connecting antecedents and conclusions by ⇒ as follows: REDUCT IN THE OBJECT-ORIENTED ROUGH SET MODELS.

Algorithm 1 :
Dynamic Reduct in the Object-Oriented Rough Set Model.DynamicRedOORS(OORS, ,  nts) OORS, the original object-oriented rough set model; ,  the dynamic reduct threshold; nts, the number of iterations.R    T  calculateAllReducts(OORS) for j=1..nts be the well defined class, name, object structures respectively.Let C  C , N  N , O  O , called sub-object- oriented rough set model of OORS .Let   OORS  be the set of all sub-object-oriented rough set model of OORS .
be the well defined class, name, object structures, respectively.Let   OORS RED denotes the set which contains all reducts of OORS and   S OOR RED  denotes the set which includes all Reducts of S OOR  .A object-oriented rough set model at least contains one reduct, which is just itself, so the set of reduct in the object-oriented rough set model is not empty.

O
be the well defined class, name, object structures, respectively.Let   OORS  be the set of all sub-object-oriented rough set model of OORS, called sub-objectoriented rough set model of OORS.The feature core of object-oriented rough set models in static reduct is

Algorithm 2 :
Core of the object-oriented rough set model.Input: object-oriented rough set model OORS.Output: the core of the object-oriented rough set model.T  calculateAllReducts (OORS).Core  finding intersection of elements of T. of all sub-object-oriented rough set model of OORS , the feature core of sub-object-oriented rough set models in static Reduct is CORE  
be the well defined class, name, object structures, respectively.Let   OORS  be the set of all sub-object-oriented rough set model of OORS, Generalized Dynamic Core the of object-oriented rough set model OORS .
and each oO is called an object.The relation X satisfy the following property: http://ijacsa.thesai.org/ O