Rough Set - Advanced Artificial Intelligence - page 396

several reducts. In some sense, if there is only one Q-reduct of knowledge P , then

knowledge is determined, that is, there is only one approach to categorize

objects to the concept of Q. When there are multiple Q-reducts of knowledge P,
there are multiple approaches to categorize objects to the concept of Q. When the
core is empty, it will aggravate this uncertainty.

P

11.2.3 Dependency of Knowledge

The dependency of knowledge can be formally defined as follows.

Definition 11.11 Suppose K=(U, R) is an approximate space, and P 、 Q ⊆ R .

(1) Knowledge Q is dependent on knowledge P, or knowledge P can deduct to

knowledge Q , iff IND( P ) ⊆ IND( Q ), denoted as P ¼ Q ;

(2) Knowledge P and Q are equivalent, iff P ¼ Q and Q ¼ P , that is,

IND( P )=IND( Q ), denoted as P = Q . Apparently, P=Q iff IND( P )=IND( Q );

(3) Knowledge P and Q are independent, iff P ¼ Q or Q ¼ P neither holds,

denoted as P ≠ Q.

The dependency of knowledge can hold partly. That is to say, knowledge P
can deduct a part of knowledge of Q, or knowledge Q is only partly dependent on
knowledge P. Partial dependency (partial deducible) can be defined by the
positive region of knowledge. In the following, we formally define partial
dependency.
Definition 11.12

Suppose K=(U, R) is a knowledge base, and P 、 Q ⊂ R, we call
knowledge Q is dependent on P with the dependency degree k (0 ≤ k ≤ 1),
denoted as P ¼ k Q, iff

k = γ P (Q)=card(POS P (Q))/card(U) (11.8)

(1) if k =1, then knowledge Q is called totally dependent on knowledge Q,

denoted as P ¼ 1 Q or P ¼ Q ;

(2) if 0< k <1, then knowledge Q is called partly dependent on knowledge Q;

(3) if k =0, then knowledge Q is called totally independent on knowledge Q.

Above idea can be explained as the ability to categorize objects. More
specifically, if = 1, then all elements on the universe can be used to categorize

objects to the concept of Q using knowledge

k

k ≠ 1only the elements
within the positive region of the universe can be used to categorize objects to the

P . If