D EPENDENCIES IN K NOWLEDGE B ASE By: Akhil Kapoor Manandeep Singh Bedi

R ECAP Reduct and Core – is essential part of knowledge if it suffice to define all basic concept in knowledge. Obtained by omitting some operations and relations from a given set G = {X1, X2, X3, X4} Red(G) = {X1,X2} if it able to define basic concept implied by G. Core is the most important part of knowledge. Hence, we may define :

What is R definable ? Let and R is equivalence relation, then we say X is R definable if X is union of R basic categories. What is IND(R)? IND(R) = IND (R-{ R }) where R be a family of equivalence relations and R R This means R is dispensable in R. And if the above equation doesn’t hold true then R is indispensable in R.

4.1 I NTRODUCTION Theorizing next to classification is the second most important aspect when drawing inferences about the world. Developing theories is based on inference rules of the form “if …. then….” In context we define how another knowledge can be induced from a given knowledge. More precisely, knowledge Q is derivable from knowledge P, if all elementary categories of Q can be defined in terms of some elementary categories of knowledge P. In other words, if Q is derivable from P we say that Q depends on P or P => Q. this will be considered in more detail in CH-7. More about the notion of dependency can be found in Novotny et al. (1988,1989,1990) and Pawlak (1985). And in the article of Buszkowski et al. (1986) – mainly to investigate relationship between dependencies between relational databases and those considered here.

4.2 D EPENDENCY OF K NOWLEDGE This chapter will mainly focus on semantic aspects of dependency. Formally dependency can be defined as shown below: Let K = (U,R) be a knowledge base and then we have the following : 1. Knowledge Q depends on knowledge P iff 2. Knowledge P and Q are equivalent, denoted as, iff P=>Q and Q=>P. 3. Knowledge P and Q are independent, denoted as, iff neither P=>Q nor Q=>P. Obviously, if and only if IND(P) = IND(Q).

E XAMPLE 1 The following example will demonstrate the definition dependency. Suppose we are given knowledge P and Q with the following partitions: U/P = {{1,5},{2,8,3},{4},{6},{7}} U/Q = {{1,5},{2,7,8},{3,4,6}} U/R = {{1,2,5},{7,8},{6}} U/P intersection R = {1,5},{8},{2},{7},{6},{3},{4} PUQ = {{1,5},{2,6,7,8,3,4}} Hence, and consequently P=>Q.

P ROPOSITION 4.1 The following conditions are equivalent: 1. P=>Q 2. IND(P U Q) = IND(P) 3. POSp(Q) = U 4. PX = X for all X E U/Q Where P X denotes IND(P)X. Proposition 4.1 demonstrates that if Q depends on P then knowledge Q is superfluous within the knowledge base in the sense that knowledge PUQ and P provide the same characterization of objects.

Proposition 4.2 The following are also important properties of dependencies. If P is a Reduct of Q, then P=>Q-P and IND(P) = IND(Q) Proposition 4.3 a) If P is dependent then there exists a subset such that Q is a Reduct of P. b) If and P is independent, then all basic operations in P are pair wise independent. c) If and P is independent then every subset R of P is independent

P ROPOSITION If P=>Q, and, then P’=>Q 2. If P=>Q, and then P=>Q’ 3. P=>Q and Q=>R imply P=>R 4. P=>R and Q=>R imply P U Q=>R 5. P=>R U Q imply P=>R and P=>Q 6. P=>Q and Q U R=>T imply P U R=>T 7. P=>Q and R=>T imply P U R => Q U T

P ARTIAL DEPENDENCY OF KNOWLEDGE The derivation (dependency) can also be partial, which means that only part of knowledge Q is derivable from knowledge P. the partial derivability can be defined using the notion of the positive region of knowledge. Defining the partial derivability formally. Let K = (U,R) be the knowledge base and Where knowledge Q depends in a degree k( 0= = 1) from knowledge P, symbolically P =>k Q, if and only if where card denotes cardinality of the set.

C ASES OF ‘ K ’ If k = 1 Q totally depends from P If k>0 and K<1 Q partially depends from P If k = 0 Q is independent from P More precisely: if k = 1, then all elements of the universe can be classified to elementary categories of U/Q by using knowledge P. If k!= 1, only those elements of the universe which belong to the positive region can be classified to categories of knowledge Q, employing knowledge P. if k = 0, none of the elements of the universe can be classified using knowledge P - to elementary categories of knowledge Q.

D EGREE OF D EPENDENCY : If P=> k Q, then the positive region of the partition U/Q induced by Q covers k x 100 percent of all objects in the knowledge base. On the other hand, only those objects belonging to the positive region of the partition can be uniquely classified. This means that k x 100 percent of objects can be classified into blocks of partition U/Q employing knowledge P. Thus the coefficient can be understood as a degree of dependency between Q and P The measure k of dependency does not capture how this partial dependency is actually distributed among classes U/Q.

C OEFFICIENT where X U/Q which shows how many elements of each class of U/Q can be classified by employing knowledge P. Thus the two numbers give us full information about the "classification power" of the knowledge P with respect to the classification U/Q.

Summary Dependencies, in particular partial dependencies, in a knowledge base are basic tools when drawing conclusions from basic knowledge, for they state some of the relationships between basic categories in the knowledge base.