Dependencies in Knowledge Base

1. Dependencies in Knowledge Base By: AkhilKapoor Manandeep Singh Bedi

2. Recap • 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 :

3. 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. 4.1 Introduction • 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.

5. 4.2 Dependency of Knowledge • 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 : • Knowledge Q depends on knowledge P iff • Knowledge P and Q are equivalent, denoted as , iff P=>Q and Q=>P. • Knowledge P and Q are independent, denoted as , iff neither P=>Q nor Q=>P. Obviously , if and only if IND(P) = IND(Q) .

6. Example 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.

7. Proposition 4.1 • The following conditions are equivalent: • P=>Q • IND(P U Q) = IND(P) • POSp(Q) = U • 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.

8. 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 • If P is dependent then there exists a subset such that Q is a Reduct of P. • If and P is independent, then all basic operations in P are pair wise independent. • If and P is independent then every subset R of P is independent

9. Proposition 4.4 • If P=>Q, and , then P’=>Q • If P=>Q, and then P=>Q’ • P=>Q and Q=>R imply P=>R • P=>R and Q=>R imply PUQ=>R • P=>RUQ imply P=>R and P=>Q • P=>Q and QUR=>T imply PUR=>T • P=>Q and R=>T imply PUR => QUT

10. Partial 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=< k >= 1) from knowledge P, symbolically P =>k Q, if and only if where card denotes cardinality of the set.

11. Cases 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.

12. Degree of Dependency: • If P=>kQ, 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.

13. Coefficient 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.

14. 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.