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Chapter 4

Chapter 4. Dependency of Knowledge Edited by Shengyu Li (#105). P => Q. All elementary categories of Q can be defined in terms of some elementary categories of knowledge P , Q depends on P . (Chapter 4, page 39). Example:. Given knowledge P and Q with the following partitions:

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Chapter 4

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  1. Chapter 4 Dependency of Knowledge Edited by Shengyu Li (#105)

  2. P=>Q All elementary categories of Q can be defined in terms of someelementary categories of knowledge P, Q depends on P. (Chapter 4, page 39)

  3. Example: 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}} Category classes: What is the relationship between P and Q? Q: Q1 = {1, 5} Q2 = {2, 7, 8} Q3 = {3, 4, 6} P: P1 = {1, 5} P2 = {2, 8} P3 = {3} P4 = {4} P5 = {6} P6 = {7} Q1=P1 Q2=P2 U P6 Q3= P3 U P4 U P5

  4. RECALL: All elementary categories of Q can be defined in terms of some elementary categories of knowledge P, Q depends on P. Q1=P1 Q2=P2 U P6 Q3= P3 U P4 U P5 P =>Q

  5. Denoted knowledge P using element to indicate knowledge Q by using color: Q: P: • Q1= P1 • Q2= P2 U P6 • Q3= P3 U P4 U P5 P=>Q

  6. Continue with proposition: IND (PUQ) = IND (P)

  7. What is PUQ? U = { 1, 2, 3, 4, 5, 6, 7 } U/P = {{1,5},{2,8},{3},{4},{6},{7}} = { P1, P2, P3, P4, P5, P6 } U/Q = {{1,5},{2,7,8},{3, 4, 6}} = { Q1, Q2, Q3 } P: P1 = {1, 5} P2 = {2, 8} P3 = {3} P4 = {4} P5 = {6} P6 = {7} Q: Q1 = {1, 5} Q2 = {2, 7, 8} Q3 = {3, 4, 6}

  8. What is PUQ? PUQ = {{1,5},{2,8},{3},{4},{6},{7}} = P What does PUQ mean: Example: If P represents toy's age group, Q represents toy's color code: PUQ represents toy's color or age group: e.g. I need to get the toy which is either red or ages 6-10.

  9. What does IND(P) mean? RECALL: Ch1, page 5: If P R and P ≠ Ф then IND(P) = ∩ P (intersection of all equivalence relations belongs to (P)) U/P = {{1,5},{2,8},{3},{4},{6},{7}} PUQ = {{1,5},{2,8},{3},{4},{6},{7}} = P • IND (PUQ) = IND (P)

  10. Continue with proposition: POSp(Q) = U

  11. POSp(Q) = U • Recall: Chapter 2, p10 ~ 11 POSp(Q) = PQ = U{Y U/P: Y Q} • Example: U = { 1, 2, 3, 4, 5, 6, 7 } U/P = {{1,5},{2,8},{3},{4},{6},{7}} = {P1, P2, P3, P4, P5, P6} U/Q = {{1,5},{2,7,8},{3, 4, 6}} = {Q1, Q2, Q3} P1, P2, P3, P4, P5, P6 U/P P1, P2, P3, P4, P5, P6 Q P1 U P2 U P3 U P4 U P5 U P6 = U POSp(Q) = U

  12. Continue with proposition: If P => Q, and P’ P, then P’ => Q

  13. If P => Q, and P’ P, then P’ => Q

  14. If P => Q, and Q’ Q, then P => Q’

  15. P => R and Q => R, imply P U Q => R

  16. P => R U Q imply P => R and P => Q

  17. P => Q and Q U R => T imply P U R => T

  18. P => Q, and R => T imply P U R => Q U T

  19. Partial Dependency of Knowledge • Partial dependency means only part of knowledge Q is derivable from knowledge P. • Let K = ( U, R) be the knowledge base and P, Q is subset of R. • Knowledge Q depends in a degree k (0 ≤ k ≤ 1) from knowledge P, P => kQ, iff: • k = (cardPOSp{Q})/(cardU) • Degree of dependency between Q and P

  20. Partial Dependency of Knowledge • k = 1: Q totally depends from P • 0 < k < 1: Q roughly (partially) depends from P • k = 0: none of the elements of universe be classified using knowledge P to elementary categories of knowledge Q.

  21. Example 2: • Compute the degree of dependency of knowledge Q from knowledge P where the corresponding partitions are the following: U/Q = { X1, X2, X3, X4, X5 }, U/P = { Y1, Y2, Y3, Y4, Y5, Y6 } Q: X1 = {1} X2 = {2, 7} X3 = {3, 6} X4 = {4} X5 = {5,8} P: Y1 = {1,5} Y2 = {2, 8} Y3 = {3} Y4 = {4} Y5 = {6} Y6 = {7} PX1 = Ø PX2 = Y6 = {7} PX3 = Y3 U Y5 = {3, 6} PX4 = Y4 = {4} PX5 = Ø POSp(Q) = Y3 U Y4 U Y5 U Y6 = {3, 4, 6, 7} (Only these elements can classified into blocks of dependency between Q and P.) k = 4/8 = 0.5 (degree of dependency between Q and P)

  22. THANK YOU!

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