Chapter 5 Knowledge Representation

1. Chapter 5Knowledge Representation ID: 106 Name: Yue Lu CS267 Fall 2008 Instructor: Dr. T.Y.Lin

2. Contents • Introduction • Example • Formal Definition • Significance of Attributes • Discernibility Matrix

3. Introduction • Issue of knowledge representation in the framework of concepts • Tabular representation of knowledge represent equivalence relations • Such a table will be called Knowledge Representation System (KRS)

4. Knowledge Representation System (KRS) • KRS can be viewed as a data table • Columns are labeled by attributes • Rows are labeled by objects • Each attribute we associate an equivalence relation • Each table can be viewed as a notation for a certain family of equivalence relations

5. Example of KRS

6. Formal Definition • Knowledge Representation System is a pair S=(U,A) • U - is a nonempty, finite set called the universe • A - is a nonempty, finite set of primitive attributes • Every primitive attribute a ∈ A is a total function a : U → Va is the set of values of a, called the domain of a • With every subset of attributes B ⊆ A, we associate a binary relation IND(B), called an indiscernibilty relation and defined thus: • IND(B)={(x, y)∈ U2 :for every a ∈ B, a(x)=a(y)}

7. U = {A1, A2, A3, A4, A5, A6, A7} • A = {size, animality, color} • V = { (small, medium, large), (bear, dog, cat, horse), (black, brown) }

8. IND (size) = { (A1, A4), (A2, A5), (A3, A6, A7)} • IND (animality) = { (A1, A2), (A3), (A4), (A5, A6, A7) } • IND (color) = { (A1, A2, A4, A5, A6), (A3, A7) }

9. IND (size, animality) = { (A1), (A2), (A3), (A4), (A5), (A6, A7) } • IND (size, color) = { (A1, A4), (A2, A5), (A3, A7), (A6) } • IND (animality, color) = {(A1, A2), (A3), (A4), (A5, A6), (A7) }

10. IND (size, animality, color) = { (A1), (A2), (A3), (A4), (A5), (A6), (A7) }

11. U = {1,2,3,4,5,6,7,8} • A = {a, b, c} • V = {0, 1, 2}

12. U/IND(a)= {(1,4,5), (2,8), (3,6,7)} • U/IND(b)= {(1,3,5),(2,4,7,8),(6)} • U/IND(c)= {(1,5),(2,7,8),(3,4,6)}

13. U/IND(a)= {(1,4,5), (2,8), (3,6,7)} • U/IND(b)= {(1,3,5),(2,4,7,8),(6)} • U/IND(c)= {(1,5),(2,7,8),(3,4,6)}

14. U/IND(a)= {(1,4,5), (2,8), (3,6,7)} • U/IND(b)= {(1,3,5),(2,4,7,8),(6)} • U/IND(c)= {(1,5),(2,7,8),(3,4,6)}

15. U/IND(c)= {(1,5),(2,7,8),(3,4,6)} • U/IND(a,b) = {(1,5),(2,8),(3),(4),(6),(7)} • U/IND(a,b,c) = U/IND(a,b) • IND(a,b) ⊂ IND(c); {a,b} => {c} • CORE(A) = {a,b}; REDUCT(A) = {a,b}

16. Significance of Attributes • KRS is different from relational table • emphasis not on data structuring and manipulation, but on analysis of dependencies in the data • Closer to the statistical data model

17. Discernibility Matrix • S = (U, A), U={X1, X2, …, Xn} • A discernibility matrix of S is a symmetric n × n matrix with entries • Cij = {a ∈ A | a(xi) ≠ a(xj)} for i, j = 1,…,n • CORE(A) = {a ∈ A : Cij=(a), for some i,j }

18. Example

19. 5 ×5 matrix • A={a,b,c,d} • CORE(A)={b}

20. Conclusion • Representing Knowledge using data table • Columns are labelled with attributes • Rows with object of the universe • With each group of columns we associate an equivalence relation

21. THANK YOU