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# Chapter 5

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### Chapter 5

1. 1. Knowledge Representation Chapter 5
2. 2. Contents <ul><li>Introduction </li></ul><ul><li>Examples </li></ul><ul><li>Formal Definition </li></ul><ul><li>Significance of attributes </li></ul><ul><li>Discernibility Matrix </li></ul>
3. 3. 1.Introduction <ul><li>Partition (classification) can be viewed as a semantic definition of knowledge. </li></ul><ul><li>We need syntactic representation of knowledge to represent equivalent relations in symbolic form suitable for computer processing. </li></ul><ul><li>The syntactic representation can be data table that will be called Knowledge Representation System (KRS) or Information System </li></ul><ul><li>In the data table, </li></ul><ul><ul><ul><li>Columns are labeled by attributes </li></ul></ul></ul><ul><ul><ul><li>Rows are labeled by objects </li></ul></ul></ul>
4. 4. <ul><li>With each attribute, we can associate an equivalence relation </li></ul><ul><li>Example: </li></ul><ul><ul><li>“ Color ” attribute classifies all objects of the Universe into categories of objects having the same color, such as red, green, blue. </li></ul></ul><ul><li>Each table can be viewed as a notion for a certain family of equivalence relations </li></ul>
5. 5. 2.Examples Table 1 Objects in the information System are U1,…,U2. Attributes are Headache, Muscle pain, Temp, Flu
6. 6. [ Table 2 ] : Digits display unit in a calculator assumed to represent a characterization of “hand written” digits 1 1 0 1 1 1 1 9 1 1 1 1 1 1 1 8 0 0 0 0 1 1 1 7 1 1 1 1 1 0 1 6 1 1 0 1 1 0 1 5 1 1 0 0 1 1 0 4 1 0 0 1 1 1 1 3 1 0 1 1 0 1 1 2 0 0 0 0 1 1 0 1 0 1 1 1 1 1 1 0 G f e d c b a U a b c d e f g Objects in the information System are 1,…,9 Attributes are a, b, c, d, e, f, g
7. 7. 3.Formal Definition <ul><li>KRS is a pair S=( U,A ), where </li></ul><ul><ul><ul><li>U – is a nonempty, finite set called the universe. </li></ul></ul></ul><ul><ul><ul><li>A – is a nonempty, finite set of primitive attributes. </li></ul></ul></ul><ul><ul><ul><li>a A is a total function a : U V a </li></ul></ul></ul><ul><ul><ul><ul><li>Where V a is the set of values called the domain of a </li></ul></ul></ul></ul><ul><ul><ul><li>Every subset of attributes , we associate a binary relation IND(B) defined as </li></ul></ul></ul>Primitive means single element set Ex: a A Every subset will called an attribute.
8. 8. <ul><li>The value a(x) assigned by the primitive attribute a to the object x can be viewed as a name of primitive category of a to which x belongs (i.e. equivalence class of IND (a) containing x), that is to say a(x) is the name of [x] IND (a) . </li></ul><ul><li>Example: </li></ul><ul><ul><li>If a(x)=2 </li></ul></ul><ul><ul><li>That means that x </li></ul></ul><ul><ul><li>belongs to class 2 </li></ul></ul><ul><ul><li>and will take its value </li></ul></ul><ul><ul><li>here if a(x)=2 </li></ul></ul><ul><ul><li>x could be x 2, x3, x4, </li></ul></ul><ul><ul><li>each of them is on class 2 and take </li></ul></ul><ul><ul><li>Its value </li></ul></ul>Class 1 Class 2 Class 3 X1 X2 X4 X5 X6 X3
9. 9. Knowledge Representation System Vs Knowledge bases <ul><li>There is a one-to-one correspondence between KRSs and KBs. </li></ul><ul><li>KRS S=( U,A ) may be viewed as a description of KB K= ( U, R) </li></ul><ul><li>KB </li></ul><ul><ul><li>K=( U, R) </li></ul></ul><ul><ul><li>If , U/R ={X 1 ,…,X k } </li></ul></ul><ul><li>KRS </li></ul><ul><ul><li>S=( U,A ) </li></ul></ul><ul><ul><li>The set of attributes A belongs every attribute a: U V aR ,such that V aR = {1,…,k} and a R (x)=i iff for i=1,…,k </li></ul></ul>Attribute in KRS = Equivalence relation Rows of a table = Name of categories
10. 10. 3.Significance of attributes <ul><li>Compute whether all the attributes are of the same “ strength ” or not. </li></ul><ul><li>In order to find out the significance of a specific attribute it seems reasonable to drop the attribute from the table and see how the classification will be changed without this attribute. </li></ul><ul><li>If removing the attribute will change the classification it means that its significance is high – in the opposite case, the significance should be low . </li></ul>
11. 11. Measurement of the significance <ul><li>As a measure of the significance of the subset of attributes with respect to the classification induced by a set of attributes C, we will mean the difference </li></ul>
12. 12. Example <ul><li>R={a, b, c}, decision D={d} </li></ul><ul><li>U/{a}={{1,2,3}{4,5,6}} </li></ul><ul><li>U/{b}={{1,2,3,4,6}{5} </li></ul><ul><li>U/{c}={{1,4}{2,5}{3,6}} </li></ul><ul><li>U/R={{1}{4}{2}{5}{3}{6}} </li></ul><ul><li>U/D={{1,4,5}{2,3,6}} </li></ul><ul><li>U/{a,b}={1,2,3}{4,6}{5}} </li></ul><ul><li>U/{a,c}={{1}{2}{3}{4}{5}{6}} </li></ul><ul><li>U/{b,c}={{1,4}{2}{3,6}{5}} </li></ul>
13. 13. Example Cont’ <ul><li>POS R (D) ={{1,4,5}{2,3,6}}={1,2,3,4,5,6} </li></ul><ul><li>POS R-{a} (D) ={{1,4,5}{2,3,6}}={1,2,3,4,5,6} </li></ul><ul><li>POS R-{b} (D) ={{{1,4,5}{2,3,6}}={1,2,3,4,5,6} </li></ul><ul><li>POS R-{c} (D) ={{5}}={5} </li></ul>∴ the attribute c is most significant, since it most changes the positive region of U/IND(D) significance of attribute ‘b’ : significance of attribute ‘c’ : significance of attribute ‘a’ :
14. 14. Remark! <ul><li>KRS and Relational DB: </li></ul><ul><ul><li>In KRS, </li></ul></ul><ul><ul><ul><li>All objects are explicitly represented and the attribute values, i.e., the table entries have associated explicit meaning as features or properties of the objects. </li></ul></ul></ul><ul><ul><ul><li>Actual dependencies existing in data, and data reduction, which is closer to statistical data model. </li></ul></ul></ul><ul><ul><li>In Relational Model </li></ul></ul><ul><ul><ul><li>It is not interested in the meaning of the information stored in the table. </li></ul></ul></ul><ul><ul><ul><li>Emphasis is placed on efficient data structuring and manipulation. </li></ul></ul></ul>
15. 15. 5.Discernibility Matrix <ul><li>Advantage : </li></ul><ul><ul><li>It enables simple computation of the core and reducts. </li></ul></ul><ul><li>Definition </li></ul><ul><li>Let S = (U, A) be a decision table, </li></ul><ul><li>with U={x 1 , x 2 , .., x n } , An discernibility Matrix M(S), n x n matrix defined thus : </li></ul><ul><li>The core can be defined now as the set of all single element entries of the discernibility matrix, </li></ul><ul><li>A reduct of A is the subset of attributes that discerns all objects discernable by the whole set of attributes. </li></ul>C ij is the set of all the condition attributes that classify objects x i and x j into different classes.
16. 16. Example Table 1 Knowledge Representation System U a b c d 1 0 1 2 0 2 1 2 0 2 3 1 0 1 0 4 2 1 0 1 5 1 1 0 2
17. 17. Table 2 Discernibility Matrix 1 5 a,d b,c,d b a,c,d 5 a,b,c,d a,b,d a,c,d 4 b,c,d a,b,c 3 a,b,c,d 2 4 3 2 1
18. 18. Table 3 CORE reducts : {a, b} {d, b}