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Chapter 5

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  • 1. Knowledge Representation Chapter 5
  • 2. Contents
    • Introduction
    • Examples
    • Formal Definition
    • Significance of attributes
    • Discernibility Matrix
  • 3. 1.Introduction
    • Partition (classification) can be viewed as a semantic definition of knowledge.
    • We need syntactic representation of knowledge to represent equivalent relations in symbolic form suitable for computer processing.
    • The syntactic representation can be data table that will be called Knowledge Representation System (KRS) or Information System
    • In the data table,
        • Columns are labeled by attributes
        • Rows are labeled by objects
  • 4.
    • With each attribute, we can associate an equivalence relation
    • Example:
      • “ Color ” attribute classifies all objects of the Universe into categories of objects having the same color, such as red, green, blue.
    • Each table can be viewed as a notion for a certain family of equivalence relations
  • 5. 2.Examples Table 1 Objects in the information System are U1,…,U2. Attributes are Headache, Muscle pain, Temp, Flu
  • 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. 3.Formal Definition
    • KRS is a pair S=( U,A ), where
        • U – is a nonempty, finite set called the universe.
        • A – is a nonempty, finite set of primitive attributes.
        • a A is a total function a : U V a
          • Where V a is the set of values called the domain of a
        • Every subset of attributes , we associate a binary relation IND(B) defined as
    Primitive means single element set Ex: a A Every subset will called an attribute.
  • 8.
    • 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) .
    • Example:
      • If a(x)=2
      • That means that x
      • belongs to class 2
      • and will take its value
      • here if a(x)=2
      • x could be x 2, x3, x4,
      • each of them is on class 2 and take
      • Its value
    Class 1 Class 2 Class 3 X1 X2 X4 X5 X6 X3
  • 9. Knowledge Representation System Vs Knowledge bases
    • There is a one-to-one correspondence between KRSs and KBs.
    • KRS S=( U,A ) may be viewed as a description of KB K= ( U, R)
    • KB
      • K=( U, R)
      • If , U/R ={X 1 ,…,X k }
    • KRS
      • S=( U,A )
      • 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
    Attribute in KRS = Equivalence relation Rows of a table = Name of categories
  • 10. 3.Significance of attributes
    • Compute whether all the attributes are of the same “ strength ” or not.
    • 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.
    • If removing the attribute will change the classification it means that its significance is high – in the opposite case, the significance should be low .
  • 11. Measurement of the significance
    • 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
  • 12. Example
    • R={a, b, c}, decision D={d}
    • U/{a}={{1,2,3}{4,5,6}}
    • U/{b}={{1,2,3,4,6}{5}
    • U/{c}={{1,4}{2,5}{3,6}}
    • U/R={{1}{4}{2}{5}{3}{6}}
    • U/D={{1,4,5}{2,3,6}}
    • U/{a,b}={1,2,3}{4,6}{5}}
    • U/{a,c}={{1}{2}{3}{4}{5}{6}}
    • U/{b,c}={{1,4}{2}{3,6}{5}}
  • 13. Example Cont’
    • POS R (D) ={{1,4,5}{2,3,6}}={1,2,3,4,5,6}
    • POS R-{a} (D) ={{1,4,5}{2,3,6}}={1,2,3,4,5,6}
    • POS R-{b} (D) ={{{1,4,5}{2,3,6}}={1,2,3,4,5,6}
    • POS R-{c} (D) ={{5}}={5}
    ∴ 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. Remark!
    • KRS and Relational DB:
      • In KRS,
        • 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.
        • Actual dependencies existing in data, and data reduction, which is closer to statistical data model.
      • In Relational Model
        • It is not interested in the meaning of the information stored in the table.
        • Emphasis is placed on efficient data structuring and manipulation.
  • 15. 5.Discernibility Matrix
    • Advantage :
      • It enables simple computation of the core and reducts.
    • Definition
    • Let S = (U, A) be a decision table,
    • with U={x 1 , x 2 , .., x n } , An discernibility Matrix M(S), n x n matrix defined thus :
    • The core can be defined now as the set of all single element entries of the discernibility matrix,
    • A reduct of A is the subset of attributes that discerns all objects discernable by the whole set of attributes.
    C ij is the set of all the condition attributes that classify objects x i and x j into different classes.
  • 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. 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. Table 3 CORE reducts : {a, b} {d, b}