Knowledge representation 2. Knowledge Representation using structured objects.
Chapter 5 Knowledge Representation
description
Transcript of Chapter 5 Knowledge Representation
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Chapter 5Knowledge Representation
ID: 106Name: Yue LuCS267 Fall 2008Instructor: Dr. T.Y.Lin
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Contents
Introduction Example Formal Definition Significance of Attributes Discernibility Matrix
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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)
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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
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Example of KRS
UA1A2A3A4A5A6A7
SizeSmallMediumLargeSmallMediumLargeLarge
AnimalityBearBearDogCat HorseHorseHorse
ColorBlackBlackBrownBlackBlackBlackBrown
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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)}
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UA1A2A3A4A5A6A7
SizeSmallMediumLargeSmallMediumLargeLarge
AnimalityBearBearDogCat HorseHorseHorse
ColorBlackBlackBrownBlackBlackBlackBrown
U = {A1, A2, A3, A4, A5, A6, A7} A = {size, animality, color} V = { (small, medium, large), (bear,
dog, cat, horse), (black, brown) }
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UA1A2A3A4A5A6A7
SizeSmallMediumLargeSmallMediumLargeLarge
AnimalityBearBearDogCat HorseHorseHorse
ColorBlackBlackBrownBlackBlackBlackBrown
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) }
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UA1A2A3A4A5A6A7
SizeSmallMediumLargeSmallMediumLargeLarge
AnimalityBearBearDogCat HorseHorseHorse
ColorBlackBlackBrownBlackBlackBlackBrown
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) }
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UA1A2A3A4A5A6A7
SizeSmallMediumLargeSmallMediumLargeLarge
AnimalityBearBearDogCat HorseHorseHorse
ColorBlackBlackBrownBlackBlackBlackBrown
IND (size, animality, color) = { (A1), (A2), (A3), (A4), (A5), (A6), (A7) }
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U = {1,2,3,4,5,6,7,8} A = {a, b, c} V = {0, 1, 2}
U12345678
a10211220
b0101 0211
c21002011
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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)}
U12345678
a10211220
b0101 0211
c21002011
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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)}
U12345678
a10211220
b0101 0211
c21002011
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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)}
U12345678
a10211220
b0101 0211
c21002011
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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}
U12345678
a10211220
b0101 0211
c21002011
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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
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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 }
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Example
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
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5 ×5 matrix
A={a,b,c,d} CORE(A)={b}
1 2 3 4 5
1 ∅
2 a,b,c,d
∅
3 a,b,c b,c,d ∅
4 a,c,d a,b,d a,b,c,d
∅
5 a,c,d b b,c,d a,d ∅
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
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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
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THANK YOU