Chapter 5Knowledge Representation

ID: 106Name: Yue LuCS267 Fall 2008Instructor: Dr. T.Y.Lin

Contents

Introduction Example Formal Definition Significance of Attributes Discernibility Matrix

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)

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

Example of KRS

UA1A2A3A4A5A6A7

SizeSmallMediumLargeSmallMediumLargeLarge

AnimalityBearBearDogCat HorseHorseHorse

ColorBlackBlackBrownBlackBlackBlackBrown

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)}

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) }

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) }

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) }

UA1A2A3A4A5A6A7

SizeSmallMediumLargeSmallMediumLargeLarge

AnimalityBearBearDogCat HorseHorseHorse

ColorBlackBlackBrownBlackBlackBlackBrown

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

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

U12345678

a10211220

b0101 0211

c21002011

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

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

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

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

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

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 }

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

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

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

THANK YOU