Models for Inexact Reasoning Fuzzy Logic – Lesson 5 Fuzzy Relations

Models for Inexact Reasoning Fuzzy Logic – Lesson 5

Fuzzy Relations

Master in Computational Logic Department of Artificial Intelligence

Crisp Relations

• Crisp relations represent the presence or absence of

– Association – Interaction

between the elements from two or more sets

• Example

– M = {John, Mark}, W = {Mary, Sonya}

– John is Mary’s husband, Sonya is Mark’s wife

Crisp Relations

• A relation among crisp sets is a crisp subset

( ^{1, 2, , N} ) ^{1 2 N}

R X X K X ⊆ X × X × ×K X

• Crisp relations can be defined using characteristic functions

( ^{1, 2, ,} ) ^{1, 1, 2, ,}

0,

N N

iff x x x R

R x x x

otherwise

 ∈

=  K K

• Tuples in the relation identify elements related to one another

Example

• X = {U.S., France, Spain, U.K., Germany}

• Y = {Dollar, Pound, Euro}

• R = Association between a country and its currency

U.S. France Spain U.K. Germany

Dollar 1 0 0 0 0

Pound 0 0 0 1 0

Euro 0 1 1 0 1

Fuzzy Relations

• Characteristic functions of crisp relations can be generalized to allow degrees of membership

• A fuzzy relation is a fuzzy set defined over the cartesian product of crisp sets

• Fuzzy relations can be defined using membership functions

• The membership grade denotes the strength of the relationship between the elements of the tuple

Example

• C = {NYC, Paris, Beijing, Madrid}

Distance (Km) NYC Paris Beijing Madrid

NYC 0 5850 11019 5779

Paris 5850 0 8238 1050

Beijing 11019 8238 0 9241

Madrid 5779 1050 9241 0

Projections of a Fuzzy Relation

• Let R be a fuzzy relation defined over X₁×X₂×…×X_N

• Let E = {X₁, X₂, …, X_N} and Y⊂E

• Let Y={X_i, X_j}, i, j ≤ N, i≠j

• We define S_Y(y), y∈X_i×X_j as:

{^{X Xi, j}}

(

) ^{

^}

S y y = z K z ∈ R z = y z = y

• Thus, we define the projection of a relation R over a subset Y⊂E as follows:

( )

( )

z SY y

R Y y R z

 ↓  = ∈

 

Example

• Let X₁={0, 1}, X₂={0, 1}, X₃={0, 1, 2}

• Let R be a fuzzy relation defined as follows:

X₁ X₂ X₃ R(x₁, x₂, x₃)

0 0 0 0.4

0 0 1 0.9

0 0 2 0.2

0 1 0 1.0

0 1 1 0

0 1 2 0.8

X₁ X₂ X₃ R(x₁, x₂, x₃)

1 0 0 0.5

1 0 1 0.3

1 0 2 0.1

1 1 0 0

1 1 1 0.5

1 1 2 1.0

• Calculate ^_^{R ↓}{^{X X1, 2}}^_

(

)

Cylindric Extensions of a Fuzzy Relation

• Cylindric extensions can be seen as inverse operations to projections

• Let E = {X₁, X₂, …, X_N} and Y⊂E

• Let R be a fuzzy relation defined over the cartesian product over all sets in Y

• The cylindric extension of R to set E-Y is defined as:

[ R ↑ − E Y ]( < x

, K , x

> = ) R y ( )

– If Y = {X_i, X_j} then y = < x_i, x_j >

Example

• Let X₁={0, 1}, X₂={0, 1}, X₃={0, 1, 2}

• Let R be a fuzzy relation defined as follows:

X₂ X₃ R(x₂, x₃)

0 0 0.5

0 1 0.9

0 2 0.2

1 0 1

1 1 0.5

1 2 1

• Calculate ^_^{R ↑} { }^{X1 }_

(

)

Cylindric Closure

• It is not always possible to recover the original relation from the cylindric extension of one of its projections

– Information is lost when a fuzzy relation is replaced by any of its projections

• Sometimes it can be reconstructed from the intersection of a set of its projections

– This intersection is called “the cylindric closure”

– Not always possible to fully recover the relation

Example

• It is not always possible to recover the original relation from the cylindric extension

• There is no guaranty that a cylindric closure exists, either

Exercise (Homework)

1. Calculate all the different projections over the relation in slide 6 (Km distances)

2. Calculate the cylindric extension for each projection

3. Determine if it is possible to recover the original relation (i.e., if a cylindric closure exists)

Binary Fuzzy Relations

• Binary relations are generalized mathematical functions

• The main difference:

– Relations may assign to each value of X two or more elements from Y

• Thus, some basic operations over functions also apply to binary relations

Domain of a Binary Fuzzy Relation

• We define the domain of a binary fuzzy relation R(X,Y) as the fuzzy set:

( )

( ) max , Dom R x y Y R x y

= ∈

• Example:

{ }^{0,1 ,} {^{0,1, 2}}

0 1 0 0.3 0.7

1 1 0.4

2 0.6 0

X Y

R

= =

 

 

=  

 

 

( ) 1 / 0 .7 / 1 Dom R x = +

Range of a Binary Fuzzy Relation

• The range of a binary fuzzy relation is defined as the fuzzy set:

( ) max ( , )

x X

Ran R y R x y

= ∈

• Example:

{ }^{0,1 ,} {^{0,1, 2}}

0 1 0 0.3 0.7

1 1 0.4

2 0.6 0

X Y

R

= =

 

 

=  

 

 

( ) .7 / 0 1/ 1 .6 / 2 Ran R x = + +

Height of a Binary Fuzzy Relation

• The height of R(x, y) is a number defined by

( ) max max ( , )

y Y x X

h R R x y

∈ ∈

=

• h(R) is the largest membership grade in the relation

{ }^{0,1 ,} {^{0,1, 2}}

0 1 0 0.3 0.7

1 1 0.4

2 0.6 0

X Y

R

= =

 

 

=  

 

 

( ) 1 h R =

Inverse of a Fuzzy Binary Relation

• The inverse of given fuzzy relation R is defined as follows

1( , ) ( , ) R⁻ y x = R x y

• Example:

{ }^{0,1 ,} {^{0,1, 2}}

0 1 0 0.3 0.7

1 1 0.4

2 0.6 0

X Y

R

= =

 

 

=  

 

 

1

0 1 2

0 0.3 1 0.6

1 0.7 0.4 0 R⁻ =  

 

Composition of Binary Fuzzy Relations

• Given two relations R₁(X, Y) and R₂(Y, Z) with a common set (Y), we define their standard

composition as:

[ ^{1 2}] ( ^{1 2} )

( , ) ( , ) max min ( , ), ( , R x z R R x z y Y R x y R y z

= o = ∈  

• Properties of THIS composition (max, min):

• Associative

• R^-1(z,x)=R₂^-1(z, y) • R₁^-1(y, x)

• It is not commutative!!!

Easing the calculation of compositions J

• Calculating a compound relation is just the same as performing matrix multiplication

– We just swap:

• The product for the min

• The sum for the max

• Example:

1

2

0 1 2 0 0.2 1 0 ( , )

1 0.4 0 0.7 1 3 0 0.2 0.1 ( , ) 1 0.4 0

2 1 0

R x y

R y z

 

=  

 

 

 

=  

1 3 0 0.4 0.1 ( , )

1 0.7 0.1 R x z =  

 