M odels for Inexact Reasoning Fuzzy Logic Lesson 5 Fuzzy Relations - - PowerPoint PPT Presentation

M odels for Inexact Reasoning Fuzzy Logic – Lesson 5 Fuzzy Relations

M aster 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, M ark}, W = {M ary, Sonya} – John is M ary’s husband, Sonya is M ark’s wife

Crisp Relations

• A relation among crisp sets is a crisp subset

( )

1 2 1 2

, , ,

N N

R X X X X X X ⊆ × × × K K

• Crisp relations can be defined using characteristic

functions

( )

1 2 1 2

1, , , , , , , 0,

N N

iff x x x R R x x x

• therwise

 ∈ =   K K

• Tuples in the relation identify elements related to
• ne 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 Pound 1 Euro 1 1 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, M adrid}

Distance (Km) NYC Paris Beijing M adrid NYC 5850 11019 5779 Paris 5850 8238 1050 Beijing 11019 8238 9241 M adrid 5779 1050 9241

Projections of a Fuzzy Relation

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

×XN

• Let E = {X1, X2, …

, XN} and Y⊂E

• Let Y={Xi, Xj}, i, j ≤ N, i≠j
• We define S

Y(y), y∈Xi×Xj as:

{ } (

) {

}

1 ,

, , , | ,

i j

i j N i i j j X X

S y y z z R z y z y = ∈ = = K

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

( )

( )

( )

max

Y

z S 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) 0.4 1 0.9 2 0.2 1 1.0 1 1 1 2 0.8 X1 X2 X3 R(x1, x2, x3) 1 0.5 1 1 0.3 1 2 0.1 1 1 1 1 1 0.5 1 1 2 1.0

• Calculate

{ } (

)

1 2 1 2

, , R X X y y   ↓  

Cylindric Extensions of a Fuzzy Relation

• Cylindric extensions can be seen as inverse
• perations to projections
• Let E = {X1, X2, …

, XN} 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:

1

[ ]( , , ) ( )

N

R E Y x x R y ↑ − < > = K

– If Y = {Xi, Xj} then y = < xi, xj >

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) 0.5 1 0.9 2 0.2 1 1 1 1 0.5 1 2 1

• Calculate

{ } (

)

1 1 2 3

, , R X x x x   ↑  

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
• riginal 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 ,

y Y

Dom R x R x y

∈

=

• Example:

{ } { }

0,1 , 0,1,2 1 0.3 0.7 1 1 0.4 2 0.6 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 1 0.3 0.7 1 1 0.4 2 0.6 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 1 0.3 0.7 1 1 0.4 2 0.6 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 1 0.3 0.7 1 1 0.4 2 0.6 X Y R = =     =      

1

1 2 0.3 1 0.6 1 0.7 0.4 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 ( , ), ( ,

y Y

R x z R R x z R x y R y z

∈

= =    

• 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

1 2 0.2 1 ( , ) 1 0.4 0.7 1 3 0.2 0.1 ( , ) 1 0.4 2 1 R x y R y z   =         =       1 3 0.4 0.1 ( , ) 1 0.7 0.1 R x z   =    