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 ( ) ⊆ × × × , , K , K R X X X X X X 1 2 1 2 N N • Crisp relations can be defined using characteristic functions  ∈ 1, , , K , iff x x x R ( ) =  1 2 N , , K , R x x x 1 2 N  0, otherwise • 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, M adrid} Distance (Km) NYC Paris Beijing M adrid NYC 0 5850 11019 5779 Paris 5850 0 8238 1050 Beijing 11019 8238 0 9241 M adrid 5779 1050 9241 0

Projections of a Fuzzy Relation • Let R be a fuzzy relation defined over X 1 ×X 2 ×… ×X N , X N } and Y ⊂ E • Let E = {X 1 , X 2 , … • Let Y={X i , X j }, i, j ≤ N, i ≠ j Y (y), y ∈ X i ×X j as: • We define S } ( ) { } = ∈ = = , , K , | , S y y z z R z y z y { 1 i j N i i j j , X X i j • Thus, we define the projection of a relation R over a subset Y ⊂ E as follows: ( ) ( )   ↓ = max R Y y R z   ( ) ∈ z S y Y

Example • Let X 1 ={0, 1}, X 2 ={0, 1}, X 3 ={0, 1, 2} • Let R be a fuzzy relation defined as follows: X 1 X 2 X 3 R(x 1 , x 2 , x 3 ) X 1 X 2 X 3 R(x 1 , x 2 , x 3 ) 0 0 0 0.4 1 0 0 0.5 0 0 1 0.9 1 0 1 0.3 0 0 2 0.2 1 0 2 0.1 0 1 0 1.0 1 1 0 0 0 1 1 0 1 1 1 0.5 0 1 2 0.8 1 1 2 1.0 } ( ) {   ↓ • Calculate , , R X X y y   1 2 1 2

Cylindric Extensions of a Fuzzy Relation • Cylindric extensions can be seen as inverse operations to projections , X N } and Y ⊂ E • Let E = {X 1 , X 2 , … • 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: ↑ − < > = [ ]( , K , ) ( ) R E Y x x R y 1 N – If Y = {X i , X j } then y = < x i , x j >

Example • Let X 1 ={0, 1}, X 2 ={0, 1}, X 3 ={0, 1, 2} • Let R be a fuzzy relation defined as follows: X 2 X 3 R(x 2 , x 3 ) 0 0 0.5 0 1 0.9 0 2 0.2 1 0 1 1 1 0.5 1 2 1 } ( ) {   ↑ • Calculate , , R X x x x   1 1 2 3

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 R x y ∈ y Y • Example: { } { } = = 0,1 , 0,1,2 X Y 0 1   0 0.3 0.7   Dom R x = + ( ) 1/ 0 .7 /1 = 1 1 0.4 R     2  0.6 0 

Range of a Binary Fuzzy Relation • The range of a binary fuzzy relation is defined as the fuzzy set: = ( ) max ( , ) Ran R y R x y ∈ x X • Example: { } { } = = 0,1 , 0,1,2 X Y 0 1   0 0.3 0.7   Ran R x = + + = ( ) .7 / 0 1/1 .6 / 2 1 1 0.4 R     2  0.6 0 

Height of a Binary Fuzzy Relation • The height of R(x, y) is a number defined by = ( ) max max ( , ) h R R x y ∈ ∈ y Y x X • h(R) is the largest membership grade in the relation { } { } = = 0,1 , 0,1,2 X Y 0 1   0 0.3 0.7 h R =   ( ) 1 = 1 1 0.4 R     2  0.6 0 

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

Composition of Binary Fuzzy Relations • Given two relations R 1 (X, Y) and R 2 (Y , Z) with a common set (Y), we define their standard composition as: [ ] ( ) = =   ( , ) o ( , ) max min ( , ), ( , R x z R R x z  R x y R y z  1 2 1 2 ∈ y Y • Properties of THIS composition (max, min): • Associative -1 (z, y) • R 1 • R -1 (z,x)=R 2 -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: 0 1 2   0 0.2 1 0 = ( , )   R x y 1 3 1 1  0.4 0 0.7    0 0.4 0.1 = ( , )   R x z 1 3 1  0.7 0.1    0 0.2 0.1   = ( , ) 1 0.4 0 R y z   2   2  1 0 