Fuzzy Logic

By Manoj Harsule

OverviewOverview

• A Little History• Fuzzy Logic – A Definition• Fuzzy set theory• Introduction to fuzzy set• Fuzzy Relations

A little History

In the 1960’s Lotfi A. Zadeh Ph.D,. University of California, Berkeley, published an obscure paper on fuzzy sets . His unconventional theory allowed for approximate information and uncertainty when generating complex solutions; a process that previously did not exist.

Fuzzy Logic has been around since the mid 60’s but was not readily excepted until the 80’s and 90’s. Although now prevalent throughout much of the world, China, Japan and Korea were the early adopters

WHAT IS FUZZY LOGIC?WHAT IS FUZZY LOGIC?

Definition of fuzzy

Fuzzy – “not clear, distinct, or not precise;

uncertain”

Definition of fuzzy logic

A form of knowledge representation suitable for

notions that cannot be defined precisely, but which

depend upon their contexts.

TRADITIONAL REPRESENTATION OF LOGIC

Slow (Low) Fast (High)

Speed = 0 Speed = 1

FUZZY LOGIC FUZZY LOGIC REPRESENTATIONREPRESENTATION

For every problem must represent in terms of fuzzy sets.

Slowest

Fastest

Slow

Fast

[ 0.0 – 0.25 ]

[ 0.25 – 0.50 ]

[ 0.50 – 0.75 ]

[ 0.75 – 1.00 ]

Introduction to Introduction to Fuzzy Set TheoryFuzzy Set Theory

Fuzzy SetsFuzzy Sets

Types of UncertaintyTypes of Uncertainty

• Stochastic uncertaintyStochastic uncertainty– E.g., rolling a diceE.g., rolling a dice

• Linguistic uncertaintyLinguistic uncertainty– E.g., low price, tall people, young ageE.g., low price, tall people, young age

• Informational uncertaintyInformational uncertainty– E.g., credit worthiness, honestyE.g., credit worthiness, honesty

Crisp or Fuzzy LogicCrisp or Fuzzy Logic

• Crisp LogicCrisp Logic– A proposition can be A proposition can be truetrue oror falsefalse only.only.

• Ajay is a student (true)Ajay is a student (true)• Smoking is healthy (false)Smoking is healthy (false)

– The degree of truth is The degree of truth is 0 or 10 or 1..

• Fuzzy LogicFuzzy Logic– The degree of truth is The degree of truth is between 0 and 1between 0 and 1..

• Raj is young (0.3 truth) Raj is young (0.3 truth) • Amol is smart (0.9 truth) Amol is smart (0.9 truth)

Crisp SetsCrisp Sets

• Classical sets are called crisp setsClassical sets are called crisp sets– either an element either an element belongsbelongs to a set or to a set or

not, i.e.,not, i.e.,

• Member Function of crisp setMember Function of crisp set

x A or x A

0( )

1A

x Ax

x A

( ) 0,1A x

P

Crisp SetsCrisp Sets

P : the set of all people.

Y : the set of all young people. YY

( ) 25,ageYoung y y x x P

1

y

( )Young y

25

Fuzzy SetsFuzzy SetsCrisp sets ( ) 0,1A x

( ) [0,1]A x

1

y

( )Young y

Example

Definition:Fuzzy Sets and Membership Functions

If U is a collection of objects denoted generically by x, then a fuzzy set A in U is defined as a set of ordered pairs:

( , ( ))AA x x x U membership

function

: [0,1]A U

U : universe of discourse.

Example (Discrete Universe)

{1,2,3,4,5,6,7,8}U # courses a student may take in a semester.

(1,0.1) (2,0.3) (3,0.8) (4,1)

(5,0.9) (6,0.5) (7,0.2) (8,0.1)A

#appropriate courses taken

0.5

1

02 4 6 8

x : # courses

( )A x

Example (Discrete Universe)

{1,2,3,4,5,6,7,8}U # courses a student may take in a semester.

(1,0.1) (2,0.3) (3,0.8) (4,1)

(5,0.9) (6,0.5) (7,0.2) (8,0.1)A

appropriate # courses taken

Alternative Representation:

1 2 3 40.1/ 0.3 / 0.8 / 1.0 / 0.9 / 0.5 / 0.2 / 0.1/5 6 7 8A

Example (Continuous Universe)

possible agesU : the set of positive real numbers

( , ( ))BB x x x U

4

1( )

501

5

B xx

about 50 years old

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100

x : age

( )B x

4505

1

1 xRB x

Alternative Representation:

Alternative NotationAlternative Notation

( , ( ))AA x x x U

U : discrete universe

U : continuous universe

( ) /i

A i ix U

A x x

( ) /AUA x x

Note that and integral signs stand for the union of membership grades; “

/ ” stands for a marker and does not imply division.

Mem

bers

hip

valu

e

height

1

0

Membership Functions (MF’s)

• A fuzzy set is completely characterized A fuzzy set is completely characterized by a membership function.by a membership function.– a a subjectivesubjective measure. measure.– notnot a a probabilityprobability measure. measure.

“tall” in Asia

“tall” in USA“tall” in Aus

5’10”

Fuzzy Partition

• Fuzzy partitions formed by the Fuzzy partitions formed by the linguisticlinguistic values “values “youngyoung”, “”, “middle agedmiddle aged”, and “”, and “oldold”:”:

Introduction to Fuzzy Set Theory

Set-Theoretic Operations

Set-Theoretic OperationsSet-Theoretic Operations

• SubsetSubset

• ComplementComplement

• UnionUnion

• IntersectionIntersection

( ) ( ), A BA B x x x U

( ) max( ( ), ( )) ( ) ( )C A B A BC A B x x x x x

( ) min( ( ), ( )) ( ) ( )C A B A BC A B x x x x x

( ) 1 ( )AAA U A x x

Set-Theoretic Operations

A BA B

A BA B

AA

A BA B

PropertiesProperties

A AInvolutionA B B A

Commutativity A B B A

A B C A B C Associativity A B C A B C

A B C A B A C Distributivity A B C A B A C

A A A Idempotence A A A

A A B A Absorption A A B A

A B A B De Morgan’s laws

A B A B

Properties

• The following properties are The following properties are invalidinvalid for fuzzy sets:for fuzzy sets:

– The laws of contradictionThe laws of contradiction

– The laws of excluded middleThe laws of excluded middle

A A

A A U

Other Definitions for Set Operations

• UnionUnion

• IntersectionIntersection

( ) min 1, ( ) ( )A B A Bx x x

( ) ( ) ( )A B A Bx x x

Other Definitions for Set Operations

• UnionUnion

• IntersectionIntersection

( ) ( ) ( ) ( ) ( )A B A B A Bx x x x x

( ) ( ) ( )A B A Bx x x

Generalized Union/Intersection

• Generalized IntersectionGeneralized Intersection

• Generalized UnionGeneralized Union

t-norm

t-conorm

T-Norm

:[0,1] [0,1] [0,1]T

Or called triangular norm.

1.1. SymmetrySymmetry

2.2. AssociativityAssociativity

3.3. MonotonicityMonotonicity

4.4. Border ConditionBorder Condition

( , ) ( , )T x y T y x

( ( , ), ) ( , ( , ))T T x y z T x T y z

1 2 1 2 1 1 2 2, ( , ) ( , )x x y y T x y T x y

( ,1)T x x

T-ConormT-Conorm

:[0,1] [0,1] [0,1]S

Or called s-norm.

1.1. SymmetrySymmetry

2.2. AssociativityAssociativity

3.3. MonotonicityMonotonicity

4.4. Border ConditionBorder Condition

( , ) ( , )S x y S y x

( ( , ), ) ( , ( , ))S S x y z S x S y z

1 2 1 2 1 1 2 2, ( , ) ( , )x x y y S x y S x y

( ,0)S x x

Fuzzy Relations

Review Fuzzy Relations

Aa1

a2

a3

a4

B

b1

b2

b3

b4

b5

Binary Relation (Binary Relation (RR))R A B

1 1 1 3 2 5

3 1 3 4 4 2

( , ), ( , ), ( , )

( , ), ( , ), ( , )

a b a b a bR

a b a b a b

1 0 1 0 0

0 0 0 0 1

1 0 0 1 0

0 1 0 0 0

RM

1 1a Rb 1 3a Rb 2 5a Rb

3 1a Rb 3 4a Rb 4 2a Rb

The Real-Life Relation

• x is close to y – x and y are numbers

• x depends on y– x and y are events

• x and y look alike– x and y are persons or objects

• If x is large, then y is small– x is an observed reading and y is a corresponding

action

Fuzzy RelationsFuzzy Relations

A fuzzy relation R is a 2D MF:

( ,,( , ) ( , )) |R x yx y x yR X Y

Example (Approximate Equal)

( ,,( , ) ( , )) |R x yx y x yR X Y

{1,2,3,4,5}X Y U

1 0.8 0.3 0 0

0.8 1 0.8 0.3 0

0.3 0.8 1 0.8 0.3

0 0.3 0.8 1 0.8

0 0 0.3 0.8 1

RM

1 0

0.8 1( , )

0.3 2

0

R

u v

u vu v

u v

otherwise

A fuzzy relation defined on X an Z.

Max-Min Composition

X Y ZR: fuzzy relation defined on X and Y.

S: fuzzy relation defined on Y and Z.

R 。 S: the composition of R and S.

( , ) max min ( , ), ( , )R S y R Sx z x y y z

( , ) ( , )y R Sx y y z

Example

1 0.1 0.2 0.0 1.0

2 0.3 0.3 0.0 0.2

3 0.8 0.9 1.0 0.4

R a b c d0.9 0.0 0.3

0.2 1.0 0.8

0.8 0.0 0.7

0.4 0.2 0.3

S

a

b

c

d

1 0.4 0.2 0.3

2 0.3 0.3 0.3

3 0.8 0.9 0.8

R S

0.1 0.2 0.0 1.00.9 0.2 0.8 0.4min0.1 0.2 0.0 0.4max

( , ) max min ( , ), ( , )S R v R Sx y x v v y

Max-Product CompositionMax-Product Composition

( , ) max ( , ) ( , )R S y R Sx z x y y z

A fuzzy relation defined on X an Z.

X Y ZR: fuzzy relation defined on X and Y.

S: fuzzy relation defined on Y and Z.

R 。 S: the composition of R and S.

Max-min composition is not mathematically tractable, therefore other compositions such as max-product composition have been suggested.

ProjectionProjectionR

XR R X YR R Y

Dimension Reduction

ProjectionProjectionR

XR R X YR R Y

R

XR R X YR R Y

XR R X YR R Y max ( , ) /RX y

x y xmax ( , ) /RY xx y y

( ) max ( , )YR R

xy x y ( ) max ( , )

XR Ry

x x y

Dimension Reduction

Cylindrical Extension Cylindrical Extension Dimension Expansion

A : a fuzzy set in X.

C(A) = [AXY] : cylindrical extension of A.

( ) ( ) | ( , )AX YC A x x y

( ) ( , ) ( )C A Ax y x

Types of Fuzzy Relations

• Reflexive– Irreflexive– Antireflexive– Epsilon Reflexive

• Symmetric– Asymmetric– Antisymmetric

XxxxR allfor 1),(

XxxxR somefor 1),(

XxxxR allfor 1),(

XxxxR allfor ),(

XxxyRyxR allfor ),(),(

XxxyRyxR somefor ),(),(

XyxyxxyRyxR , allfor 0 ),( and 0),(

Types of Fuzzy Relations

• Transitive (max-min transitive)

– Non-transitive: For some (x,z), the above do not satisfy.– Antitransitive:

• Example: X = Set of cities, R=“very far” Reflexive, symmetric, non-transitive

X allfor )],(),,(min[max),(

x,zzyRyxRzxRYy

X allfor )],(),,(min[max),(

x,zzyRyxRzxRYy

Types of Fuzzy Relations

• Transitive Closure– Crisp: Transitive relation that contains R(X,X)

with fewest possible members– Fuzzy: Transitive relation that contains

R(X,X) with smallest possible membership– Algorithm:

TRR

RRRR

RRRR

'

''

'

:Stop 3.

1 step togo and make , If .2

).( .1

Types of Fuzzy Relations

• Fuzzy Equivalence or Similarity Relation– Reflexive, symmetric, and transitive– Decomposition:

– Partition Tree[0,1]}|) ({(R)

:partitions ofSet

relation. eequivalenc crisp a is

]1,0[

R

R

RR

Types of Fuzzy Relations

• Fuzzy Compatibility or Tolerance Relation– Reflexive and symmetric– Maximal compatibility class and complete cover

• Compatibility class• Maximal compatibility class: largest compatibility class• Complete cover: Set of maximal compatibility classes

– Maximal alpha-compatibility class– Complete alpha-covers– Note: Relation from distance metrics forms tolerance

relation in clustering.

Ryx,such that of Subset XA

BibliographyBibliography

• J. R. Jang, C. Sun, E. Mizutani, “Neuro-J. R. Jang, C. Sun, E. Mizutani, “Neuro-Fuzzy and Soft Computing: A Fuzzy and Soft Computing: A Computational Approach to Learning and Computational Approach to Learning and Machine Intelligence, Prentice HallMachine Intelligence, Prentice Hall

• Slides and notes: Slides and notes: http://equipe.nce.ufrj.br/adriano/fuzzy/bibliogr-ic.htm