Fuzzy Logic and Design

7/31/2019 Fuzzy Logic and Design

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CS 288: Fuzzy Logic & Design

M.Tech. (Computer Science & Engg.)

Uttarakhand Technical University


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What is a fuzzy set?

Crisp Set: An element either belongs or does notbelong to a set.

Example People in a town divided into two sets set of all males and set of all females.

Fuzzy Set: An element belongs to a set with certaindegree of belongingness.

Example People in a town divided into two sets set of young people and set of old people.


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Crisp logic

Crisp logic: 2-level logic TRUE / FALSE or YES /NO or 1 / 0

That is, discrete binary truth value.

So, hard decision logic.

Crisp sets based on crisp logic.

But, crisp logic does not apply to most real-world

situations.

Example No precise boundary between young andold people.


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Concept of fuzzy logic

Fuzzy logic: Multi-level logic conceived by Zadehin 1965.

Continuous range of truth value, i.e., extent of truth

(or falsity) in the range 0 to 1

So, soft decision logic as much as the humanreasoning, e.g., very young, moderately young, notso young, and so on.

Fuzzy sets based on fuzzy logic.

Crisp logic (set) is a special case of fuzzy logic (set).


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Why fuzzy logic?

Limited precision.

A way of handling uncertainty.

Quantitative methods of handling qualitative issues.

To introduce human-like thinking in computers.

Applications:

Control systems Pattern recognition

Decision making


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Fuzzy set definition

IfXis a collection of objects denoted generically byx, then a fuzzy setA inXis a set of ordered pairsgiven as

X= Universal set

x= Set element

A = Fuzzy set;

= Membership grade;

Xxxx

AA )(

)(xA

XA

1,0)( xA


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Membership function

Membership grade is the degree ofbelongingness of elementxin the fuzzy setA.

That is, it is the measure of the extent by whichxsatisfies the property of fuzzy setA.

Membership function: User defined function forcalculating the membership grade

Probabilistic measure

Possibilistic measure

)(xA


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Membership function

Membership vs probability measure:

Membership grade = degree of belongingness

Probability measure = chance of belongingness

Example Probability thatxis young is 0.8;membership ofxin set of young people is 0.8

Multi-dimensional membership grade: Membership

grade based on multiple criteria

Example Membership grade for a person infuzzy set TALL depends on his height and age.

NA x 1,0)(


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Some definitions

Support: The support of a fuzzy setA in theuniversal setXis the crisp set that contains all theelements ofXthat have non-zero membership gradeinA.

Power set: The set of all possible fuzzy subsets ofX.

Empty fuzzy set: whose support is a null set.

Height: The height of a fuzzy set is the largestmembership grade attained by any element in theset.

Normalization: A fuzzy set is normalized when atleast one of its elements attain the maximumpossible membership grade which is generally one.


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Some definitions

-cut: An -cut of a fuzzy set is a crisp set that

contains all the elements whose membership gradeis at least equal to .

It implies

Level set: The crisp set of all membership gradevalues, including 0.

2121 ifAA

XxxAA somefor)(


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Some definitions

Convex fuzzy set: A fuzzy set is convex if and only ifeach of its -cuts is a convex set.

Scalar cardinality: Sum of the membership grades ofall elements.

Fuzzy cardinality: It is the fuzzy set defined as

AAA

~

]1,0[,)1(,)(),(min)( srxsrx AAA


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Operations on fuzzy sets

Set inclusion:

Set equivalence:

Proper subset:

Set complement

Set union

Set intersection

XxxxBA BA ),()(if

XxxxBA BA ),()(if

)()(such thatand

),()(if

xxx

XxxxBA

BA

BA


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Set complement

The complement operation is defined by a function

C: [0,1] [0,1] and the complement of a fuzzy set

A is given as

Axiomatic requirements:

C(0) = 1, C(1) = 0

Ifa < b then C(a) C(b) C(.) is a continuous function

C(.) is involutive, i.e., C( C(a) ) = a

xxC

AA )(


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Complement functions

Sugeno class:

Yager class:

Standard complement:

,1,

1

1)(

a

aaC

,0,1)( 1 WaaC WWW

1or0for,1)( WaaC


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Set union

The union operation is defined by a function

U: [0,1] [0,1] [0,1] and the union of fuzzy sets

A and B is given as


U(0,0) = 0, U(0,1) = U(1,0) = U(1,1) = 1

Commutative: U(a,b) = U(b,a) Monotonic: Ifa p, b q then U(a,b) U(p,q)

Associative: U( U(a,b), c) = U(a, U(b,c))

xxxUBABA )(),(


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Set union

U(.) is continuous function

Idempotent: U(a,a) = a

Union functions:

Standard: max(a,b)

Algebraic sum: a + b ab

Bounded sum: min(1, a+b)

Drastic union: Umax(a,b) = a,ifb=0= b, ifa=0

= 1, otherwise


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Set intersection

The intersection operation is defined by a function

i: [0,1] [0,1] [0,1] and the intersection of fuzzy

setsA and B is given as


i(1,1) = 1, i(0,0) = i(0,1) = i(1,0) = 0

Commutative:i(a,b) = i(b,a) Monotonic: Ifa p, b q then i(a,b) i(p,q)

Associative: i( i(a,b), c) = i(a, i(b,c))

xxxiBABA )(),(


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Set intersection

i(.) is continuous function

Idempotent: i(a,a) = a

Intersection functions: Standard: min(a,b)

Algebraic product: ab

Bounded difference: max(0, a+b1)

Drastic intersection: imin(a,b) = a,ifb=1

= b, ifa=1

= 0, otherwise


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Properties of fuzzy set operations

Commutative:

Associative:

Idempotent:

Distributive:

Identity:

ABBAABBA ;

)()()(

);()()(

CABACBA

CABACBA

AXAAA ;

CBACBA

CBACBA

)()(

;)()(

AAAAAA ;


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We have

Absorption:

De Morgans Laws:

Involution:

Equivalence formula:

XXAA ;

ABAAABAA )(;)(

BABABABA ;

AA

BABABABA


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Note the following which are different fromconventional crisp set:

(Law ofnon-contradiction)

(Law ofexcluded middle)

AA

XAA


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Fuzzy set decomposition

Representations of fuzzy sets by crisp sets

For every in the level set ofA, find the -cut.

To obtain backA:

From every -cut obtained above, form a fuzzy

set as

Then,

Ax

xAx

xA~

A

AA

~


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Extension principle

Mapping fuzzy subsets ofXto fuzzy subsets ofYviaa function f.

If more than one element maps to the sameelement yin Y, the maximum of the membershipgrades of these elements inA is taken as themembership grade ofyin f(A).

YAfxfxfxf

Af

XAxxxA

YXf

n

n

n

n

)(;)(

...)()(

)(

;...

:

2

2

1

1

2

2

1

1


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Extension principle

If no element inXis mapped to y, membershipgrade ofyis zero.

Let,

Then membership grade ofyin f(A1,A1, ,An) isequal to the minimum of the membership grades of

xk inAk, for k= 1 to n.

YyxxxfYXXXf

n

n

,.....,,,.....,,:

21

21


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Fuzzy arithmetic

Addition, subtraction, multiplication, division,maximum, minimum, exponentiation, logarithm aredefined.

Types of numbers:

Scalars integers, real numbers

Intervals exact value not known but the boundscan be established.

Fuzzy numbers uncertain numbers with a

knowledge of range of possible values and valuethat is more possible than others, e.g.,approximately 5.


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Fuzzy numbers

It is a fuzzy set with different degree of closeness toa crisp number.

Membership function ought to be normal and

convex.

All fuzzy set operations are applicable to fuzzynumbers

Intersection, union, -cuts, extension, etc.

Operation similar to arithmetic operations are alsoapplicable


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Linguistic variables

When fuzzy numbers are connected to linguisticconcepts, such as terms like very small, small,medium, and so on.

Linguistic variable characterized by:

Name of the variable

Set of linguistic terms

Universal set

Syntactic rule

Semantic rule


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Linguistic variables


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Interval number

For an intervalA = [a, b]:

Widthw(A) = b a

Magnitude |A| = max( |a|, |b| )

ImageA= [b, a]

InverseA1 = [1/b, 1/a]

For two intervalsA = [a, b] and P=[p, q]:

EqualityA = Pwhen a=p, b=q

InclusionA subset ofB ifp a b q

Distanced(A,B) = max( |ap|, |bq| )


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Arithmetic operations on intervals

For intervalsA and P, and operatorwe define

Division,A/ P, is not defined when 0 is an elementin P.

The result of an arithmetic operation on closedintervals is again a closed interval.

/,.,,*

PpAapaPA ,**


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Arithmetic operations on intervals

Addition:

Subtraction:

Multiplication:

Division:

Note that:

However,

],[ qbpaPA

],[ pbqaPAPA

qbpbqapaqbpbqapaPA .,.,.,.max,.,.,.,.min.

q

b

p

b

q

a

p

a

q

b

p

b

q

a

p

aPAPA ,,,max,,,,min./ 1

]1,1[/and]0,0[ AAAA

AAAA /1and0


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Properties of interval operations

Commutative

Associative

Identity

Distributive:

Sub-distributive:

Inclusion monotonicity:

CABACBACcBbcb

..).(,everyfor0.If

CABACBA ..).(

QPBAQPBA

QPBAQPBA

QBPA

//;..

;

,If


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Fuzzy number and fuzzy interval

A fuzzy number is a fuzzy set on

such that

A is normal (height(A) = 1)

-cut of A is a closed interval for all in the

range (0, 1]

The support of A is bounded

Since all -cuts are closed intervals, every fuzzynumber is a convex fuzzy set.

Membership function is continuous.

1,0: A


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Fuzzy number and fuzzy interval

Comparison of a real number and a crisp intervalwith a fuzzy number and a fuzzy interval.


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Arithmetic on fuzzy numbers

Definition based on cutworthiness:

Definition based on extension principle:

BAxxBA

BABA

**

**

1,0

)(),(minmax)(*

* yxz BAyxz

BA


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Addition:


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Subtraction:


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Multiplication:


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Division:


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MIN and MAX operators

For intervals:

For fuzzy numbers:

),max(),,max(),(MAX

),min(),,min(),(MIN

qbpaPA

qbpaPA

)(),(minsup)(,MAX

)(),(minsup)(,MIN

),max(

),min(

yBxAzBA

yBxAzBA

yxz

yxz


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MIN vs min operators


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MAX vs max operators


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Properties of MIN and MAX

Commutative

Associative

Idempotent

Absorption

Distributive


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Interval equations

A +X= P

X= PA is not the solution except when

A = [a, a]

The solution is

X= [p a, q b]

The above solution exists iff

p a q b


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Fuzzy number equations

The solution to a fuzzy equationA*X= B is obtainedby solving a set of interval equationsX, one foreach nonzero in the set

Final solution

The solution forA +X= B exists iff

p a q bwhereA = [a, b] and P = [p,

q], for all

p ap a q b q bfor

BA

]1,0(

Xxx

X


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Fuzzy number equations

Similarly, for fuzzy number equationsA.X= B, X=B /A is not the solution

Solution exists iff

p / a q / bwhereA = [a, b] and P = [p,

q], for all

p / a q / bp/ a q / bfor


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Crisp relation

Crisp relation:

Example:

X={dollar, pound, rupee}

Y= {USA, Canada, Britain, India}

Relation = (currency, country)

Then R = { (dollar,USA), (dollar, Canada),(pound,Britain), (rupee,India) }

YXRYyXxxRyyxR ,,,,


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Fuzzy relation

Fuzzy relation:

Example:

X={New York (NY), Paris (P)}

Y= {Beijing (B), New York (NY), London (L)}

Relation = Very far

Then R = { 1/(NY,B); 0.6/(NY,L); 0/(NY, NY);0.9/(P,B); 0.7/(P,NY); 0.3/(P,L) }

YXRYyXxyxRR

,,

,


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Some terminologies

Domain of a relation

Range of a relation

Height of a relation

-cut of a relation

Inverse of a relation


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Properties of relation

Reflexive

Else, irreflexive and -reflexive in fuzzy relation

Symmetric

Asymmetric and anti-symmetric

Transitive:

Max-min and max-product transitive in fuzzyrelation

Non-transitive and anti-transitive


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Properties of fuzzy relation

xyyxXyx

yxxyyx

Xyxxyyx

xxXx

Xxxx

Xxxx

Xxxx

RR

RR

RR

R

R

R

R

,,,,:Asymmetric

0,,0,:symmetric-Anti

,,,,:Symmetric

0,,:eIrreflexiv

,1,:reflexive-Anti

,0,:reflexive-

,1,:Reflexive


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Properties of fuzzy relation

Non-transitive: neither transitive nor anti-transitive.

Xzxzyyxzx

Xzxzyyxzx

Xzxzyyxzx

Xzxzyyxzx

RRXy

R

RRXy

R

RRXyR

RRXy

R

,,,,max,

:transitive-antiproduct-Max

,,,,max,

:ansitiveproduct tr-Max,,,,,minmax,

:transitive-antimin-Max

,,,,,minmax,

:tivemin transi-Max


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Poset and lattice

Partial order: reflexive, anti-symmetric andtransitive.

Partially ordered set (poset)

Maximal and minimal elements

Greatest and least elements

Upper and lower bounds of subsets

Greatest lower bound

Least upper bound Lattice: A poset whose every 2-element subset has

GLB and LUB in the poset.


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Fuzzy measures

Fuzzy measure assigns a value [0, 1] to each crispsubsets of the universal set signifying the degree ofevidence or belief that a particular element belongsto the subset.

Axioms:

Boundary condition:

Monotonicity:

1,0 Xgg

BgAgBA


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Plausibility measure

Associated with each belief measure is a plausibilitymeasure defined as

Alternatively,

APlABel

ABelABelAPl

1

1

1

....)1(..........

....

21

1

21

APlAPl

AAAPl

AAPlAPlAAAPl

n

n

ji

ji

i

in


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Probability measure

Probability assignment is a mapping function:

m: P(X) [0, 1]

such that m() = 0 and P(X)m(A) = 1

Observations:

Not necessarily m(X)=1

Not necessarily m(A) m(B) when setA issubset or equal to set B.

No relationship between m(A) and m() required.


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Probability measure

Computation of belief and plausibility:

Focal element: Ifm(A) > 0, thenA is called a focalelement ofm.

Fis the set of focal elements.

(F,m) is called body of evidence.

ABAB

BmAPlBmABel )()()()(


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Probability measure

Total ignorance:

Single support function: m is a single support

function focused atA if

AAPlPl

XAABelXBel

XAXPAAmXm

1)(,0)(

0)(,1)(

),(0)(,1)(

ABXBBm

sXmsAm

,,0)(

1)(,)(


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Probability measure

Combining evidence: standard method (Dempstersrule of combination)

AAm

ACmBm

CmBm

Am

CB

ACB

,0)(

,)()(1

)()(

)(

12

21

21

12


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Possibility measure

Possibility measure is particular cases of plausibilityand belief measures.

The focal elements of a body of evidence are

nested. The associated belief and plausibility measures are

called consonants.

Properties:

)(),(max

)(),(min

BPlAPlBAPl

BBelABelBABel

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Fuzzy Logic and Design