part 6 fuzzy logic 1. classical logic 2. multivalued logics 3. fuzzy propositions 4. fuzzy...

PART 6Fuzzy Logic

1. Classical logic2. Multivalued logics3. Fuzzy propositions4. Fuzzy quantifiers5. Linguistic hedges

FUZZY SETS AND

FUZZY LOGICTheory and Applications

6. Inference from conditional fuzzy propositions7. Inference from conditional and qualified propositions8. Inference from quantified propositions

FUZZY SETS AND

FUZZY LOGICTheory and Applications

33

Classical logic

• Inference rules Various forms of tautologies can be used for

making deductive inferences. They are referred to as inference rules. Examples :

.syllogism) cal(hypotheti )())()((

tollens),(modus ))((

ponens), (modus ))((

cacbba

abab

bbaa

44

Classical logic

• Existential quantifier Existential quantification of a predicate P(x) is

expressed by the form

"There exists an individual x (in the universal set X of the variable x) such that x is P". We have the following equality:

),()( xPx

)()()( xPxPxXx

V

55

Classical logic

• Universal quantifier Universal quantification of a predicate P(x) is

expressed by the form.

“For every individual x (in the universal set) x is P". Clearly, the following equality holds:

),()( xPx

( ) ( ) ( )x X

x P x P x

66

Classical logic

• General quantifier Q

The quantifier Q applied to a predicate P(x),

x X, as a binary relation

where α, β specify the number of elements of X for which P(x) is true or false, respectively. Formally,

|},| ,|){( Xβαα, βα, β NQ

.|}false is )(|{|

|,} trueis )(|{|

xPXx

xPXx

77

Multivalued logics

88

Multivalued logics

• n-valued logics For any given n, the truth values in these

generalized logics are usually labelled by rational numbers in the unit interval [0, 1]. The set Tn of truth values of an n-valued logic is thus defined as

These values can be interpreted as degrees of truth.

.11

1,

1

2 ,

1

2 ,

1

1 ,

1

00

n

n

n

n

nnnTn

99

Multivalued logics

Lukasiewicz uses truth values in Tn and defines the primitives by the following equations:

.||1

),1 ,1min(

), ,max(

), ,min(

,1

baba

abba

baba

baba

aa

1010

Multivalued logics

Lukasiewicz, in fact, used only negation and implication as primitives and defined the other logic operations in terms of these two primitives as follows:

).()(

,

,)(

abbaba

baba

bbaba

1111

Fuzzy propositions

• Unconditional and unqualified proposition The canonical form of fuzzy propositions of this

type, p, is expressed by the sentence

where V is a variable that takes values v from some universal set V, and F is a fuzzy set on V that represents a fuzzy predicate, such as tall, expensive, low, normal, and so on.

, is : Fp V

1212

Fuzzy propositions

Given a particular value of V (say, v), this value belongs to F with membership grade F(v). This membership grade is then interpreted as the degree of truth, T(p), of proposition p. That is,

for each given particular value v of variable V in proposition p. This means that T is in effect a fuzzy set on [0,1], which assigns the membership grade F(v) to each value v of variable V.

)()( vFpT

1313

Fuzzy propositions

1414

Fuzzy propositions

In some fuzzy propositions, values of variable V are assigned to individuals in a given set / . That is, variable V becomes a function V : / → V, where V ( i ) is the value of V for individual i in V. The canonical form must then be modified to the form

. where, is )( : IiFip V

Fp is :V

1515

Fuzzy propositions

• Unconditional and qualified proposition Propositions p of this type are characterized by

either the canonical form

or the canonical form

(8.8) , is } is Pro{ : PFp V

, is is : SFp V

1616

Fuzzy propositions

In general, the degree of truth, T(p), of any truth-qualified proposition p is given for each v V by the equation

An example of a truth-qualified proposition is the proposition "Tina is young is very true."

.))(()( vFSpT

1717

Fuzzy propositions

1818

Fuzzy propositions

Let us discuss now probability-qualified propositions of the form (8.8). For any given probability distribution f on V, we have

and, then, the degree T(p) to which proposition p of the form (8.8) is true is given by the formula

;)()(} is Pro{

Vv

vFvfFV

))()(()(

Vv

vFvfPpT

1919

Fuzzy propositions

As an example, let variable V be the average daily temperature t in °F at some place on the Earth during a certain month. Then, the probability-qualified proposition

p : Pro { temperature t (at given place and time) is around 75 °F } is likely

may provide us with a meaningful characterization of one aspect of climate at the given place and time.

2020

Fuzzy propositions

2121

Fuzzy propositions

• Conditional and unqualified proposition Propositions p of this type are expressed by the

canonical form

where X, Y are variables whose values are in sets X, Y, respectively, and A, B are fuzzy sets on X, Y, respectively.

, is then , is If : BAp YX

2222

Fuzzy propositions

These propositions may also be viewed as propositions of the form

where R is a fuzzy set on X x Y that is determined for each x X and each y Y by the formula

where J denotes a binary operation on [0, 1] representing a suitable fuzzy implication.

, is , RYX

,)]( ),([) ,( yBxAyxR J

2323

Fuzzy propositions

Here, let us only illustrate the connection for one particular fuzzy implication, the Lukasiewicz implication

This means, for example, that T(p) = 1 when X = x1 and Y = y1; T(p) = .7 when X = x2 and Y = y1 and so on.

).1 ,1min() ,( baba J

.,1,5.

,1,7.,1,1Then

.15. and 18.1.Let

2313

22122111

21321

yxyx

yxyxyxyxR

yyBxxxA

2424

Fuzzy propositions

• Conditional and unqualified proposition Propositions of this type can be characterized by

either the canonical form

or the canonical form

where Pro {X is A | Y is B} is a conditional probability.

, is is then , is If : SBAp YX

, is } is | is Pro{ : PBAp YX

2525

Fuzzy quantifiers

• First Kind - Ⅰ There are two basic forms of propositions that

contain fuzzy quantifiers of the first kind. One of them is the form

where V is a variable that for each individual i in a given set / assumes a value V(i), F is a fuzzy set defined on the set of values of variable V, and Q is a fuzzy number on R.

, is such that in are There: F(i)Ii'sQp V

2626

Fuzzy quantifiers

Any proposition p of this form can be converted into another proposition, p', of a simplified form,

where E is a fuzzy set on a given set / that is defined by the composition

, are There:' E'sQp

. allfor ))(()( IiiFiE V

2727

Fuzzy quantifiers

For example, the proposition

p : "There are about 10 students in a given class whose fluency in English is high“

can be replaced with the proposition

p’ : "There are about 10 high-fluency English-speaking students in a given class."

Here, E is the fuzzy set of "high-fluency English-

speaking students in a given class."

2828

Fuzzy quantifiers

Proposition p' may be rewritten in the form

where W is a variable taking values in R that represents the scalar cardinality, W = |E|,

and,

, is :' Qp W

IiIi

iFiEE ))(()(|| V

|).(|)'()( EQpTpT

2929

Fuzzy quantifiers

Example :

p : There are about three students in / whose fluency in English, V( i ), is high.

Assume that / = {Adam, Bob, Cathy, David, Eve}, and V is a variable with values in the interval [0, 100] that express degrees of fluency in English.

3030

Fuzzy quantifiers

3131

Fuzzy quantifiers

• First Kind - Ⅱ Fuzzy quantifiers of the first kind may also

appear in fuzzy propositions of the form

where V1, V2 are variables that take values from sets V1, V2, respectively, / is an index set by which distinct measurements of variables V1,V2 are identified (e.g., measurements on a set of individuals or measurements at distinct time instants), Q is a fuzzy number on R, and F1, F2 are fuzzy sets on V1, V2 respectively.

, is )( and is )(such that in are There: 21 FiFiIi'sQp 21 VV

3232

Fuzzy quantifiers

Any proposition p of this form can be expressed in a simplified form,

where E1, E2 are

, :' 21 'sE'sEQp

. allfor ))(()(

))(()(

2 22

1 11

IiiFiE

iFiE

V

V

3333

Fuzzy quantifiers

Moreover, p’ may be interpreted as

we may rewrite it in the form

where W is a variable taking values in R and W = | E1 ∩ E2|.

.) and ( are There :' 21 'sEEQp

, is :' Qp W

3434

Fuzzy quantifiers

Using the standard fuzzy intersection, we have

Now, for any given sets E1 and E2 ,

.)()'()( WQpTpT

,))](( )),((min[ 2 21 1

Ii

iFiF VVW

3535

Fuzzy quantifiers

• Second Kind These are quantifiers such as "almost all,"

"about half," "most," and so on. They are represented by fuzzy numbers on the unit interval [0, 1].

Examples of some quantifiers of this kind are shown in Fig. 8.5.

3636

Fuzzy quantifiers

3737

Fuzzy quantifiers

Fuzzy propositions with quantifiers of the second kind have the general form

where Q is a fuzzy number on [0, 1], and the meaning of the remaining symbols is the same as previously defined.

, is )(such that in

are there is )(such that Iin Among:

22

11

FiIQi's

Fii'sp

V

V

3838

Fuzzy quantifiers

Any proposition of the this form may be written in a simplified form,

where E1, E2 are fuzzy sets on X defined by

, are :' 21 'sQE'sQEp

. allfor ))(()(

,))(()(

2 22

1 11

IiiFiE

iFiE

V

V

3939

Fuzzy quantifiers

we may rewrite p’ in the form

where

for any given sets E1 and E2.

, is :' Qp W

.||

||

1

21

E

EE W

Ii

Ii

iF

iFiF

))((

))](( )),((min[

1 1

2 21 1

V

VVW

4040

Linguistic hedges

• Linguistic hedges Given a fuzzy predicate F on X and a modifier h

that represents a linguistic hedge H, the modified fuzzy predicate HF is determined for each x X by the equation

This means that properties of linguistic hedges can be studied by studying properties of the associated modifiers.

)).(()( xFhxHF

4141

Linguistic hedges

Every modifier h satisfies the following conditions:

ns.compositio are so then k),strong(wea are and both

if moreover, and, modifiers also are with and

with of nscompositio ,modifier another given .4

versa; viceand weak is then strong, is if .3

function; continuous a is .2

.1)1( and 0)0( .1

1

gh

gh

hgg

hh

h

hh

4242

Linguistic hedges

A convenient class of functions that satisfy these conditions is the class

where α R+ is a parameter by which individual modifiers in this class are distinguished and a [0, 1]. When α < 1, hα is a weak modifier; when α > 1, hα is a strong modifier; h1 is the identity modifier.

,)( aah

43

Inference from conditional propositions

For classical logic

Assume that the variables are related by an arbitrary relation on X × Y, not necessarily a function.

Given X = u and a relation R, we can infer that Y B, where B = { y Y |<x, y> R } (Fig. 8.7a).

Similarly, given X A, we can infer that Y B, where B = { y Y |<x, y> R, x A } (Fig. 8.7b).

44

Inference from conditional propositions

Observe that this inference may be expressed equally well in terms of characteristic functions XA, XB, XR of sets A, B, R respectively, by the equation

. allfor )] ,( ),(min[sup)( Yyyxxy RAXx

B

XXX

45

Inference from conditional propositions

46

Inference from conditional propositions

For fuzzy logic Assume that R is a fuzzy relation on X x Y, and A', B' are fuzzy sets on X and

Y, respectively. Then, if R and A' are given, we can obtain B' by the equation

which is a generalization by replacing the

, allfor )] ,( ),('min[sup)(' YyyxRxAyBXx

47

Inference from conditional propositions

characteristic functions with the corresponding membership functions.

It can also be written in the matrix form as

called the compositional rule of inference.

, '' RAB

48

Inference from conditional propositions

49

Inference from conditional propositions

Viewing proposition p as a rule and proposition q as a fact, the generalized modus ponens is expressed by the following schema:

B'

A'

BA

is :Conclusion

is :Fact

is then , is If :Rule

Y

X

YX

50

Inference from conditional propositions

Example 8.1

51

Inference from conditional propositions

52

Inference from conditional propositions

Another inference rule in fuzzy logic, which is a generalized modus tollens, is expressed by the following schema:

In this case, the compositional rule of inference has the form

A'

B'

BA

is :Conclusion

is :Fact

is then , is If :Rule

X

Y

YX

)]. ,( ),('min[sup)(' yxRyBxAYy

53

Inference from conditional propositions

Example 8.2

54

Inference from conditional propositions

The generalized hypothetical syllogism is expressed by the following schema:

X, Y, Z are variables taking values in sets X, Y, Z, respectively, and A, B,C are fuzzy sets on sets X, Y, Z, respectively.

CA

CB

BA

is then , is If :Conclusion

is then , is If :2 Rule

is then , is If :1 Rule

ZX

ZY

YX

55

Inference from conditional propositions

Given R1, R2, R3, obtained by these equations, we say that the generalized hypothetical syllogism holds if

which again expresses the compositional rule of inference. This equation may also be written in the matrix form

)]. ,( ), ,(min[sup) ,( 213 zyRyxRzxRYy

.213 RRR

56

Inference from conditional propositions

Example 8.3

57

Inference from conditional and qualified propositions

• Given a conditional and qualified fuzzy proposition p of the form

p : If X is A, then Y is B is S, (8.46)

where S is a fuzzy truth qualifier, and a fact is in the form "X is A’," we want to make an inference in the form “Y is B’."

58

Inference from conditional and qualified propositions

• The method of truth-value restrictions is based on a manipulation of linguistic truth values. It involves the following four steps.

59

Inference from conditional and qualified propositions

60

Inference from conditional and qualified propositions

• Example 8.4

61

Inference from conditional and qualified propositions

62

Inference from conditional and qualified propositions

63

Inference from conditional and qualified propositions

• Theorem 8.1 Let a fuzzy proposition of the form (8.46) be given,

where S is the identity function (i.e., S stands for true), and let a fact be given in the form "X is A'," where

for all and some x0 such that A(x0)= . Then, the inference “Y is B‘ " obtained by the method of truth-value restrictions is equal to the one obtained by the generalized modus ponens, provided that we use the same fuzzy implication in both inference methods.

)(')('sup 0)(:

xAxAaxAx

]1 ,0[a a

64

Inference from qualified propositions

• Given n quantified fuzzy propositions of the form

where Qi is either an absolute quantifier or a relative quantifier, and Wi is a variable compatible with the quantifier Qi, what can we infer from these propositions?

),( is : niii ip NQW

65

Inference from qualified propositions

• Quantifier extension principle Assume that the prospective inference is expressed in

terms of a quantified fuzzy proposition of the form

The principle states the following: if there exists a function f : Rn →R such that W = f (W1, W2, …, Wn) and Q = f (Q1, Q2, …, Qn), where the meaning of f (Q1, Q2, …, Qn) is defined by the extension principle, then we may conclude that p follows from p1, p2, …, pn.

. is : QWp

66

Inference from qualified propositions

• Intersection / product syllogism

where E, F, G are are fuzzy sets on a universal set X, Q1 and Q2 are relative quantifiers (fuzzy numbers on [0, 1]), and Q1 • Q2 is the arithmetic product of the quantifiers.

'sGFE'sp

G's'sFEp

F'sE'sp

) and ( are :

are ) and ( :

are :

21

22

11

QQ

Q

Q

67

Inference from qualified propositions

Propositions p1, p2, and p may be expressed in the form

where W1 = Prop(F / E), W2 = Prop(G / E∩F), and W = Prop(F∩G / E).

, is :

, is :

, is :

2 2 2

1 1 1

QW

QW

QW

p

p

p

68

Inference from qualified propositions

• Consequent conjunction syllogism

where E, F, G are are fuzzy sets on a universal set X, Q1 and Q2 are relative quantifiers and Q is a relative quantifier given by

'sGFE'sp

G'sE'sp

F'sE'sp

) and ( are :

are :

are :

22

11

Q

Q

Q

];) ,MIN([ )]1 ,0(MAX[ 2121 QQQQQ

69

Inference from qualified propositions

that is, Q is at least MAX(0, Q1 + Q2 - 1) and at most MIN(Q1, Q2). Here, MIN and MAX are extensions of min and max operations on real numbers to fuzzy numbers (Sec. 4.5).

7070

Exercise 6

• 6.4

• 6.8

• 6.9