Klir G.J. Uncertainity and Information. Foundations of Generalized Information Theory

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Cylindric extensions of the other 2-dimensional and 1-dimensional projections

are deﬁned in a similar way. This deﬁnition of cylindric extension for fuzzy

relations is a cutworthy generalization of the classical concept of cylindric

extension.

Given any set of projections of a given relation R, the standard fuzzy inter-

section of their cylindric extensions (expressed by the minimum operator) is

called a cylindric closure of the projections. This is again a cutworthy concept.

Regardless of the given projections, it is guaranteed that their cylindric closure

contains the fuzzy relation R.

EXAMPLE 7.7. To illustrate the concepts introduced in this section, let us

consider a binary relation R on X ¥ Y, where X = {x

, x

,...,x

} and Y =

,...,y

}

, which is deﬁned by the matrix

RxyzRxy

RxyzRx

EXY XY

EX X

,, , ,

,, .

()

282 7. FUZZY SET THEORY

yyyyy y yyyy

123 4 5 6 78910

02 00 10 00 00 00 10 00 00 00

10 00 00 03 00 04 00 00 10 00

00 00 08 00 04 00 01 0

R =

... . . . .. ..

... . . . ....

..... ...000000

00 01 00 00 00 00 00 09 07 05

01 00 05 00 00 06 00 00 00 00

00 10 00 00 02 00 10 00 00 05

..... .....

... . . . .. ..

..... .....

The projections of R into X and Y are obtained for all x

Œ X and for all y

Y by the equations

respectively. It is convenient to express them as vectors

Rx Rxy

Ry Rxy

()

max , ,

xxxx xx

yyyyyyyyyy

123456

12345678910

10 10 08 09 06 10

10 10 10 03 04 06 10 09 10 05

[]

......,

...........

Now, the cylindric extensions R

and R

are obtained for all pairs ·x

, y

ÒŒ

·X ¥ YÒ by the equations

and their matrix representations are

Rxy Rx

Rxy Ry

EX i j X i

EY i j Y j

()

7.5. FUZZY RELATIONS 283

yyy y y y y y yy

123 4 5 6 78 910

10 10 10 10 10 10 10 10 10 10

08 08 08 08 08 08 08 0

R =

..........

...........

..........

80808

09 09 09 09 09 09 09 09 09 09

06 06 06 06 06 06 06 06 06 06

10 10 10 10 10 10 10 10 10 10

yyyy y y y y y yy

123 4 5 6 78 910

10 10 10 03 04 06 10 09 10 05

10 10 10 03 04 06 10

R =

... . . . . . ..

... . . . .009 10 05

10 10 10 03 04 06 10 09 10 05

...

... . . . . . ..

˙˙

The cylindric closure of projections R

and R

, Cyl{R

, R

}, is then obtained

by taking the standard intersection of these cylindric extensions. That is,

for all pairs ·x

, y

ÒŒ·X ¥ YÒ. We obtain

Cyl R R x y R x y R x y

x Y i i EX i j EY i j

,,min ,,,

{}()

()(){}

yyyyyyyyyy

Cyl R R

12345 678910

10 10 10 03 04 06 10 09 10 05

08 08 08 03 04 0

.. ........

.. . . .

{}

.. . . . .

.. ... .....

.. ........

608080805

09 09 09 03 04 06 09 09 09 05

06 06 06 03 04 06 06 06 06 05

10 10 10 03 04 06 10 09 10 05

˙˙

We can see that R 債 Cyl{R

, R

}, which is always the case, but this recon-

structed relation from the projections R

and R

is in this case much larger

than the actual relation R. This means that information about R that is con-

tained in the projection about R is in this case highly deﬁcient.

7.5.2. Compositions, Joins, and Inverses

Consider two binary fuzzy relations P and Q that are deﬁned on set X ¥ Y

and Y ¥ Z, respectively. Any such relations, which are connected via the

common set Y, can be composed to yield a relation on Y ¥ Z. The standard

composition of these relations, which is denoted by P

Q, produces a relation

R on X ¥ Z deﬁned by the formula

(7.33)

for all pairs ·x, zÒŒX ¥ Z.

Other deﬁnitions of a composition of fuzzy relations, in which the min and

max operations are replaced with other t-norms and t-conorms, respectively,

are possible and useful in some applications. All compositions are associative:

However, the standard fuzzy composition is the only one that is cutworthy.

A similar operation on two connected binary relations, which differs from

the composition in that it yields a 3-dimensional relation instead of a binary

one, is known as the relational join. For the same fuzzy relations P and Q, the

standard relational join, P *Q, is a 3-dimensional relation on X ¥ Y ¥ Z deﬁned

by the formula

(7.34)

for all triples ·x, y, zÒŒX ¥ Y ¥ Z. Again, the min operation in this deﬁnition

may be replaced with another t-norm. However, the relational join deﬁned by

Eq. (7.34) is the only one that is cutworthy.

The inverse of a binary fuzzy relation R on X ¥ Y, denoted by R

-1

, is a rela-

tion on Y ¥ X such that

for all pairs ·y, xÒŒY ¥ X. When R is represented by a matrix, R

-1

is repre-

sented by the transpose of this matrix. This means that rows are replaced with

columns and vice versa. Clearly,

holds for any binary relation.

()

Ryx Rxy

()

Rxyz P Q xyz PxyQyz,,

, , min , , ,

()

()()

()(){}

PQ Q P QQoo oo

()

Rxz P Q xz Px y Qyz

, , max min , , ,

()

()()

()(){}

284 7. FUZZY SET THEORY

It is convenient to perform compositions of binary fuzzy relations in terms

of their matrix representations. Let

be matrix representations of fuzzy relations P, Q, R in Eq. (7.33). Then, using

this matrix notation, we can write

where

(7.35)

Observe that the same elements of matrices P and Q are used in the calcula-

tion of matrix R as would be used in the regular matrix multiplication, but the

product and sum operations are replaced, respectively, with the min and max

operations.

EXAMPLE 7.8. To illustrate the standard composition of fuzzy relations by

their matrix representations, let symbols P, Q, R have the same meaning as in

Eq. (7.33), where X = {x

, x

}, Y = {y

, y

}, and Z = {z

, z

}, and

consider the following example of the matrix representation of equation

Q = R:

Following Eq. (7.35), we have, for example,

0070305040

02 03

11 11 11 12 21 13 31

24 21 14 22 24 23 34

. max min , ,min , ,min ,

min , . ,min . , . ,min . ,

. max min , ,min , ,min ,

min . , .

()

{}{ }{}{}

{}{ }{}}

()

{}{ }{}{}

rpqpqpq

{{}{}{}

,min . , . ,min . , . .05 04 0309

00 03 04

02 05 03

08 00 00

07 07 10

07 00 00 03 06

05 05 10 04 00

00 07 02 09 00

...

.....

˙˙

.....

03 04 03 04 00

05 05 05 04 02

07 00 00 03 06

07 07 07 09 06

rpq

ij jk

{}

max min , .

[] ,rpq

ik ij jk

[][ ]

PQ R=

[]

pq r

ij jk ik

, , and

7.5. FUZZY RELATIONS 285

When the join operation deﬁned by Eq. (7.34) is applied to the same fuzzy

relations, P and Q, we obtain a 3-dimensional relation. Equation (7.34) can be

expressed in terms of the associated arrays as

where

The resulting 3-dimensional array R may conveniently be represented by the

following four matrices, each associated with one element of set X:

rpq

ijk ij jk

{}

min , .

rpq

ijk ij jk

[]

[][ ]

286 7. FUZZY SET THEORY

y y yy y yy y yy y y

1 2 31 2 31 2 31 2 3

00 03 00

00 03 04

00 03 02

00 03 04

00 00 00

02 05 00

00 05 03

00 0

...

.. .

...

502

02 04 03

02 00 00

07 00 00

00 00 00

03 00 00

06 00 00

07 05 00

00 05 0

...

00 07 02

03 04 09

06 00 00

Equations (7.33), which describe R = P

Q are called fuzzy relation equations.

Normally, it is assumed that P and Q are given and R is determined by Eq.

(7.33). However, two inverse problems play important roles in many applica-

tions. In one of them, R and P are given and Q is to be determined; in the

other one, R and Q are given and P is to be determined. Various methods for

solving these problems exactly as well as approximately have been developed.

It should also be mentioned that cutworthy fuzzy counterparts of the

various classical binary relations on X ¥ X, such as equivalence relations, and

the various ordering relations, have been extensively investigated. However,

many types of fuzzy relations on X ¥ X that are not cutworthy have been inves-

tigated as well and found useful in many applications.

7.6. FUZZY LOGIC

The term “fuzzy logic,” as currently used in the literature, has two distinct

meanings. It is viewed either in a narrow sense or in a broad sense. Fuzzy logic

in the narrow sense is a generalization of the various many-valued logics, which

have been investigated in the area of symbolic logic since the beginning of the

20th century. It is concerned with development of syntactic aspects (based on

the notion of proof ) and semantic aspects (based on the notion of truth) of a

relevant logic calculus. In order to be acceptable, the calculus must be sound

(provability implies truth) and complete (truth implies provability).

Fuzzy logic in the narrow sense is important since it provides theoretical

foundations for fuzzy logic in the broad sense. The latter is viewed as a system

of concepts, principles, and methods for reasoning that is approximate rather

than exact. It utilizes the apparatus of fuzzy set theory for formulating various

forms of sound approximate reasoning with fuzzy propositions. It is fuzzy logic

in this broad sense that is primarily involved in dealing with fuzziﬁed uncer-

tainty theories.

To establish a connection between fuzzy set theory and fuzzy logic, it is

essential to connect degrees of membership in fuzzy sets with degrees of truth

of fuzzy propositions. This can only be done when the degrees of membership

and the degrees of truth refer to the same objects. Let us consider ﬁrst the sim-

plest connection, in which only one fuzzy set is involved.

Given a fuzzy set A, its membership degree A(x) for any x in the underly-

ing universal set X can be interpreted as the degree of truth of the associated

fuzzy proposition “x is a member of A.” Conversely, given an arbitrary propo-

sition of the simple form “x is A,” where x is from X and A is a fuzzy set that

represents an inherently vague linguistic term (such as low, high, near, fast),

its degree of truth may be interpreted as the membership degree of x in A.

That is, the degree of truth of the proposition is equal to the degree with which

x belongs to A.

This simple correspondence between membership degrees and degrees of

truth, which conforms well to our intuition, forms a basis for determining

degrees of truth of more complex propositions. Moreover, negations, con-

junctions, and disjunctions of fuzzy propositions are deﬁned under this corre-

spondence in exactly the same way as complement, intersections, and unions

of fuzzy sets, respectively.

7.6.1. Fuzzy Propositions

Now let us examine basic propositional forms of fuzzy propositions.To do that,

we need a convenient notation. Let X denote a variable whose states (values)

are in set X, and let A denote a fuzzy set deﬁned on X that represents an

approximate description of the state of variable X by a linguistic term such as

low, medium, high, slow, fast.

Using this notation, the simplest fuzzy propositions (unconditional and

unqualiﬁed) are expressed in the canonical propositional form,

in which the fuzzy set A is called a fuzzy predicate. Given this propositional

form, a fuzzy proposition, f

(x), is obtained when a particular state (value) x

from X is substituted for variable X in the propositional form. That is,

fx x A

()

=:,X is

fx A

()

:,X is

7.6. FUZZY LOGIC 287

where x Œ X, is a particular fuzzy proposition of propositional form f

.For

simplicity, let f

(x) also denote the degree of truth of the proposition “x is A.”

This means that the symbol f

denotes a propositional form as well as a func-

tion by which degrees of truth are assigned to fuzzy propositions based on the

form. This double use of the symbol f

does not create any ambiguity, since

there is only one function for each propositional form that assigns degrees of

truth to individual propositions subsumed under the form. In this case, the

function is deﬁned for all x ŒX by the simple equation

That is, f

(x) is true to the same degree to which x in a member of fuzzy

set A.

The propositional form f

can be modiﬁed by qualifying the claims for the

degree of truth of the associated fuzzy propositions. Two types of qualiﬁed

propositional forms are recognized: a truth-qualiﬁed propositional form

and a probability-qualiﬁed propositional form. These two forms also can be

combined.

The truth-qualiﬁed propositional form, f

T(A)

, is obtained by adding a truth

qualiﬁer, T, to the basic form of f

. That is,

where T is a fuzzy number or interval on [0, 1] that represents a linguistic term

by which the meaning of truth in fuzzy propositions associated with this propo-

sitional form is qualiﬁed. Linguistic terms such as very true, fairly true, barely

true, false, fairly false, or virtually false are typical examples of linguistic truth

qualiﬁers.

To obtain the degree of truth of a truth-qualiﬁed proposition f

T(A)

, we need

to compose the membership function A with the membership function of the

truth qualiﬁer T. That is,

(7.36)

for all x ŒX.

EXAMPLE 7.9. Consider truth-qualiﬁed fuzzy propositions of the form

In this example, X is the variable “the rate of inﬂation” (in %), A is a fuzzy

set representing the predicate “low” (L), T is a fuzzy set representing the truth

qualiﬁer “fairly true” (FT). The relevant membership functions of L and FT

are shown in Figure 7.7. Now assume that the actual rate of inﬂation is 2.5%.

Then, L(2.5) = 0.75 and FT(0.75) = 0.86. Hence

()

:The rate of inﬂation is low is fairly true.

fxTAx

()

() ()()

fAT

()

:,X is is

fx Ax

()

288 7. FUZZY SET THEORY

7.6. FUZZY LOGIC 289

12345678

= 2.5

L(2.5) = .75

M(2.5) = .25

L: low

M: medium

r : Inflation Rate in %

(a)

0.6 0.8

Absolutely fals

m = 0.25

T(m)

Absolutely true

0.56

0.07

0.25

0.51

0.75

0.86

m: Membership Degree

(b)

m = 0.75

False

Fairly false

Very false

Very true

Fairly true

True

0 0.2

0.4

Figure 7.7. Illustration of truth-qualiﬁed unconditional fuzzy propositions.

When we change the predicate or the truth qualiﬁer or both, the degree of

truth of the proposition is affected. For example, when only the truth qualiﬁer

is changed to “false” (F) the degree of truth becomes F(0.75) = 0.25.

The probability-qualiﬁed propositional form, f

P(A)

, is obtained by adding

a probability qualiﬁer, P, to the basic form f

and replacing “X is A” with

“Probability of X is A.” That is,

where P is a fuzzy number or interval deﬁned on [0, 1], called a probability

qualiﬁer, which represents an approximate description of probability by a lin-

guistic term, such as likely, very likely, extremely likely, around 0.5. Given this

propositional form, a proposition is obtained for each particular probability

measure, Pro, deﬁned on subsets of X. That is,

When X is ﬁnite,

(7.37)

where p is a given (known) probability distribution function on X. When

X = [x, x

] 債⺢,

(7.38)

where q is a given probability density function on X. Equations (7.37) and

(7.38) are standard formulas for computing probabilities of fuzzy events

(Section 8.4).

EXAMPLE 7.10. Consider probability-qualiﬁed fuzzy propositions of the

form

where A is a fuzzy number shown in Figure 7.8a that represents the fuzzy pred-

icate “around 80°F,” as understood in a given context, and P is a fuzzy number

shown in Figure 7.8b that represents the linguistic term “unlikely.” Assume

now that the probabilities p(t) for the individual temperatures, obtained

from relevant statistical data over many years, are given in Table 7.1a.To

determine the degree of truth of the proposition, we calculate ﬁrst Pro(A) by

Eq. (7.37):

fAP

()

:,Probability of temperature (at a given place and time) is is

Pro A A x q x dx

()

()()

Pro A A x p x

()

()()

f Pro A Pro A P

()

()()()

:: .is

fAP

()

:,Probability of is isX

fFTLFT

()

()()

()

=25 25 075 086.....

290 7. FUZZY SET THEORY

Now, applying this probability value to the probability qualiﬁer “unlikely” in

Figure 7.8b, we obtain 0.7. Hence,

Fuzzy propositions that are probability-qualiﬁed as well as truth-qualiﬁed

have the form

To obtain the degree of truth of any proposition of this form, we need to

perform two compositions of functions:

where Pro(A) is calculated either by Eq. (7.37) or by Eq. (7.38).

An important type of fuzzy proposition, which is essential for knowledge-

based fuzzy systems, is the conditional fuzzy proposition. This type of propo-

sition is based on the propositional form

where X and Y are variables whose states are in sets X and Y, respectively.

These propositions may also be expressed in an alternative, but equivalent

form

fAB

:, ,If is then isXY

fpTPProA

TPA

()

[]

()

[]

{}

fAPT

TPA

()

[]

:.Probability of is is isX

fpPProAP

()

()()

()

=015 07...

Pro A

()

=¥+¥+¥+¥+¥

+¥ + ¥ =

025 014 05 011 075 004 1 002 075 0001

05 0006 025 0004 015

...... ...

.. . . ..

7.6. FUZZY LOGIC 291

A(t)

t [ F]

78 80

82 84

Very

likely

Likely

Unlikely

0.7

0.29

0.25

0.5

0.75

0.15

Very unlikely

(a)

(b)

Figure 7.8. Illustration of probability-qualiﬁed unconditional fuzzy proposition.