Given the fuzzy rule in equation (4.27) and the data x ¼ A0 , we wish to draw a

conclusion about Slow. If the conclusion is y ¼ B0 , we can apply the generalized

modus ponens in (4.24) compute a new fuzzy number B0 for Speed. To do this

we must ¬rst choose an fuzzy logical implication operator I(x, y) giving the impli-

cation relation between A(x) and B( y). I(x, y) could be any function that will reduce

to the classical values for implication in Table 3.2 when x and y are 0 or 1, so we

could use any of the formulas given in equations (3.7) “(3.9) or any other fuzzy

implication operator.

There are many fuzzy implication operators from which to choose; Klir and Yuan

(1995, p. 309) list fourteen. Choice, according to Klir and Yuan (1995), will depend

on the application. Our application is clear; we wish to employ the implication oper-

ator in the generalized fuzzy modus ponens. Let R be the fuzzy relation (A ! B).

We pick the simplest (Gaines “Rescher), de¬ned by equation (4.28),

R(x, y) ¼ tv(A(x) ! B( y)) ¼ 1 if tv(A(x)) tv(B( y)), else R(x, y) ¼ 0 (4:37)

Membership functions of fuzzy numbers A, A0 , and B.

Figure 4.4

TEAM LinG - Live, Informative, Non-cost and Genuine !

68 FUZZY LOGIC, FUZZY SETS, AND FUZZY NUMBERS: II

and obtain B0 by composing A0 with R, using

B0 ( y) ¼ supx {(A0 (x), R(x, y))} (4:38)

where sup denotes supremum, the smallest number that is not exceeded by the argu-

ments. Carrying out the calculation for 0 x 20, we obtain the B0 shown in

Figure 4.5.

We can see immediately that B0 is nowhere less than 0.5 on the entire real line.

Clearly, using the centroid method to defuzzify a membership function that extends

from 2in¬nity to þin¬nity with non-zero membership is not possible. The problem

is not caused by the particular implication operator chosen; any implication operator

that reduces to the classical for crisp operands has a similar problem.

Another problem is the property of consistency. We say a method of fuzzy

reasoning is consistent if whenever A0 ¼ A, we get the conclusion B 0 ¼ B; that is,

if the data matches the antecedent exactly, the conclusion must match the conse-

quent exactly. However, approximate reasoning may, or may not, be consistent. It

depends on the implication operator. For some it is consistent and for other impli-

cation operators it is not consistent. Klir and Yuan (1995, p. 309) list 14 implication

operators, of which 7 do not possess consistency. [The Gaines “ Rescher implication

in equation (4.28) is consistent.]

A third problem is that if A0 > A ¼ 1 using Tm , we get B0 ( y) ¼ 1 for all y; if the

data A0 and the speci¬cation A are disjoint, the conclusion is the universal set. We

have been assuming that I(0, y) ¼ 1 for all y, which is true for most of the usual

implication operators (Klir and Yuan, 1995). You are also asked to check this

result in the problems. Because of these reasons, and others discussed in Chapter 8,

we will not use approximate reasoning in our fuzzy expert system. What is used in

practice is not an implication operator, but a fuzzy AND (t-norm). Although of

Figure 4.5 Membership functions of fuzzy numbers A, A0 , B, and B0 . B0 is obtained from A,

A0 , and B by Approximate Reasoning using Gaines “ Rescher implication.

TEAM LinG - Live, Informative, Non-cost and Genuine !

69

4.4 HEDGES

Figure 4.6 Membership functions of fuzzy numbers A, A0 , B, and B0 . B0 is obtained by

composing A0 with (A AND B).

course a t-norm is not an implication at all, the min t-norm used in this context is

sometimes called a “Mamdani implication”, from its use by Mamdani in fuzzy

control.

The above discussion assumed that approximate reasoning was based on fuzzy

implications that reduce to the classical for crisp operands. Fuzzy inference may

be based on other fuzzy relations R. Going back to equation (4.28), we could

de¬ne R by

R(x, y) ¼ min(A(x), B( y)) (4:39)

We can compose A0 with this R to obtain B0 . If we do, we get the perfectly reason-

able result shown in Figure 4.6. In fact, this general type of fuzzy relation based on

t-norms is used almost universally in fuzzy control.

Approximate reasoning may be extended to more complex antecedents and to

blocks (multiple) of IF-the rules (Klir and Yuan, 1995). However, we shall not

present further results in this book.

4.4 HEDGES

Hedges are modi¬ers, adjectives, or adverbs, which change truth values. Such

hedges as “about”, “nearly”, “roughly”, and so on, are used in fuzzy expert

systems to make writing rules easier and to make programs more understandable

for users and domain experts. The term was originated by Zadeh (1972), and

hedges have been developed and used to great effect by Cox (1999), which is

highly recommended. Hedges are indispensible to the builder of fuzzy expert

systems in the real world. There are several types of hedges, of which we will con-

sider the two most important.

TEAM LinG - Live, Informative, Non-cost and Genuine !

70 FUZZY LOGIC, FUZZY SETS, AND FUZZY NUMBERS: II

TABLE 4.1 Hedges that Create Fuzzy Numbers

Hedge Spread, þ/2 % of central value at

membership 0.5

nearly 5%

about 10%

roughly 25%

crudely 50%

One type of hedge, applied to scalar numbers, changes the scalar to a fuzzy

number with dispersion depending on the particular term used. Thus, “nearly 2”

is a fuzzy number with small dispersion, and “roughly 2” has a considerably

wider spread. The precise meaning of the hedge term will vary from one expert

system shell to another.

In FLOPS, each hedge term is associated with a percent of the central value, and

speci¬es the spread of the fuzzy number from the central value to the 0.5 truth value

point. [The reason the spread is speci¬ed at the 0.5 truth value rather than the support

is that normal (bell-shaped, Gaussian) fuzzy numbers in theory have in¬nite support,

and in practice a large support depending on the precision of the ¬‚oating-point

numbers in a particular implementation.]

FLOPS hedges of this type are not untypical of those employed by Cox. The hedge

terms and corresponding membership function spread are given in Table 4.1, and are

A fuzzy 5 created by various hedges.

Figure 4.7

TABLE 4.2 Hedges to Modify Truth Values

Hedge Power to which truth value is raised

slightly cube root

somewhat square root

very square

extremely cube