The bounded sum and difference operators should be used when combining

5.2

A and NOT A, since they are maximally negatively associated.

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386 ANSWERS

The min “ max operators should be used when combining A and A, since

5.3

they are maximally positively associated.

a. Fuzzy control techniques are used in shaping membership functions, and

5.4

are based on many years of experience with current operators; introdu-

cing different operators would invalidate much of current working tech-

niques. Further, the precise way of determining the correlation coef¬cient

between two membership functions is not yet agreed upon. However, if

the membership functions being combined cross at no greater than 0.5

membership, use of the bounded sum “ difference operators produces a

much more intuitive smooth result.

b. Yes. Other operators than the bounded sum “difference produce counter-

intuitive results, and can make programming dif¬cult. This is especially

true when making approximate numerical comparisons such as A less

than OR equal to B.

a. Zero. This is a Boolean comparison, and the temperature is 78, not 75.

5.5

The truth value of Temperature is 0.6; the truth values of the comparison

is zero; and the truth value of the literal 78 is 1. The antecedent truth

value is min(0.6, 0, 1) ¼ 0.6.

b. 0.6. The truth value of Temperature is 0.6; the truth value of the compari-

son is 1; truth value of the literal 78 is 1. Antecedent truth value is

min(0.6, 1, 1) ¼ 0.6.

c. 0.6. Since ,X. is a variable, it is assigned the value and truth value of

Temperature. Again, the truth value of Temperature is 0.6; truth value of

the comparison is 1; the truth value of ,X. is 0.6. Antecedent truth

value is min(0.6, 1, 0.6) ¼ 0.6.

d. 0.356. Truth value of the entire fuzzy set is one by default; truth value of

the comparison is one; truth value of Large is its grade of membership,

0.356. Antecedent truth value is min(1, 1, 0.356) ¼ 0.356.

e. One. Truth value of Temperature.cf is one by default; truth value of the

comparison is 1; and truth value of the literal 0.5 is also 1. Antecedent

truth value is min(1, 1, 1) ¼ 1.0.

a. From the graph in Figure Question 5.6 it appears that the truth value of

5.6

A ¼ B is 0.25.

b. First, we construct the fuzzy number ,B, NOT B OR ,B. This

number is

Fuzzy numbers A and ,B:

Figure Answer 5:6b

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387

ANSWERS

Since The highest intersection point of A and ,B is one, the truth

value of the comparison is 1.

c. First we construct the fuzzy number . ¼ B, b.B OR B

Fuzzy numbers A and .= B:

Figure Answer 5:6c

From the graph, it appears that the truth value of A . ¼ B is 0.25.

At ¬rst blush, it might seem that A and B are semantically inconsistent, since

5.7

they have different names. However, this is not true; A and B have no known

prior relation to each other, and therefore they are not semantically incon-

sistent. Use the default logic.

Unlike Question 5.7, A and B now have a prior relationship; the both describe the

5.8

same entity. In this case, the bounded sum“difference operators are appropriate.

The problem lies in the requirement of prior knowledge of conditional prob-

5.9

abilities. If we have such knowledge, Bayesian methods are rock solid;

otherwise, they are theoretically shaky.

5.10 Theoretical fuzzy logic deals with two measures of uncertainty: possibility

and necessity. Dempster “ Shafer methods have two analogous measures:

plausibility and credibility. Possibility and plausibility both measure the

extent to which the data fail to refute an hypothesis; necessity and credibility

both measure the extent to which the data support an hypothesis.

5.11 If we have no knowledge at all, the possibility of a hypothesis is 1, since there

are no data to refute it; its necessity is 0, since there are no data to support it.

Possibility measures the extent to which the data refute a conclusion;

6.1

necessity measures the extent to which the data support a conclusion.

In the lack of any evidence, Nec(A) ¼ 0, and Pos(A) ¼ 1.

6.2

a. Zero.

6.3

b. Initializing grades of membership to 0 permits subsequent rules that tend

to support a particular classi¬cation to increase its grade of membership.

If we initialized to 1, we would have to write only rules that refute this

classi¬cation with non-monotonic reasoning rather than rules that tend

to establish it, a much less intuitive task.

Monotonic reasoning permits truth values to increase, but not decrease; non-

6.4

monotonic reasoning permits truth values to increase or decrease; and down-

ward monotonic reasoning permits truth values to decrease but not increase.

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388 ANSWERS

Under monotonic reasoning, that instruction would be rejected; the datum™s

6.5

truth value would remain unaltered.

When we ¬nd out that a datum should be wholly or partially invalidated.

6.6

When modifying membership functions prior to defuzzi¬cation.

6.7

Monotonic reasoning.

6.8

a. Yes.

6.9

b. The TMSoff command permits non-monotonic reasoning to be employed;

the TMSon command restores monotonic reasoning. In defuzzifying,

default inference is downward monotonic.

2 3

7.1 a.

0 0 0

0:25 0:75 0:75 6 7

A0 ¼ , , , ½Ai AND Bj ¼ 4 0 0:5 0:5 5

a1 a2 a3

0 0:5 1

B0 ¼ A0 W ½Ai > Bj

8 9

> max(min(0:25, 0), min(0:75, 0), min(0:75, 0)), >

< =

¼ max(min(0:25, 0), min(0:75, 0:5), min(0:75, 0:5)),

> >

: ;

max(min(0:25, 0), min(0:75, 0:5), min(0:75, 1))

¼ {max(0, 0, 0), max(0, 0:5, 0:5), max(0, 0:5, 0:75)}