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107

6.6 TESTS OF PROCEDURES

Desirable Inference Properties a

TABLE 6.1A

Desirable Properties

Downward

A0 A0 ¼ A B0

A B NonMonotonic Monotonic Monotonic

1. 2. 3. 1. 2. 3. 1. 2. 3.

0 0 0 0 0 Y Y Y Y Y Y Y Y Y

0 0 1 0 0 Y ” Y Y ” Y Y ” Y

0 1 0 0 1 N Y Y Y Y Y Y N Y

0 1 1 0 1 N ” Y Y ” Y Y ” Y

1 0 0 0 0 Y Y Y Y Y Y Y Y Y

1 0 1 1 1 Y ” ” Y ” ” Y ” ”

1 1 0 0 1 Y Y N Y Y Y Y N N

1 1 1 1 1 Y ” ” Y ” ” Y ” ”

a

Method A: monotonic rule: B0 ¼ (A0 ¼ A) OR B.

6.6.3 Tests of Methods Against Desirable Properties

We now test these four methods on rule (6.14) in the simple case of a fuzzy set with

one member, with grades of membership of A, B, and A0 0 or 1, with results in

Table 6.1A “ D. By using crisp values, we leave open the de¬nitions of AND and

OR as any valid t-norm and t-conorm, and of any implication operator that

reduces to the classical for crisp values. (“Desirable properties” are de¬ned in

Section 6.6.1.)

Methods A, B, and C all satisfy the desirable properties for the type of reasoning

they are designed to handle, and fail to satisfy all desirable properties for any other

reasoning type. They are not interchangeable.

TABLE 6.1B Desirable Inference Properties a

Desirable Properties

Downward

0 0 0

A ¼A

A B A B NonMonotonic Monotonic Monotonic

1. 2. 3. 1. 2. 3. 1. 2. 3.

0 0 0 0 0 Y Y Y Y Y Y Y Y Y

0 0 1 0 0 Y ” Y Y ” Y Y ” Y

0 1 0 0 0 Y Y Y N N N N N N

0 1 1 0 0 Y ” Y Y ” N Y ” N

1 0 0 0 0 Y Y Y Y Y Y Y Y Y

1 0 1 1 1 Y ” ” Y ” ” Y ” ”

1 1 0 0 0 Y Y Y Y N N Y N N

1 1 1 1 1 Y ” ” Y ” ” Y N N

a

Method B: non-monotonic rule: B0 ¼ (A0 ¼ A).

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108 INFERENCE IN AN EXPERT SYSTEM I

Desirable Inference Properties a

TABLE 6.1C

Desirable Properties

Downward

A0 A0 ¼ A B0

A B NonMonotonic Monotonic Monotonic

1. 2. 3. 1. 2. 3. 1. 2. 3.

0 0 0 0 0 Y Y Y Y Y Y Y Y Y

0 0 1 0 0 Y ” Y Y ” Y Y ” Y

0 1 0 0 0 Y Y Y N N N Y Y Y

0 1 1 0 0 Y ” Y N ” N Y ” Y

1 0 0 0 0 Y Y Y Y Y Y Y Y Y

1 0 1 1 0 N ” ” Y ” ” Y ” ”

1 1 0 0 0 Y Y Y N N N Y Y Y

1 1 1 1 1 Y ” ” Y ” ” Y ” ”

a

Method C: downward monotonic rule: B0 ¼ (A0 ¼ A AND B).

The approximate reasoning method fails to satisfy all desirable properties for any

type of reasoning: monotonic, non-monotonic, or downward monotonic. Of particu-

lar concern is the failure to satisfy property 3. If A0 and A are disjoint so that

(A0 ¼ A) is 0, it is quite possible to produce a B0 that is everywhere 1, a very counter-

intuitive result.

If the t-norm (Mamdani “implication”) A AND B is used instead of the fuzzy

implication in the approximate reasoning method, all properties are satis¬ed for

downward monotonic reasoning. This is important, since Castro (1995) has demon-

strated that this formulation produces universal approximators for any t-norm used

for the AND operator. However, since the Mamdani “implication” does not reduce

to the classical implication for crisp values, it is not properly speaking an implication

Desirable Inference Properties a

TABLE 6.1D

Desirable Properties

Downward

0 0

A B A A!B B NonMonotonic Monotonic Monotonic

1. 2. 3. 1. 2. 3. 1. 2. 3.

0 0 0 1 0 Y Y Y Y Y Y Y Y Y

0 0 1 1 1 N ” N Y ” N N ” N

0 1 0 1 0 Y Y Y N N Y Y Y Y

0 1 1 1 1 N ” N Y ” Y Y ” N

1 0 0 0 0 Y Y Y Y Y Y Y Y Y

1 0 1 0 0 N ” ” Y ” ” Y ” ”

1 1 0 1 0 Y Y Y N N N Y Y Y

1 1 1 1 1 Y ” ” Y ” ” Y ” ”

a

Method D (approximate reasoning): B0 ¼ A0 AND (A IMPL B).

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109

6.6 TESTS OF PROCEDURES

at all. In the procedure de¬ned for approximate reasoning in (6.25), the fuzzy

relation (A ! B) is replaced by (A AND B), becoming

B0 ¼ A0 (A AND B) (6:26)

The method of composing A0 with a fuzzy relation between A and B de¬ned by a

t-norm is precisely equivalent to our Method C, downward monotonic inference,

B0 ¼ (A0 ¼ A) AND B:

We have demonstrated that for crisp truth values for A, A0 , and B, the non-mono-

tonic rule method satis¬es all properties for non-monotonic logic; the monotonic rule

method satis¬es all properties for monotonic logic; the downward non-monotonic

rule method satis¬es all properties for downward non-monotonic logic; and the

approximate reasoning method fails to satisfy all desirable properties for any reason-

ing type. We will be content with the simple assertion that the rule methods also

satisfy the corresponding properties for fuzzy truth values, and will forego the proof.

Note that in the de¬nitions of methods A, B, and C for determining B0 from A, A0 ,

and B summarized in 6.6.2, the antecedent involves only A0 and A; the consequent

involves only B0 and B. Only the antecedent in approximate reasoning method D has

a clause that involves both A and B; as we have shown, the approximate reasoning

method fails to meet the desirable properties, and hence has little use in an expert

system. In an expert system, it is then possible to compute the consequent con¬dence

in two steps, the ¬rst of which depends only on the antecedent, and the second of

which couples this to the consequent.