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101

6.3 NATURE OF INFERENCE IN A FUZZY EXPERT SYSTEM

permitted or not depends on its existing truth value, the truth value of the new value,

and the type of inference being used. Creation of new data is always permitted; the

only question here is the con¬dence that will be assigned to any values speci¬ed in

the new datum™s value. Since truth values are so important here, we ¬rst take up the

modi¬cation and assignment of truth values; following that, modi¬cation of the

value of data will be discussed.

Since inference is carried out by rules, we ¬rst review the structure of fuzzy rules.

The most compact form of a rule is

IF (P) THEN (Q) (6:1)

in which P (the antecedent) is a fuzzy proposition, and Q (the consequent) is a set of

instructions to be carried out if the rule is ¬red.

The antecedent proposition will make some assertion(s) about data. These asser-

tions may be very simple (that a given datum exists, or even more simply an asser-

tion whose truth value is always true). A somewhat less simple assertion is that the

truth value of a datum is true. Very commonly, an antecedent proposition will

involve a comparison between value or truth value of a datum and that of either a

literal or another datum. Most commonly the antecedent will be a complex pro-

position made up of two or more elemental propositions connected by AND, OR,

and NOT connectives. In all cases, evaluation of the antecedent will yield a

single truth value.

There are many types of consequent instructions. In this section, we are interested

only in those instructions that modify data.

There are two kinds of data modi¬cation. In the ¬rst kind, which applies only to

single-valued attributes of types integer, ¬‚oat and string, both the datum and its truth

value may be changed. In the second kind, which applies to both single- and multi-

valued attributes, only truth values are modi¬ed. Without loss of generality we may

consider rules that modify truth values; if a modi¬cation of a truth value is per-

mitted, modi¬cation of the value of a single-valued attribute is also permitted.

A rule that modi¬es truth values may be written as

IF (P) THEN (B0 ¼ B) (6:2)

P is the antecedent as before; B is an existing datum; and B0 is a revised datum, with

a different truth value. Our special interest is in deciding whether or not modi¬cation

of a truth value is permitted

A further development of rule form (6.2) re¬‚ects an antecedent that compares

data values. In that case, we have

IF (A0 ¼ A) THEN (B0 ¼ B) (6:3)

in which A and B are speci¬ed prior values, and A0 is an observed new value, usually

different from the prior value of A. If A is a fuzzy set, then A0 must be de¬ned on the

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102 INFERENCE IN AN EXPERT SYSTEM I

same universe as A; similarly, if B is a fuzzy set, then B0 must be de¬ned on the same

universe as B.

6.4 MODIFICATION AND ASSIGNMENT OF

TRUTH VALUES

If the antecedent truth value of rule (6.2) is P, the general principle is that we execute

the consequence B with con¬dence P. Let the modi¬ed truth value of B be B0 . We

have three possible types of inference: monotonic, non-monotonic, and downward

monotonic. In monotonic inference, the truth value B0 may increase or remain

unchanged, but not decrease; in non-monotonic inference, the truth value may

increase, decrease, or remain unchanged; and in downward monotonic inference,

the truth value may decrease or remain unchanged but not decrease.

The rule itself is assigned a truth value, say r. (By default, r is 1.) The antecedent

con¬dence is modi¬ed before executing the consequent instructions by replacing P

with P AND r to yield the posterior con¬dence, where usually the Zadehian AND or

minimum is used. To avoid cumbersome notation in this chapter, we will assume

that whenever the antecedent truth value appears, it will have been ANDed with

the rule truth value (by default 1).

6.4.1 Monotonic Inference

In monotonic inference, we assume that the existing truth value of B is ¬rmly

based, and cannot be reduced by any new information. (As is customary in AI,

we assume that the term unquali¬ed term “monotonic” means that facts believed

true will not be invalidated, i.e., truth values of data will not be reduced.) This

amounts to the assumption that we have not made any erroneous conclusions up

till now. Any new information can only add to our con¬dence that the value of

B is valid.

Suppose we have already established that the grade of membership of B is 0.8. A

rule now ¬res with antecedent con¬dence 0.5. The argument is that since we already

know that B™s grade of membership is 0.8, the new information should be discarded,

and the rule would fail; B™s grade of membership would remain at 0.8.

A formula for B0 , the new truth value of B0 , using monotonic inference is

B0 ¼ P OR B (6:4)

where P is the antecedent truth value. P may have the form (A0 ¼ A), and P will be

the con¬dence that A0 and A are approximately equal (discussed in Chapter 3).

Suppose A and A0 represent single-valued data (integers, ¬‚oats, strings), with the

same value but different truth values A and A0 , then

P ¼ truth value(A0 ¼ A) ¼ max(A, A0 ) (6:5)

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103

6.4 MODIFICATION AND ASSIGNMENT OF TRUTH VALUES

If A and A0 are not scalar quantities, that is, discrete fuzzy sets, fuzzy numbers or

membership functions, then we can write (6.5) as

P ¼ tv(A0 ¼ A) ¼ max(A(x), A0 (x))8x in A, A0 (6:6)

Similarly, if B and B0 are not scalar quantities, we can write (6.4) as

B0 (x) ¼ max(P, B(x)) (6:7)

According to (6.7), if P is multivalued and is non-zero, B would be every non-zero

also. It appears that monotonic reasoning is then not likely to be useful in modifying

a multivalued datum.

6.4.2 Non-monotonic Inference

In non-monotonic inference, we assume that the new information supplied by the

¬ring of a new rule is more reliable than any existing information.

As before, suppose we have already established that the grade of membership of

B is 0.8. A rule now ¬res with antecedent con¬dence 0.5. Since we believe the new

information, we would reduce the truth value of B to 0.5. If, on the other hand, the

new rule ¬res with con¬dence 0.9, we would increase the truth value of B to 0.9.

For single-valued data, a formula for B0 , using non-monotonic inference is

B0 ¼ P (6:8)

where P is the con¬dence that A0 and A are approximately equal as given just above.

If B and B0 are not single-valued quantities, non-monotonic reasoning is not

useful unless A, A0 , B, and B0 are de¬ned on the same universe. In this case, we

could write (6.8) as

B0 j(x) ¼ A0 (x) AND A(x) (6:9)

but this is of doubtful utility in a fuzzy expert system. Another possibility is two use two

levels of fuzziness; that is, we assign P as the truth value of the existing fuzzy number or

membership function B. However, this case is usually covered by monotonic down-

ward reasoning, discussed next. The assignment of truth values to other truth values

has not been explored to any great extent in working fuzzy expert systems, except in

the case of downward monotonic reasoning, discussed next in Section 6.4.3.