the same; in monotonic downward reasoning, consequent truth values may decrease

or stay the same, but may not increase.

We consider a rule that modi¬es data or truth values as

if P then B0 ¼ B (6:27)

in which P is the antecedent, B is an existing truth value, and B0 is the new truth

value modi¬ed by the ¬ring of the rule. P may have the form

P ¼ (A0 ¼ A) (6:28)

where A is a speci¬ed value, and A0 is an observed value. This form is required by

the generalized modus ponens used in approximate reasoning.

In data modi¬cation by a rule, we consider three types of inference and de¬ne

these types quantitatively, in terms of the antecedent con¬dence and the truth

value of data to be modi¬ed. We also similarly de¬ne the Approximate Reasoning

method.

Monotonic reasoning: B0 = P OR B

.

Monotonic reasoning is useful when modifying values of scalar data, or

grades of membership of discrete fuzzy sets by such consequent instructions

as THEN name = "Jane" or THEN size is Small.

Non-monotonic reasoning: B0 = P

.

Non-monotonic reasoning is useful when modifying truth values directly,

especially when invalidating data previously believed to be true.

Monotonic downward: B0 = P AND B

.

For membership functions, where x is a scalar argument,

B™(x) = P AND B(x)

This type of reasoning is useful when combining the grade of membership of a

linguistic term with its membership function prior to defuzzi¬cation.

Approximate reasoning:

.

B™ = A™ o [A IMPLIES B]

in which o denotes fuzzy composition and IMPLIES denotes any fuzzy impli-

cation operator that reduces to the classical for crisp operands.

We de¬ne desirable properties for inference in an expert system for these

inference types. We then show that the de¬nitions of monotonic, non-monotonic,

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113

6.8 QUESTIONS

and monotonic downward reasoning all satisfy the desirable properties for their

reasoning types, but the approximate reasoning method fails to satisfy all desir-

able properties for any type of reasoning. We note that Mamdani inference,

which uses a fuzzy AND operator in place of an implication operator,

is precisely the same as our monotonic downward method.

6.8 QUESTIONS

What is the difference between possibility and necessity?

6.1

We have proposition A, but as yet no evidence to support it or refute it. What is

6.2

Nec(A)? Pos(A)?

We have created a fuzzy set to hold some preliminary classi¬cations of an

6.3

object.

a. To what value should the grades of membership of the fuzzy set members

be initialized?

b. If we are going to employ monotonic reasoning, what advantage does this

initialization have?

What are the differences among monotonic, non-monotonic and downward

6.4

monotonic reasoning?

We are employing monotonic reasoning, and have a ¬reable rule with a con-

6.5

sequent instruction that would decrease the truth value of a datum. What

happens to that truth value when the rule is ¬red?

Under what circumstance should we employ non-monotonic reasoning?

6.6

When should downward monotonic reasoning be employed?

6.7

What type of inference does FLOPS employ by default?

6.8

a. Can FLOPS default inference method be changed?

6.9

b. If so, how, and to what method?

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7 Inference in a Fuzzy Expert

System II: Modi¬cation of

Data and Truth Values

In Chapter 6, we de¬ned three types of fuzzy inference useful in expert systems.

These are monotonic (upward), in which consequent truth values can increase or

stay the same but cannot decrease; non-monotonic, in which truth values may

increase, decrease or stay the same; and monotonic downward, in which truth

values may decrease or stay the same, but cannot increase. Monotonic inference is

useful when modifying values of scalar data (integers, ¬‚oats and string) by such con-

sequent instructions as x ¼ 13, where x is an integer, and in inferring grades of mem-

bership in discrete fuzzy sets by such instructions as size is Small, where size

is a discrete fuzzy set of which Small is a member. Non-monotonic inference is

required when invalidating data previously believed to be true, by directly assigning

truth values or grades of membership with such instructions as x.cf ¼ 0, which sets

the truth value of x to 0, and size.Large ¼ 0, which sets the truth value of

member Large of discrete fuzzy set size to 0. Monotonic downward inference

is useful when modifying membership functions prior to defuzzi¬cation.

In this chapter, we will examine how these methods are used in modifying exist-

ing data and truth value when ¬ring rules. We will not be concerned with evaluating

the truth value of the rule antecedent; we will assume that this has been determined

and combined with the truth value of the rule itself using the methods of Chapters 3

and 4. We will refer to this combined truth value as the antecedent con¬dence to

simplify our wording. When the antecedent has been found to be suf¬ciently true,

we are ready to execute the consequent of the rule.

There are many possible consequent instructions, including creation, modi¬-

cation and input or output of data, direction of the ¬‚ow of the program by metarules,

debugging instructions, and so on. Here, we will be concerned only with instructions

that change or create values or truth values of data; the other instruction types are

ancillary to the main purpose of the program, which is to reason from data to

conclusions.

Creating new data in the consequent of a rule offers no problems. Scalar data

(integer, ¬‚oats, and strings) and fuzzy numbers are set to the new value with

(by default) the truth value set to the antecedence con¬dence. Fuzzy numbers are

Fuzzy Expert Systems and Fuzzy Reasoning, By William Siler and James J. Buckley

ISBN 0-471-38859-9 Copyright # 2005 John Wiley & Sons, Inc.

115

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116 INFERENCE IN A FUZZY EXPERT SYSTEM II

set to the speci¬ed membership function; since entire fuzzy numbers themselves do

not have a truth value, the antecedent con¬dence is ignored. Grades of membership

of discrete fuzzy sets are initialized to 0. Membership functions of linguistic terms

are set to the speci¬ed values. Default truth values for new scalar data and grades of

membership of discrete fuzzy set members may be overridden by direct assignment

of these values.