5.3

should be used?

Should the bounded sum and difference operators be used when combining

5.4

semantically inconsistent membership functions in

a. Fuzzy control applications?

b. In general-purpose fuzzy reasoning applications?

Temperature is a scalar whose value is 78 and whose truth value is 0.6. What

5.5

are the truth values of the following antecedent clauses?

a. “Temperature ¼ 75”

b. “Temperature is 78”

c. “Temperature is ,X.”

d. “size is Large”. (The grade of membership of Large in fuzzy set size is

0.356.)

e. “Temperature.cf . 0.5”?

We have two fuzzy numbers, A and B, shown below in Figure

5.6

Question 5.6.

a. What is the truth value of the proposition “A ¼ B”?

b. Of the proposition “A , B”?

c. Of the proposition “A .¼ B”?

Using the fuzzy numbers A and B in Question 5.6, we wish to construct the

5.7

fuzzy numbers A AND B and A OR B, with the min“ max logic as our

default. Should we use min “max or the bounded operators in combining

these two fuzzy numbers?

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98 COMBINING UNCERTAINTIES

Two fuzzy numbers.

Figure Question 5.6

Assume that the fuzzy numbers A and B in Question 5.5 are in fact member-

5.8

ship functions used to describe the same numeric quantity. Should we use the

default min “ max or the bounded operators in combining these two fuzzy

numbers?

What is the main problem with the use of Bayesian methods?

5.9

What is the relation between Dempster “Shafer methods and fuzzy logic?

5.10

In the lack of any knowledge about a hypothesis, what is its possibility? Its

5.11

necessity?

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6 Inference in an Expert System I

6.1 OVERVIEW

In this chapter, we are concerned with the theory of the fuzzy reasoning process;

drawing conclusions from data and rules that relate data to conclusions, under

conditions of uncertainty, ambiguity, and contradictions. This is a very dynamic

process; conclusions reached in early stages of the reasoning process may be, and

usually are, modi¬ed or invalidated as the process continues from one step to

the next.

The process of drawing conclusions from existing data is called inference; we

infer new truths from old. Of course, classical logic propositions have only two

truth values, true or false, the process of inference is simpli¬ed as compared to

fuzzy logic, where we have to be concerned not only with propositions but also

with their truth values. Accordingly, both Chapters 6 and 7 are concerned with

the ways with which we can determine new truth values.

Much of the literature on fuzzy mathematics is concerned with possibility, a

measure of the extent to which the data fail to refute a conclusion. In the real

world, we are primarily concerned with necessity, a measure of the extent to

which the data support a conclusion. The reasoning process of establishing necess-

ary conclusions is not the same as the process of establishing possible conclusions.

For example, when we initialize possible truth values in the lack of any data, we set

them to 1; but when we initialize necessary truth values in the lack of any data,

we set them to 0. When working with possibility, the fuzzy modus ponens using

the implication operator is an important tool; when working with necessity, we

employ t-norms rather than the implication operator.

Klir and Yuan (1995, Chapter 8) deal with Fuzzy Logic and various schemes of

inference. Unfortunately, their presentation is not geared to the different modes of

inference necessary in a rule-based system for fuzzy reasoning. Baldwin et al.

(1995) present a discussion of inference in a Prolog system of depth-based search

with backtracking, as used in their expert system Fril, which employs backward-

chaining Horn clauses. Both of these references deal with approaches suf¬ciently

different from that necessary in FLOPS so that in the next two chapters (Chapters

7 and 8) we essentially develop our own theory from scratch.

Chapters 6 and 7 assume we are working with necessity rather than possibility

unless otherwise stated, and are concerned with establishing the process under

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

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

99

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100 INFERENCE IN AN EXPERT SYSTEM I

which modi¬cation of data and truth values may take place. Aside from initializa-

tion, these modi¬cations will usually take place when a rule is ¬red, and are per-

formed by consequent instructions. We are speci¬cally concerned with the

question of the conditions under which we may permit these consequent instructions

to be executed. While the consequent instruction (then) “size is Small” may be

viewed as a proposition, viewing it as an instruction is perhaps pragmatically

more direct.

In the early days of FLOPS, we used the term “con¬dence” rather than the

term “truth value”, and we continue to use these terms interchangeably. Old habits

die hard.

In FLOPS, each rule has an associated truth value; by default, each rule is

assumed to have full con¬dence unless otherwise speci¬ed. A rule-¬ring threshold

is effective, either speci¬ed or set by default. In general, a rule is considered ¬reable

if the antecedent con¬dence is above the rule-¬ring threshold; if the rule is actually

¬red, the consequent instructions are carried out with a con¬dence equal to the fuzzy

AND of the antecedent con¬dence and the rule con¬dence.

6.2 TYPES OF FUZZY INFERENCE

We list three types of rule-based inference: monotonic, in which consequent truth

values may increase but not decrease; non-monotonic, in which consequent truth

values may increase or decrease; and downward monotonic, in which consequent

truth values may decrease but not increase. We de¬ne four methods of determining

consequent truth values from antecedents, including approximate reasoning. We list

three desirable properties for each of the three reasoning types. We show that the

¬rst three methods possess all desirable properties for the reasoning type for

which they were designed, but approximate reasoning fails to possess all desirable

properties for any reasoning type.

We will not be concerned with evaluating the truth value of the rule antecedent;

we will assume that this has been determined by the methods of Chapters 3 and 4.

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.

6.3 NATURE OF INFERENCE IN A FUZZY EXPERT SYSTEM

Inference in an expert system involves the modi¬cation of data, either its value or its

truth value or both, by rules. Whether modi¬cation of the value of a datum is