THEN species is setosa;

and seek to establish membership functions for our input linguistic terms, perhaps

from a training set, that will maximize the number of correct classi¬cations predicted.

It is likely that the data would permit omitting a number of rules of this type,

perhaps half, so that the actual number of rules might be around 15 or 20.

There is, however, another approach that will reduce the number of rules to three.

We achieve this by revising the linguistic terms for our input variables. We now de¬ne

these linguistic terms (members) for input discrete fuzzy sets PL,PW,SL, and SW as

setosa,versicolor, and virginica. (Ofcourse, the membership function for

setosa in linguistic variable PL would be different from the membership function

for setosa in linguistic variable PW, e.g.) Our three rules now become:

IF (PL is setosa AND PW is setosa AND SL is setosa AND SW

is setosa)

THEN species is setosa;

Determination of membership functions from test data becomes easier. We can

simply determine the distribution of the input variables for each of the three

species, and use these data to derive our membership functions.

In general, our method for reducing the number of rules to a manageable

minimum amounts to the top-down method of de¬ning the problem by ¬rst de¬ning

our outputs, then de¬ning our inputs, and their discretization in a way relevant to the

desired outputs. Combinations of the discretized inputs that do not relate to a desired

output are simply omitted from our rule set as irrelevant to our goal.

7.11 SUMMARY

Inference in an expert system is the process of drawing conclusions from data that is

deriving new data or truth values from input data and truth values. The new data may

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137

7.11 SUMMARY

be the ¬nal conclusions, or in multistep reasoning, may be intermediate conclusions

that constitute input to the next step.

7.11.1 Data Types and Their Truth Values

Available attribute data types and their truth values may be summarized as:

Integers, ¬‚oats, and strings. A single truth value is attached to any value

.

these attributes may have. This truth value is itself an attribute, accessed by

appending .cf to the attribute name.

Fuzzy numbers. Values are any number from the real line. Truth values are

.

de¬ned by a parameterized membership function that maps any real number

value onto its truth value (grade of membership). Fuzzy numbers are used

primarily in approximate comparisons in a rule antecedent. A full range of

approximate comparison operators is available corresponding to the conventional

Boolean numerical comparison operators furnished by most computer languages.

Discrete fuzzy sets. Values are de¬ned as the names of the members of the

.

fuzzy set. Each member has a single truth value, its grade of membership of

that member in the fuzzy set.

[wec4]Membership functions. If a discrete fuzzy set is a linguistic variable,

.

whose members are linguistic terms describing a numeric quantity, member-

ship functions are attached, one to each linguistic term. These functions map

any real or fuzzy number onto the grade of membership of the corresponding

linguistic term. The grade of membership of the linguistic term is the truth

value of its membership function.

7.11.2 Types of Fuzzy Reasoning

We assume that the truth value of an antecedent has been determined and combined

with the truth value of the rule itself to furnish the antecedent con¬dence P. We will

denote the truth values of data to be modi¬ed as B™, the modi¬ed truth value, and B,

the existing truth value.

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.

Monotonic reasoning. Here truth values in the consequent are nondecreasing.

.

B™ = P OR B

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

of membership of discrete fuzzy sets.

Non-monotonic reasoning. Here truth values in the consequent may increase,

.

decrease or stay the same.

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138 INFERENCE IN A FUZZY EXPERT SYSTEM II

B™ = P

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

especially when invalidating data previously believed to be true.

Monotonic downward. Here truth values in the consequent are nonincreasing.

.

B™ = P AND B

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, de¬ned as

.

B™ = A0 o [A IMPLIES B]

in which o denotes fuzzy composition and IMPLIES denotes any fuzzy

implication operator that reduces to the classical implication 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, and monotonic

downward reasoning all satisfy these desirable properties for their reasoning types, but

the approximate reasoning method fails to satisfy all desirable 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.

The question of inference in a fuzzy expert systems boils down to the de¬ning of

the way in which values and truth values of consequent data are inferred when a rule

is ¬red, knowing the prior truth values of the antecedent and of consequent data.

7.12 QUESTIONS

We have the rule “If A0 ¼ A then B0 ¼ B”, where A0 , A, B0 , and B are the

7.1

discrete fuzzy sets

0 0:5 1 0:25 0:75 0:75

0

A¼ , , , ,