a1 a2 a3 a1 a2 a3

???

0 0:5 1

, B0 ¼

B¼ , , , ,

b1 b2 b3 b1 b2 b3

Calculate B0 by

a. Composing A0 with [Ai AND Bj] (7.2).

b. B0 ¼ pB, where p is the antecedent truth value of A0 ¼ A (7.4).

What are the advantages and disadvantages of using separate rules for fuzzi-

7.2

¬cation and defuzzi¬cation of discrete fuzzy sets?

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139

7.12 QUESTIONS

We wish to classify regions of an image from several numeric measurements.

7.3

We set up linguistic variables for the input data, and a non-numeric discrete

fuzzy set if possible region classi¬cations. The input variables have been fuz-

zi¬ed. We have several rules whose consequent would set the grade of mem-

bership of classi¬cation Artifact. Two of these rules are concurrently ¬reable;

of these, one would set the truth value of Artifact to 0.7, and one would set

it to 0.4,

a. What is the default inference method?

b. What will the truth value of Artifact be after the rules have ¬red?

c. What rationale can you give for using the default inference method?

We wish to use the information from a physiological monitor to evaluate the

7.4

condition of a patient in an intensive care unit. The data include heart rate, sys-

tolic and diastolic blood pressures, percent of oxygen saturation in both arter-

ial and venous blood, and temperature. We set up linguistic variables for

each of these measurements, with ¬ve linguistic terms in each linguistic vari-

able, and fuzzify the input data. The data are collected nearly continuously

(1 sample every 2 s), so we can calculate rates of change for each input vari-

able, and set up corresponding linguistic variables for the rates of change.

Our output consists of two discrete fuzzy sets, one for present condition (good,

fair, poor, bad) and one for changes (improving. stable, deteriorating). We

write rules whose consequents are present condition (condition is good),

and other rules whose consequents are rate of change (state is deteriorating).

We have several rules that have the same consequent. Rule A would set the

grade of membership of “deteriorating” to 0.1; rule B would set its grade of

membership to 0.2; and rule C would set its grade of membership to 0.5.

Since our rules are ¬red in parallel, all three rules are ¬red concurrently. To

what value should we set the grade of membership of “deteriorating”?

We have ¬ve input variables with four possible values for each, and six output

7.5

variable values.

a. How many rules will be required using the conventional IRC method?

b. How many rules will be required by the Combs Union Rule Con¬guration

method?

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8 Resolving Contradictions:

Possibility and Necessity

Chapters 6 and 7 dealt with the modi¬cation of data and truth values under the

assumption that the truth values represent necessity, the extent to which the data

support a proposition. In this chapter, we also will treat truth values that represent

possibility, the extent to which a truth value represents the extent to which the

data fail to refute a proposition. We denote the necessity of proposition A by

Nec(A), and its possibility by Pos(A). Here, we will consider the dynamics of a

multistep reasoning process. In such a process, we often proceed ¬rst to establish

plausible preliminary conclusions that are supported by the relevant data. If the con-

clusions so reached are mutually exclusive, we then proceed to rule out preliminary

conclusions that are refuted by additional data or by existing data looked at more

deeply. In the ¬rst of these steps we consider the extent to which the data support

a conclusion, that is, necessity; in the second step, we consider whether any data

refute a proposition, that is, possibility, and any effect a change in possibility of a

proposition might have on its necessity.

While we accept the de¬nition of possibility and necessity given by Dubois

and Prade (1988), we do not accept that all the axioms used to develop conven-

tional possibility theory [Klir and Yuan (1995), Chapter 7] are valid for fuzzy

reasoning with rule-based systems. In particular, we do not accept the axiom

that the propositions involved are nested (A ) B ) C Á Á Á), nor the use of

min“ max logic when combining A and NOT A. This last objection is especially

valid since a family of fuzzy logics has been constructed that does obey these

fundamental laws (Buckley and Siler, 1998b and 1999). Since it is well known

that min “ max logic does not obey the law of the excluded middle (A and

NOT A ¼ 0), its use in combining A and NOT A to yield min(a, 1-A) is

invalid. For these reasons, the treatment of possibility and necessity developed

here differs substantially from conventional theory; hence, it is somewhat contro-

versial. Aside from theoretical considerations, the theory here has one big

advantage: it works.

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

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

141

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142 RESOLVING CONTRADICTIONS: POSSIBILITY AND NECESSITY

8.1 DEFINITION OF POSSIBILITY AND NECESSITY

Dubois and Prade de¬ne possibility and necessity as:

Necessity of a proposition (Nec) is the extent to which the data support its truth.

Possibility of a proposition (Pos) is the extent to which the data fail to refute

its truth.

The term “necessity” is closely analogous to the term credibility in Dempster“Shafer

theory, and the term “possibility” is closely analogous to their term plausibility.

Both possibility and necessity are truth values in [0, 1]. Clearly we are dealing

with two different kinds of truth values. For example, suppose that we have only

this information about proposition A: its current necessity is 0.3, and its current

possibility is 1.0; the truth value (necessity) of proposition P that supports A is

0.4, and the truth value (necessity) of proposition Q that refutes A is 0.2. Nec(A),

the necessity that A is true, is then max(0.3, 0.4) ¼ 0.4. Since Q refutes A, the

extent to which Q fails to refute A is NOT Nec(Q), or 0.8. Then, considering

these new data, Nec(A) ¼ 0.4, and Pos(A) ¼ 0.8.

If our fuzzy reasoning language permits it, we could specify the truth value of a

proposition by the two values, possibility and necessity. However, few if any current

rule-based fuzzy languages permit storing more than one truth value, usually neces-

sity. But since we can calculate the possibility of a proposition from the necessities

of refuting data, we can deal with the most important aspect of possibilities within a

necessity-based system.

8.2 POSSIBILITY AND NECESSITY SUITABLE FOR

MULTI-STEP RULE-BASED FUZZY REASONING

We now develop formulas for possibility and necessity that are suitable for complex

multistep forward-chaining rule-based fuzzy reasoning systems. (We concur with

Anderson (1993) that rule-based systems offer a very powerful way of emulating

human reasoning.) At the ith reasoning step, we denote Nec(A) as Nec(A)i , and

Pos(A) by Pos(A)i . (If a relationship is always true the i subscript is omitted.)

8.2.1 Data Types and Their Truth Values

First, we consider the way in which our system uses truth values. Most truth systems

maintain only one truth value, necessity; if we are to maintain both possibility and

necessity, each truth value consists of these two values.

Now consider the different types of data that we might have. There are scalar or

single-valued data; typically these include integers, ¬‚oats, and strings; to each of