GENERALIZING AND AND OR OPERATORS

This chapter will deal with the problem of combining truth values. In Chapter 3, we dealt with definitions of the AND, OR, and NOT operators for multivalued logics, and pointed out that many definitions of these operators can be defined, which reduce to the classical definitions for crisp logic for crisp truth values. In Chapter 4, we presented the mathematics of a family of fuzzy logics that obey the classical laws of Excluded Middle and Non-Contradiction. In this chapter, we will present a general treatment of combining truth values, with the objective of calculating the truth value of rule antecedents. In particular, we discuss use of prior association between operands as a guide to selecting which fuzzy logical operators to use.

The concept of truth-functional operators is based on the idea that the result of applying these operators will yield a value that depends only on the values of the operands. As we have seen in Chapter 3, any of the three definitions of AND and OR are truth functional, and may all give different results; however, we are not given any basis on which to make a choice among the available operators. Certainly, the Zadehian min – max operators are used more often than any other, for a variety of reasons, including some rather nice mathematical properties, and in the real world they can usually be made to work.

Let us back up and ask what properties the classical AND, OR, and NOT oper-ators posses. Among the most basic are the Law of the Excluded Middle and Law of Non-Contradiction. These can be quite simply formulated:

Excluded middle: P AND NOT P¼ false¼0 (5:1) Non-Contradiction: P OR NOT P¼true¼1 (5:2) Unfortunately, all the definitions for the AND, OR, and NOT operators except the bounded sum fail to obey these laws. While this can be disturbing to some people, many fuzzy mathematicians seem to regard it as a virtue. In Chapter 4, we developed the mathematical theory for a family of fuzzy logics that does obey these laws; in this chapter, we discuss the origin of this development and its utility.

Fuzzy Expert Systems and Fuzzy Reasoning, By William Siler and James J. Buckley ISBN 0-471-38859-9 Copyright#2005 John Wiley & Sons, Inc.

5.1.1 Correlation Logic: A Family of Fuzzy Logical Operators that Obeys Excluded Middle and Non-Contradiction Laws

We felt that the inability to make a rational choice of logical operators and the failure to obey the laws of Excluded Middle and Non-Contradiction creates a rather unsatisfactory situation. Spurred by a paper by Ruspini (1982), we investi-gated what effect prior associations between the operands of the logical operators might have on the choice of a proper set of operators. We adopted a model for fuzzi-ness presented by Klir and Yuan (1996, pp 283ff). In this model, a number of persons were asked whether a proposition were true, and were restricted to true (1) or false (0) responses. The truth value of the proposition was considered to be the average of all responses. As the number of observers increases without limit, the granularity of our estimate of the truth value vanishes; the estimate of the truth value becomes the probability that an observer would say that the proposition is true. The following treatment is taken from Buckley and Siler (1998b, 1999).

We extend Klir and Yuan’s model just above to two propositions simultaneously presented, and find the probability that an observer would report that P is true; that P AND Q is true; that P OR Q is true; and the association between the individual reports of the truth of P and of Q. Table 5.1 gives a sample of such reports.

We found that if the truth values of P and Q were positively correlated as strongly as possible, the Zadehian AND/OR operators were correct in predicting the mean value for P AND Q and P OR Q. If the individual P and Q truth values were maxi-mally negatively correlated, the bounded sum operators gave correct results. If the individual P and Q truth values were uncorrelated, the product-sum operators gave correct results, as would be expected from elementary probability theory. In Table 5.1, the correlation coefficient is 0, and the truth values obtained for P AND Q and OP OR Q are those we would expect from probability, assuming independence.

This approach yielded a family of logical operators with a single parameter; the prior correlation coefficient between the operands. We concluded that a rational

TABLE 5.1 Sample Table of Observer Reports of Truth of Two Propositions, P and Q^a

Observer P Q P AND Q P OR Q

choice among logic operators could be based on information regarding such associ-ations. The family of AND/OR operators returns the truth values of P AND Q and P OR Q given the truth values of P(a) and Q(b) and a single parameter, r, the corre-lation coefficient between prior values of a and b.

We first place a restriction on the maximum and minimum permissible values of the parameter r, ru, and rl, respectively, and from these restrictions derive a working value for r, r⁰. The reason for this is that the values of a and b may make some values for r impossible. For example, if a¼0.2 and b¼0.6 it is impossible for these values to be perfectly correlated. If the specified r is less than rl, then the formulas will use the bounded sum operators; if the specified r¼0, the formulas will use the sum-product operators; if the specified r is greater than ru, the formulas will use the Zadehian max – min operators. We first present the formulas for this family of AND/OR operators, then present some numerical examples of their performance.

Since we are interested in implementing these operators on a computer, we will present them as statements in BASIC.

ru¼(min(a, b)ab)=SQR(a(1a)b(1b))

Table 5.2 gives some typical results of applying these formulas.

Of course, this family of operators is not truth-functional, since other information is required besides the values of the operands. In many cases, that information is lacking and the Zadehian operators are a good default for expert systems, provided that we do not combine A and NOT A. They are the mathematical equivalent of the

“a chain is no stronger than its weakest link” common sense reasoning. Further, any other multivalued logic limits the complexity of the rule antecedents that can be used: If several clauses are ANDed together, the resulting truth value tends to drift down to 0; if they are ORd together, the resulting truth value tends to drift up to 1.

TABLE 5.2 Examples of Application of Logic Operator Family Using the Prior Correlation Coefficient

r a b r⁰ aANDb aORb Resulting Logic

21 0.2 0.6 20.612 0 0.8 Bounded

0 0.2 0.6 0 0.12 0.68 Product

þ1 0.2 0.6 0 0.2 1 Max – min

21 0.4 0.8 20.612 0.2 1 Bounded

0 0.4 0.8 0 0.32 0.88 Product

þ1 0.4 0.8 0.408 0.4 0.8 Max – min

5.1 GENERALIZING AND AND OR OPERATORS 87

There are two circumstances under which there is no question as to prior associ-ations. If P and NOT P are the operands, they are maximally negatively associated; if P and P are the operands, they are maximally positively correlated. If we want to use multivalued logic, and wish to retain the laws of Excluded Middle and Non-Contradiction, we can use any multivalued logic we please unless the equivalent of P and NOT P or P AND P appear in the same proposition. We can save Excluded Middle and Non-Contradiction by using the Zadehian max – min logic when com-bining P and P, and by switching to the bounded-sum operator pair when comcom-bining P and NOT P. (We might require rearranging the proposition to bring P and P together, and to bring P and NOT P together.)

In most cases, we will not have prior information available on which to base a choice of which logical operators we should employ. However, we can choose a default set of operators. Let us assume that our observers share a common back-ground. It would seem that in this case, they would tend to agree more than to dis-agree. They might have more or less strict ideas as to when to say a proposition is true, resulting in individually larger or smaller estimates as to the truth of P and of Q, but if the observer who is more strict says that P is true, it seems likely that the less strict observer would also say that P is true. (Our formulation provides for precisely this contingency.) So it seems likely that the default correlation should beþ1, yield-ing the Zadehian operators; indeed, many years of practice have shown that this choice works.