members are combined with their membership functions to yield modi¬ed member-

ship functions. Step 6: The modi¬ed membership functions of the consequent

discrete fuzzy set, a linguistic variable, are combined to yield a membership function

for the entire consequent linguistic variable. Step 7: A single numeric ¬‚oating-point

variable is derived from the combined membership functions.

In this section, we assume that Step 1, fuzzi¬cation, has already taken place.

Methods of Step 2, determining truth values of the antecedents, has been covered

in Chapters 3 “5. Steps 5 “ 7 have already been covered in Sections 7.2, 7.3, and

7.8. In this section we will be concerned with Steps 3 and 4; determination of

new grades of membership q0j of consequent fuzzy set members from antecedent

truth values, rule con¬dences and prior grades of membership of consequent

fuzzy sets in an entire rule set. We will use a notation that is suitable for an entire

rule set. As noted in Section 7.3, monotonic reasoning is indicated as a default

reasoning type for modi¬cation of grades of membership of discrete fuzzy sets.

For monotonic reasoning, that is often used when determining grades of member-

ship for discrete fuzzy sets,

Q0 ¼ (P R) OR Q (7:18)

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131

7.9 MODELING THE ENTIRE RULE SPACE

We de¬ne non-monotonic reasoning as

Q0 ¼ P R (7:19)

and monotonic downward reasoning as

Q0 ¼ (P R) AND Q (7:20)

Non-monotonic reasoning is most often used only when invalidating data; rules of

this type are extremely unlikely to ¬t into this framework. Monotonic downward

reasoning is usually incorporated in the defuzzi¬cation process; since defuzzi¬ca-

tion is deferred to the next rule-¬ring step, monotonic downward reasoning is also

unlikely to be used in this framework.

7.9.1 Conventional Method: The Intersection Rule

Con¬guration (IRC)

Conventionally, each input discrete fuzzy set is represented by members that

describe the input variable. For example, a numeric input variable velocity

might be represented by discrete fuzzy set speed with members Slow, Medium,

and Fast. A typical rule might be

IF (A1 is a1,i1 AND A2 is a2,i2 AND A3 is a3,i3 . . . ) THEN Q is qj (7:21)

in which A1, A2 , . . . are input discrete fuzzy sets and Q is an output discrete

fuzzy set.

With N input discrete fuzzy sets, each with M members, and Q members of the

output fuzzy set, we would have NM antecedents and Q consequent fuzzy set

members. If a single consequent fuzzy set member is associated with each antece-

dent, we have NM rules. In general, we could have up to NMÃ Q rules. If the R

matrix is sparse, the actual number of rules could be considerably less. In either

case, the rule set could be evaluated by (7.14), (7.15), or (7.16), depending on the

type of reasoning employed.

The well-known problem with systems using rules of this type is the exponential

growth in the number of rules as the number of input variables increases. Assume an

economical formulation in which there is only one consequent fuzzy set member

associated with each antecedent. Say each of the input discrete fuzzy sets has ¬ve

members. For two input discrete fuzzy sets, we have 25 rules; for three input sets,

we have 125 rules; for 4 input sets, we have 3125 rules, an exponential increase.

This makes it exceedingly dif¬cult or impracticable to use this type of rule with a

large number of input variables.

We now consider an alternate model of the entire rule set, quite different

from (7.21).

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132 INFERENCE IN A FUZZY EXPERT SYSTEM II

7.9.2 The Combs Union Rule Con¬guration

William Combs of the Boeing Company has devised a method to avoid the expo-

nential growth in the number of rules as the number of input variables increases

(Combs and Andrews, 1998). Combs proposes a Union Rule Con¬guration or

URC. The structure of the rules in the Combs method exploits the following equiv-

alence in the propositional calculus:

½( p1 and p2) implies q is logically equivalent to ½( p1 implies q) or ( p2 implies q)

(7:22)

Rules written in the conventional style, which Combs calls the IRC, are similar to this:

IF (U is Ai AND V is Bj AND W is Ck ) THEN Z is DRn (7:23)

where DRn denotes the output fuzzy set member assigned to the nth rule. This rule

becomes three in the simplest implementation of the Combs method URC:

IF (U is Ai ) THEN Z is DR1

IF (V is Bj ) THEN Z is DR2 (7:24)

IF (W is Ck ) THEN Z is DR3

The truth value of D is obtained by averaging DRn over N;

1X

D¼ DRn (7:25)

Nn

where N is the number of rules in which DRn appears in the consequent.

If we have N input variables X, Y, . . . , i, j, and k in range from 1 to M, and we wish

to include rules with all combinations of i, j, and k, the IRC will require MN rules. The

URC will require MÃ N rules. The number of rules in the IRC goes up exponentially

with the number of input variables N; the number of rules in the URC goes up linearly

with N. For control systems with a large number of input variables, the Combs URC

system offers considerable economy in the number of rules, as illustrated in Table 7.2.

In practice, the “OR” operator in (7.22) is de¬ned to yield the mean of the oper-

ands. More advanced formulations of the URC (Weischenk et al., 2003) deal

successfully with the questions of precise equivalence between the IRC and URC

methods by employing more than one block of rules and rule-¬ring step. Neverthe-

less, the simple formulation of the URC in (7.24) and (7.25) works quite well in

many if not most applications.

The utility of the Combs method for classi¬cation programs is less clear than for

control programs, and depends on the type of fuzzy set chosen for the input vari-

ables. If numeric input variables are fuzzi¬ed into fuzzy sets with members such

as Small, Medium, and Large [as done in Kasbov (1998, p. 219)], and these in

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133

7.9 MODELING THE ENTIRE RULE SPACE

Number of Rules Required by IRC and URC methods a

TABLE 7.2

Number of Input Variables

1 2 3 4 5

IRC URC IRC URC IRC URC IRC URC IRC URC

3 3 9 6 27 9 81 12 243 15

a

We assume that there are three members to each input variable (fuzzy set); that rules are required for each

combination of the input fuzzy set members; and that each rule has one consequent clause.

turn are used in rules similar to (7.23), the reduction in number of rules provided by

the Combs method may be realized. If, however, the membership functions are tai-

lored to the classi¬cations, with input fuzzy set members such as Class_1, Class_2,

and Class_3, the URC method may actually produce a greater number of rules than

the IRC. For example, in the famous Iris classi¬cation problem, the IRC program

iris.par requires three rules of the type

IF (in Data PL is setosa AND PW is setosa AND SL is

setosa AND SW is setosa) THEN Class is setosa;