Notes on the Combs Method

The Combs Method.

The Combs method depends on a rather simple result stemming from classical logic. The rule

'a' AND 'b' ENTAILS c

can be re-written as

(a ENTAILS c) OR (b ENTAILS c)

where ENTAILS is a Boolean operator that has it's own truth table:

a	b	A ENTAILS b
TRUE	TRUE	TRUE
TRUE	FALSE	FALSE
FALSE	TRUE	TRUE
FALSE	FALSE	FALSE

The operator 'ENTAILS' can be expressed as “IF a THEN b”. A better explanation of this would be that if 'a' is true, then 'b' must also be true. If 'a' is false then it makes no difference if 'b' is true or false.

With this rule in mind, on first glance it would appear that the third rule from the table above is incorrect.

False ENTAILS true = true.

However, this rule is actually quite logical. To help clarify the logic behind this rule, consider the following example by (Millington, I. 2006).

IF I'm-in-the-bath THEN I'm-wet

so if were in the bath(which is not empty of water), then we will obviously be wet. However, the bath may not be the only reason for being wet: getting caught in heavy rain, being splashed by a hose and so on. Therefore the above rule can still be correct when I'm-wet is true and I'm-in-the-bath is false.

What the above equates to is that we can write:

IF a AND b THEN c

or

(IF a THEN c) or (IF b THEN c)

MAKE SURE I MENTION THAT THE CONCLUSIONS FROM RULES ARE OR_ED TOGETHER, this allows us to split this new method of formatting rules into two individual rules

IF a THEN c

IF b THEN c

For clarity, the above shall be referred to as the Combs format.

Larger rules can also be expressed in this formating:

IF a1 AND . . . AND a(n) THEN c

can be re-done as

IF a1 THEN c

:

IF a(n) THEN c

This has taken us from requiring rules in all the possible state combinations to a much simpler set of rules requiring only a single state in the IF and THEN clauses. This method removes the need for multiple combinations which in turn leaves us with the number of rules equalling the number of states. This cuts the growth of rules from exponential to linear. E.g. having 10 inputs each with 5 states would result in only 50 rules compared to the unmanageable 10million rules.

When using the Combs method, all the rules have to be built from scratch. It is not possible to change a general set of rules into Combs format as truth tables using Combs format rules cannot be created or represented. While their will be examples that can convert rather easily, these are nothing more than a convenient coincidence.

As an example of this, consider the following pair of rules:

IF corner-entry AND going-fast THEN brake

IF corner-exit AND going-fast THEN accelerate

These two rules would be broken down into the following four rules:

IF corner-entry THEN brake

IF going-fast THEN brake

IF corner-exit THEN accelerate

IF going-fast THEN accelerate

Decomposing the original two rules into Combs format results in an inconsistent rule set. Rule 2 and 4 from the decomposed set contradict each other, we cannot possibly accelerate and break at the same time. Therefore which rule do we follow? The answer to this question would depend on our location within the corner.

When limiting a fuzzy system to only using the Combs format, the systems overall sophistication will unavoidably restricted. However, the tractability of rule creation allows these rules to be altered more easily (Millington, I. 2006)

It is not practical to apply the Combs method to traditional logic, the result would be incredibly restrictive. But when used with FL, where multiple fuzzy states can be active at any one time, this means that the different states can interact with each other (meaning we can both accelerate and brake at the same time, but the overall change in speed would be determined by the DOM of each state). The Combs method produces rules that are still able to produce interaction effects between all the states, even although these interactions are not explicit within the rules.