There is a point about operating on fuzzy sets that Bill feels you may not be
appreciated.
Say we have two fuzzy sets A and B, ANDed to produce fuzzy set C.
The question of inclusion of the members of A and B in C is a binary one; C
only has members which are in both in A and B. Say we have member x in both A
and B; then x is also in C.
In determining the grade of membership of x in C, we have a decision to make -
which multivalued logic should we use? There is an infinity of logics
available. Restricting this domain to what I would call reasonable (won't go
into that now), the grade of membership of x in C, say mu(x, C) can have (for
AND) a maximum if the Zadeh max-min logic is used, and a minimum if the
Lukasiewicz logic is used. (For OR, max-min gives a minimum and Lukasiewicz
gives a minimum mu(x, C). So the question now is, what logic should I choose?
As a restriction on this choice, let us make the reasonable one that we would
like to preserve the laws of excluded middle and contradiction, so that Elkan's
criticism is wiped out. In fact, we would like to preserve ALL the laws of
binary logic, as listed in Klir's book.
Jim Buckley and I have a paper coming out in the IEEE transactions on Fuzzy
stuff which proves the following:
If the quantities being operated on have no prior association, you can use any
logic you please from a single-parameter family we give which includes the
max-min (max positive association), the sum-product probabilistic logic (zero
association) and the Lukasiewicz logic (max negative association). (A and A
have max positive prior association; A and NOT-A have max negative
association.) So we are free to choose whatever logic we please in most cases.
There may be other problem-specific restrictions which would lead us to one or
another logic.
We have not proved the following, but both SF Thomas and I have come to the
same conclusion (tentative, pending proof). Suppose we are operating two
members of the same linguistic variable, say SLOW and FAST. These are
incommensurate members, and in this case we should employ the Lukasiewicz
logic.
So I tentatively think that in Bill's method, we may have the additional
problem-specific restrictions on logic choice I mentioned above. For example,
in generating x *y, we should use the probabilistic logic.
In short, the Combs method has worked on a variety of problems, and if the
above idea of combining fuzzy sets using appropriate logics, not necessarily
only max-min, is used we may have a great deal of flexibility. Certainly the
aim of avoiding the exponential increase in the number of rules as problem
complexity increases is incredibly desirable - we can definitely use more
research based on his method.