Re: Adaptative fuzzy systems and classification

WSiler (wsiler@aol.com)
Tue, 22 Sep 1998 02:41:10 +0200 (MET DST)

>I am referring to the infinitely valued logic which
>Lukasiewicz developed which is a superset of the later
>fuzzy logic.

Certainly Zadeh's fuzzy logic is later than Lukasiewicz. However, I can't see
how the Lukasiewicz logic is a superset of fuzzy logic; both are multivalued
logics, albeit differently defined.

Zadeh's logic is so well known that I feel a little silly repeateing it here,
but for completeness here it is. (p and q are truth values of propositions.)

p AND q = min(p, q)
p OR q = max(p, q)

The most commonly accepted Lukasiewicz logic is:

p AND q = max(0, p + q - 1)
p OR q = min(1, p + q)

Neither logic is a subset of the other; they are simply two instances of the
possible multivalued logics.

If we wish, we can imagine a binary process which gives rise to truth values
between zero and one. We place our two propositions on the blackboard, and then
poll a large number of students as to whether the two propositions are true are
false (no inbetweens). We take the average value for the truth p or proposition
P, and the average for the truth value q of proposition q. Suppose p comes out
to be .35, and q to be .55.

Now we look to see how many students voted by P AND Q to be true, and the
number who voted either P OR Q (or both) to be true.

So which is the superset? Neither one. The Zadeh logic has some very nice
mathematical properties, as shown years ago by Bellman and Zadeh. The
Lukasiewicz logic also has a nice property (laws of excluded middle and
contradiction are obeyed), but it also has a large-scale drawback; when ANDing
a bunch of propositions (as is often the case in expert systems) the ANDs of
multiple clauses drizzle off to zero almost immediately, and the ORs of
multiple clauses sail off to 1 equally quickly. This makes it difficult to use
the Lukasiewicz logic in working expert systems except in special cases when we
want to preserve excluded middle and contradiction, e.g A AND NOT A.

It would be nice to know to what use the "negated form of the Lukasiewicz
implication operation can be put.

Earl, you got me into this!

Bill Siler

If the individual student votes for the truth of P and the truth of Q are
positively correlated to the maximum extent possible, our computed value for (p
AND q) will turn out to be the Zadeh value, min(p, q), and our computed value
for (P OR Q) will also be the Zadeh value, max(p, q).

However, if the individual student votes for the truth of P and the truth of Q
are negatively correlated to the maximum extent possible, our computed value
for (p AND q) will turn out to be the Lukasiewicz value, max(0, p + q - 1), and
our computed value for (P OR Q) will also be the Lukasiewicz value, min(1, p +
q).

If we wish, we can imagine a process which will give rise to truth values of p
and q anywhere between zero and 1
Suppose we have a sequence of values of p and q. A little thought will show
that if

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