> If A = percentage of students failing question A and,
> B = percentage of students failing question B,
> then if
> C = percentage of students failing both questions A and B,
> C will lie in these ranges, and C < = A ZAND B,
> what is the rational for A ZAND B = min(A(x),B(x)) then ?
>
It is probably improper to use the min-max operator here. If we are using
probabilities, then (if we want to preserve the laws of excluded middle and
contradiction)the proper operator to use depends on the prior association (or
correlation) between A and B. In your case, the probability of failing question
A and of failing question B are almost certainly somewhat imperfectly
correlated.
Let me review three of the inifinity of possible operators.
The Zadeh operators, valid for max possible association are:
A ZAND B = min(A, B)
A ZOR B = max(A, B)
The probabilistic operators, valid for zero correlation, are:
A PAND B = A * B
A POR B = A + B - A * B
The Lukasiewicz operators, valid for max negative correlation, are:
A LAND B = max(0, A + B - 1)
A LOR B = min(1, A + B)
If the probability of failing question A and of failing question B are
positively correlated to the maximum extent possible, then the Zadeh operators
are the correct ones to use. For example, if all students failing question A
(say 45%) also fail question B, and some students fail question B (say 55%) but
don't fail question A, then the probability of failing both questions is the
mininum of .45 and .55, or point 45. The probabililty of failing at least one
question is the maximum of .45 and .55, or .55.
If the probability of failing question A and of failing question B are
negatively correlated to the maximum extent possible, then the Lukasiewz
operators are the correct ones to use. For example, if all students failing
question A (say 45%) do not fail question B, and some students fail question B
(say 55%) but only 10% fail question A, then the probability of failing both
questions is the maximum of 0 and (.45 + .55 -1), or 0.1. The probabililty of
failing at least one question is the minimum of 1 and (.45 + .55), or 1.
If the probability of failing question A and of failing question B are
uncorrelated then the probilistic operators are the correct ones to use. For
example, if students have a random probility of failing question A (say .45 and
also a random probiltiy of failing question B (say .55), then the probability
of failing both questions is the (.45 * .55), or 0.2475. The probabililty of
failing at least one question is (.45 + .55 - .45 * .55), or .7525.
This example is absurdly simple. However, we can extend it to more interesting
applications to fuzzy expert systems.
First, consider the probability of failing question A (.45) and of NOT failing
question A (.55). These are maximally negatively associated, so we should use
the Lukasiewicz operators, so the A LAND NOT A = max(0, (.45 + .55 - 1) or
zero, and intuitively correct answer.
Since the probability of failing question A can be used as a definition of the
truth of the statement "Student fails question A", and similarly for B to
"Student fails question B", then we can use the above ideas in evaluating the
truth of the rule:
IF (Student fails question A) and (student fails question B) THEN....
and on to more complex rules.
We can extend the idea of maximally negative association to the notion of
semantic inconsistency. For example, when combining Slow = .45 OR Medium = .25,
since Slow and Medium are semantically inconsistent we should use the Lukasiewz
OR in combining their weighted membership functions. This would probably play
merry hell with fuzzy control engineers, but for more general fuzzy expert
systems it gives intuitively reasonable results.
William Siler
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