From: Jeff Drake (jefdrake@nmsu.edu)
Date: Thu Jan 20 2000 - 22:06:56 MET

There is a fundamental problem with the formulation of the example, IF
outsideIsGreen AND insideIsRed THEN objectIsWatermelon in terms of fuzzy
logic.

1) Begin with this fuzzy rule
IF outsideIsGreen THEN objectIsWatermelon

The output of a fuzzy inference system is another fuzzy set.
To get any "crisp" value from this relationship the output must be
"defuzzified".
The crisp result using center of area (COA) or any other method will be the
same value, regardless of to what degree outsideIsGreen. This is because
the degree to which outsideIsGreen simply becomes a scaling factor w. For
example,

For a single rule with single antecedent using Mamdani's fuzzy implication
we have:

mu_out = w min mu_B

where
w is the degree to which outsideIsGreen for the sample
mu_out is the fuzzy output set, represented by its membership function
and mu_B is the fuzzy set objectIsWatermelon represented by its membership
function.

The result is one just gets a scaled version of the membership function
objectIsWatermelon. Defuzzifing this scaled version by say COA gives the
exact same crisp output, regardless of the value of w.

This same situation holds for the multiple antecedent case such as

IF outsideIsGreen AND insideIsRed THEN objectIsWatermelon

2) So what is going on?

You must have another fuzzy rule.

In classical set theory A Union not(A) = omega (the entire universe)

This is known as the "Law of the excluded middle"

In Fuzzy set theory there is no such identity!
In fuzzy set theory A Union not(A) not equal omega

So you must specify another rule in addition to your first rule such as:

IF Not(outsideIsGreen) OR Not(insideIsRed) THEN not(objectIsWatermelon)

Then the inference system can perform correctly.
(after taking AND as min as pointed out by others)

--
Jeff Drake
New Mexico State University
Electrical and Computer Eng. Dept.
P.O. Box 30001 Department 3-0
Las Cruces, New Mexico 88003-8001
(505)646-4408 My Office (505)646-3115 EE Dept.

-----Original Message-----
From: fuzzy-mail@dbai.tuwien.ac.at
[mailto:fuzzy-mail@dbai.tuwien.ac.at]On Behalf Of Will Dwinnell
Sent: Wednesday, January 19, 2000 7:23 PM
To: Multiple recipients of list

Kami ROUSSEAU wrote:
"For my M.Sc. in computer science, my subject is the development of an
expert system. I wanted to use fuzzy logic, but my director said that fuzzy
rules do not take added evidence into account. Here is the exemple that he
gave me:
(assume that the AND operator is defined as "maximum")
IF outsideIsGreen AND insideIsRed THEN objectIsWatermelon

Suppose we only have 1 fact:
a certain fruit's outside is 60% green.
we can conclude that the fruit is a watermelon with possibility 60%

Suppose we add the following evidence:
the fruit's inside is red at 40%.
after recomputing the truth value of the rule, we still get 60%.

My director says that the possibility should have increased, but it did not,
so fuzzy logic is not appropriate for this kind of reasoning.

What do you think of his argument?"

It assumes very simplistic (and strange) rules. A fuzzy system like this
involves both rules and facts. His rules are obviously of the form "IF
antecedent is X true, THEN conclusion is X true". Besides, fuzzy logic is
not limited to min/max operators. See Earl Cox's first book on fuzzy logic
for a slew of alternatives.

Will Dwinnell
predictor@compuserve.com

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