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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: Q: Implementing Fuzzy OR
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Date: Fri, 11 Apr 1997 11:20:13 GMT
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Stephen Regelous (regelous@antispam.gen.nz) wrote:
: When multiple rules assert the same fuzzy output value, I assume that
: the max operator should be used to arbitrate between the weights of
: these rules, thus providing fuzzy OR logic. <snip>

: However this approach ignores important inference information. <snip>
: In these cases
: I have found it useful to sum the rules, of course this means that some
: output values will have a weight exceeding 1.

: So my question is this: 
: Is this how the max operator is supposed to function and if so how
: should one obtain the functionality of summing rules which make the same
: assertion?

This is a very good question that occurs to most people
on first acquaintance with a fuzzy set theory or fuzzy logic
in which OR is *defined* in terms of the max operator, and
correspondingly AND is *defined* in terms of the min operator.
I agree with the incredulity you express as to whether that
is how the max operator is supposed to function, and agree
also with your common-sense engineer's solution of improvising
an integration rule (presumably normalized) in favor of the 
maximization rule which the received theory proffers.  

While the integration rule
"works", I doubt very much though whether it could be justified
within any sort of fuzzy set theory or logic.  (You actually
have stumbled on the core of the Bayesian controversy in 
statistical inference--Bayesian inference appears to work
fairly well, but philosophical and theoretical justification
is elusive.  But that's another story.)  The rule you seek,
judging from the little I could gather of the context in
which it is to be applied, is the product-sum rule, namely
a+b-ab, as another poster on the thread has already remarked.

The bounded-sum rule, min(1,a+b) is an alternative, but I would
reject that one also in this context, because it would be
appropriate only if a strong *negative* semantic consistency 
relation between the rules could be assumed, and is the 
flip-side of the max rule, which is appropriate when a
strong *positive* semantic consistency relation between the
rules could be assumed.  As you implicitly reject the latter,
I doubt whether you want to embrace the former.  (Even though
there appears to be some "summing" going on within the bounded-
sum rule, it is deceptive due to the bound, and would leave 
you unable, for example, to distinguish two rules each weighted
0.5, from two rules, each weighted 1.0.  That also is not
what you want, and I take your "summing" to be a normalized
summing, akin to Bayesian integration.)

To stay within the confines of a fuzzy set theory, albeit a
non-Zadehian one that is not limited a priori to a min/max
calculus, the appropriate rule is the product-sum rule,
which is appropriate when the rules in question may be deemed
semantically independent of each other.  

The notions of semantic consistency, positive and negative,
and semantic independence alluded to above, are exactly
as you would think them to be from an informal common-sense
viewpoint.  But a formal exploration and discussion of these
notions from within a non-Zadehian fuzzy set theory of 
semantics may be found in my _Fuzziness and Probability_ (1995).

: Thanks in advance.

: ________________________________________________________________________
: Stephen Regelous				regelous@actrix.gen.nz
: Weta Ltd - Digital Film Effects			stephen@wetafx.co.nz

Regards,
S. F. Thomas
