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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: Defining fuzzy descriptors (was  NOT and DIFF)
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Date: Fri, 28 Mar 1997 11:37:51 GMT
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WSiler (wsiler@aol.com) wrote:
: It sounds as if our two developments are indeed very close together. And
: as a matter of fact, with three or more elements jammed together with a
: splattering of parentheses, getting the correct r values for compound
: clauses is indeed an acute pain in the neck. There may be an easy way to
: do it, but we haven't found it yet. We are trying to work on a general
: parser for any amount of compound clauses, <snip>

That's one approach.  Another is implicit, or rather explicit,
in my development.  There, the whole membership *functions*
constituting the operands, not only the pointwise membership 
values, are in effect arguments to the function determining 
the pointwise values of the connective.
The correlation coefficient between membership function pairs
is posited to be the key determinant: if the correlation coefficient
is positive beyond some threshold, the relation between the two
membership functions is one of "positive semantic consistency", 
and the min/max rules (min[a,b], max[a,b]) follow.
If the correlation coefficient is negative, again beyond some 
threshold, the opposite holds, ie. there is "negative semantic
consistency", and the bounded-sum rules (max[a+b-1,0] and
min[a+b,1] hold.  (This is the rule pair that gives us back
law of excluded middle, and law of contradiction.)  
In between the two thresholds, one positive and the other 
negative, the product and product-sum rules (ab, and a+b-ab) 
hold.  This of course is an idealization, in which as the 
correlation coefficient (between the two membership *functions*
constituting the operands) ranges from -1 to +1, the rules change 
in bang-bang fashion from bounded-sum, to product/product-sum, to 
min/max.  It is also possible to have *mixtures* of these rules 
on each side of zero, in a more smoothly varying fashion, which 
of course is the general case that subsumes bang-bang transitions 
which constitute an interesting special (and extreme) case.  
Now, my point is, if the transformation function
is properly chosen, it *should* be the case that the restrictions
you have identified as needing to be fulfilled if associativity
and distributivity are to hold, would automatically be fulfilled.
It may in fact be a desideratum for deciding on the acceptability
of any transformation rule that may be posited.  If we have a
transformation rule that can be shown to be associativity/distributivity
conforming, then it would remain sufficient to consider operands
purely pairwise, rather than all-at-a-time, as becomes necessary
in your development, and which I can see could become a difficult
endeavor in general.  Seen another way, your difficulty stems 
from the lack of truth-functionality in your rules of combination.
In my development, there is again lack of truth-functionality,
but this lack is sought to be overcome by making the rules, in
some sense, membership-function-functional, to coin a phrase, which
provides the determinacy (ie. the computers can easily be told
how to do it) that is required.

Finding associativity/distributivity conforming transformation
functions would make an interesting subject for mathematical 
research, which, if I were 24 again, I might take up with relish.  
Right now, my inclination is to let the younger researchers have a stab
at it.

: Best - Bill Siler

Best regards,
S. F. Thomas


