Newsgroups: comp.ai.fuzzy
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From: sthomas@decan.com (S. F. Thomas)
Subject: Re: Max Min Functions
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References: <4tkkdq$b5u@rockall.cc.strath.ac.uk>  <4uef56$9c2@newsbf02.news.aol.com> <4upg4k$192@rockall.cc.strath.ac.uk> <hubey.840134086@pegasus.montclair.edu>
Date: Mon, 19 Aug 1996 13:12:56 GMT
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H. M. Hubey (hubey@pegasus.montclair.edu) wrote:
: Bruce Postlethwaite <Bruce@ChemEng.strath.ac.uk> writes:

: >The predictive performance of all the fuzzy models I have built is always better 
: >when Sum-product compostion is used instead of max-min. If you look at the 
: >prediction surfaces of the two types of model, the max-product system is much 
: >smoother. 

: I don't have my book with me (Klir and Yuan) so maybe what I am
: writing is off, but it seems to me that it should be possible
: to write the fuzzy OR, and AND functions in the form:

: 	AND(x,y)=t(x)*t(y)
: 	OR(x,y)=t(x)+t(y)-t(x)*t(y)

: where t(x) is the truth-valuation function. Written this way it 
: is isomorphic to probabilities.  Do such things exist, and do I
: not remember them, or do the truth-valuation not obey the above?
: I can't recall seeing anything explicitly like this in Klir &Yuan
: and Klir&Folger. I haven't seen Cox's book.  I just derived two
: truth-valuation schemes in which the above hold, and they are smooth
: and naturally reduce to bivalent logic at the boundaries.

: Are there logics like above in which one can clearly see 
: analogies to probability theory ?

(( cuts ))

I believe the answer to be yes, and not by accident, rather 
necessarily so, if one identifies the foundational concept of
the membership function with a notion of semantic likelihood
exactly akin to the likelihood functions that derive from
a probability model.  If one develops fuzzy set theory from
such a foundation, the rules for union and intersection may
be *derived*, rather than merely postulated.  There remain
an infinity of rule pairs (for intersection and union), but 
there is an overarching logic for the construction of these 
rule pairs, linked with the empirically testable notion of 
the semantic consistency coefficient.  I addressed this question 
once before on the newsgroup, and I repeat that posting below:

 +----------------------------------------------------------
 |The general law is of the form (Thomas, 1995; p.117):
 |
 |            { (1-t).a.b + t.min[a,b],           1 >= t >= 0
 | a AND b  = { 
 |            { (1+t).a.b - t.max[0,a+b-1],      -1 <= t < 0
 |
 |where a and b represent the membership functions of two
 |fuzzy sets A and B ranging on the same universe of
 |discourse, and  -1 <= t <= 1 is a semantic consistency
 |coefficient that depends only on the membership functions
 |a and b.  
 | <snip>
 |The semantic consistency
 |coefficient t is postulated to be a function of the 
 |correlation coefficient between the two membership
 |functions.  
 |
 |As Ellen Hisdal has pointed out in a contribution
 |to this thread, and also in various of her papers
 |which others have cited, when the membership 
 |functions are identified with probabilities of
 |word usage, it is possible to *derive* appropriate
 |rules of combination, rather than to 
 |postulate them directly.  It is in this fashion that the
 |general law mentioned above is derived.  As will
 |be evident, there is in fact an infinity of laws,
 |depending upon the value of t, but three cases
 |are of interest, and yield results already quite familiar
 |from the literature:
 |
 | a AND b = a.b	(t=0 -- semantic independence)
 |
 | a AND b = min[a,b] 	(t=1 -- positive semantic consistency)
 |
 | a AND b = max[0,a+b-1] (t=-1 -- negative semantic consistency)
 |
 |There are corresponding rules for union, as follows:
 |
 | a OR b = a+b-a.b	(t=0)
 |
 | a OR b = max[a,b]	(t=1)
 |
 | a OR b = min[1,a+b]	(t=-1)
 |
 |It should be noted that the min-max rules (t=1) and 
 |the bounded-sum rules (t=-1) are intertwined through the 
 |negation postulate and the correlation coefficient: they 
 |are flip sides of the same coin, so to speak, not
 |separate, arbitrary laws."
 +----------------------------------------------------------

I hope this is helpful.  Incidentally, this formulation allows
the laws of contradiction and excluded middle to be restored
to the fuzzy set theory, without having to sacrifice the core
concept of fuzziness.  It is *not* an attempt merely to reduce
fuzzy to probability.

: -- 
: Regards, Mark

: http://www.csam.montclair.edu/~hubey         hubey@pegasus.montclair.edu


Regards,
S. F. Thomas

