From: mailbox@dmitry-kazakov.de   
      
   On Sun, 15 Jan 2006 20:07:56 +0100, ml wrote:   
      
   > there exist numerous t-norm operators such as min/Zadeh operator,   
   > algebraic product/probabilistic operator and Lukasiewicz operator.   
   > I've read that   
   > if A contains B or B contains A, then u(A and B) = min( u(A), u(B) )   
   > if A and B are independent,  then u(A and B) = u(A)* u(B)   
   > if A and B are mutually exclusive, then u(A and B) = max (0, u(A)+u(B)-1)   
      
   Well the above is true when u is an additive measure (like Pr is). But   
      
   1) if you fix u then there is nothing to talk about and Lebesgue integral   
   is the answer.   
      
   2) No "norm" can be defined in general case for an additive measure.   
      
   > Here, the dependency between A and B seems to play a important role to   
   > help choosing the proper t-norm operator.   
      
   Well, ideally, it is the application domain, which should determine the   
   measure u and thus the "norm".   
      
   > My questions are:   
   >   
   > 1. Do people usually use dependency as criterion to decide which t-norm   
   > should be used?   
      
   I don't think so. Example: the probability theory u=Pr is successfully used   
   for dependent events. Dependency might make things very difficult or even   
   unsolvable, but it would be silly to use this as a motivation for changing   
   u. Consider a CPU vendor, which would use MOV instruction instead of ADD,   
   because the latter seems to be too complex to implement!   
      
   > 2. What about the situation where the dependency between A and B is   
   > unknown? Which t-norms should be used then?   
      
   None, except the possibility measure u=pos, which does not require   
   knowledge of dependency.   
      
   >     I've heard that the "product operator" makes the "member functions   
   > vary more smoothly" than the "min operator", and therefore should be   
   > prefered. It that true?   
   >   
   > 3. What about other t-norms like Einstein and Hamacher?  Does dependency   
   > play a role to distinguish these operators? Are there any unversal   
   > criteria to compare different t-norms?   
      
   I think that the major criterion is completeness. That is - if you have   
   u(A) and u(B), then you can tell something about u(AUB) and u(A^B) for a   
   wide enough range of situations. Even more challenging is to perform   
   inference and so gain knowledge from evidences. This means also, that the   
   results are interpretable in terms of the measure of choice. For example,   
   mathematical statistics fulfills this requirement, because all answers are   
   given as probabilities or estimations of. Whether the input probabilities   
   are multiplied, divided or exponentiated is of minor interests as long as   
   the result is consistent.   
      
   The bottom line is, "norm", if any, is rigidly fixed by the set measure.   
   Or, technically, if you have some algebra of sets A, then a set measure   
   u:A->[0,1] unambiguously translates set operations into some numeric   
   operations on [0,1]. Nothing to choose. End of story.   
      
   P.S. Of course, as it historically happened with probabilities, one can   
   start with numeric operations + and * on chances, and then come to sigma   
   algebra and Pr. But there is no doubt that the notion of measure is more   
   fundamental here.   
      
   P.P.S. My personal belief is that among real-valued measures, there can be   
   nothing really interesting beyond Pr (=> +,*) and pos/nec (=> max, min). If   
   we want more, we should look rather after fuzzy-valued or set-valued   
   measures.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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