From: mailbox@dmitry-kazakov.de   
      
   On Tue, 11 Dec 2007 16:36:10 +0100, bv wrote:   
      
   > Like in the tutorial and many other books, one assumes the rule set to be   
   > complete. For instance, if one uses 2 variables each being attached to a   
   > Fuzzy Set made of 3 Fuzzy States, on expects to have 9 rules like   
   > IF V1_FSi is XXX AND V2_FSj is YYY THEN ...   
   >   
   > I have always wondered what consideration must be made when using an   
   > incomplete rule set (e.g. 7 "AND" rules), and/or a rule sets using other   
   > operators i.e. using OR, NOT ?   
      
   These are two separate issues.   
      
   1. If you use an incomplete set, [i.e. formally, there exist points in the   
   input space (Cartesian product of variable's domains) for which no rule   
   from the set reaches the truth level 1], then under standard inference   
   rules you might get a non-normal distribution in the output space (the   
   outcomes). That is - no outcome reaches the truth level 1. This is not   
   uncommon, using triangular linguistic variables has the same [undesired]   
   effect.   
      
   When truth levels are possibilities then, this is a contradiction. [A   
   distribution of possibilities always reaches 1 somewhere = you'll have a   
   problem in interpreting what you get.]   
      
   Same is true if variables are intuitionistic. An incomplete set may yield a   
   contradictory outcome. For example: possibility could exceed necessity.   
      
   Otherwise it depends on the empiric norm you use.   
      
   2. There is nothing special in OR. NOT, XOR etc under standard inference.   
   Assume:   
      
      not X in A <=> X in ~A   
      
   and apply de Morgan's laws to normalize the rules to the conjunctive normal   
   form. Here ~A is the complement set of A. Of course, standard rules honor   
   de Morgan's laws. But again, if it is non-standard, then the meaning   
   depends on the norm.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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