From: mailbox@dmitry-kazakov.de   
      
   William Siler wrote:   
      
   > "Dmitry A. Kazakov"  wrote in message   
   > news:...   
   >> EarlCox wrote:   
   >>   
   >> >That aside, there is a BIG   
   >> > difference between the idea of "unknown" as a testable state of   
   >> > variable and the idea that the value of a fuzzy outcome cannot be known   
   >> > until all the rules are fired. Intermediate values of the   
   >> > under-generation outcome fuzzy set are simply erroneous until then.   
   >>   
   >> Aha. This is a very important point we disagree upon. Erroneous =   
   >> "contradictory" which is not "unknown". More precisely, "unknown" does   
   >> not imply (include) "contradictory". So my point is that by substituting   
   >> "unknown" for any "known" value one cannot achive a contradictory result.   
   >> The result might be not enough certain, but never contradictory. In other   
   >> words we might get "reject", but never "error".   
   >   
   > It seems to me that Dmitry is using an improper definition for   
   > erroneous. Let me distringuish between "contradictory" and   
   > "ambiguous". Suppose our consequent datum is a discrete fuzzy set.   
   > After firing the rules, suppose that more than one member of this   
   > fuzzy set has a non-zero grade of membership. Is this an ambiguity or   
   > a contradiction? Depends on whether the members of the fuzzy set are   
   > mutually exclusive. If they are not, as in {Fast Medium Slow} we have   
   > a desirable ambiguity, but not a contradiction, an no error. It they   
   > are mutually exclusive, as in {Ford Chevrolet BMW} we have a   
   > contradiction, and should do something about it. We have a   
   > contradiction, but not an error.   
      
   No it is also ambiguity, provided that the truth values are possibilities   
   (sic!). Consider, the following statement: "I saw a car, it was possibly   
   Ford or Chevrolet". Is that contradictory? No it is just uncertain,   
   ambiguous. Formally:   
      
   Uncertainty (do not know which):   
      X & notX = true   
   Contradiction (neither):   
      X V notX = false   
      
   An example of a contradictory expression about the car type is:   
      
   "provided that any car belongs to Ford U Chevrotet U BWM, this car is   
   neither of mentioned"   
      
   Under error I understand a "meta-error", which cannot be expressed in the   
   terms of the first-order language. That is a logical error or an inability   
   to apply the logical rules correctly etc. To deal with that we need a   
   meta-language, as we do using English to talk about it.   
      
   As I understood Earl Cox means that firing rules out of order is that kind   
   of error. My point is that in a correctly built system based on   
   intuitionistic logic, it is not. And moreover in such a system any partial   
   result will not be contradictory in the sense I have described.   
      
   And returning to the initial question, whether it is possible to introduce   
   derived variables in a rule-based system, the answer is definitely yes. I   
   can always resort the rules having no circular dependencies, so that the   
   firing order will be correct in the sense of Earl Cox. It is a pure   
   software problem, by any means solvable.   
      
   > Earl states that if we only fire one of several concurrently fireable   
   > rules, we can get an erroneous result. Surely if firing one rule gives   
   > {Slow/0.5 Medium/0 Fast/0), and firing the next rule gives {Slow/0.5   
   > Medium/0.9 Fast/0}, the state of the discrete fuzzy set after firing   
   > the first rule is incorrect. But where is the contradiction?   
      
   The contradiction is that the first rule states that Medium is impossible,   
   while it is not. So either the rule is wrong or it uses contradictory data.   
   It is also important to clarify the meaning of Medium/0. Usually it is just   
   an estimation of the truth value for Medium. Let it be a lower estimation,   
   then everything is OK.   
      
   > I'm   
   > afraid that I agree with Earl; when firing several rules concurrently,   
   > the result cannot be considered to be a correct representation of the   
   > state of knowledge until all these rules have been fired. After they   
   > have all been fired, we may have a contradiction, as in {Ford/0.8   
   > Chevrolet/0.5 BMW/0.3}, but that is not incorrect; it may be a valid   
   > picture of the state of affairs up to this point.   
   >   
   >> If we have 4 rules about price, it formally means that we want to   
   >> evaluate price under their conditions:   
   >>   
   >> price | A & B & C & D   
   >>   
   >> This is what you mean talking about simultaneity. But this does not   
   >> prevent us from having:   
   >>   
   >> price | A   
   >   
   > No, it does not. But it means that if we know A and B and C ..., the   
   > result of firing the rule "if A then price" does not correctly   
   > represent the state of knowledge. In this sense, the result is   
   > erroneous.   
      
   "The state of knowledge" in this case is A. So the result is correct in the   
   sense that it does not contradict to any result obtained using another   
   "state of knowledge", as long as both states do not contradict each other.   
   So the theory is not self-contradictory.   
      
   > What this does is focus attention on the necessity for having some   
   > theory to deal with getting a correct consequent truth value when   
   > firing a bunch of rules concurrently. The theoreticians have been of   
   > no help to me at all; in the book on automated fuzzy reasoning I'm   
   > just finishing there are two chapters on the theory of how to handle   
   > this problems in different situations; I had to roll my own theory,   
   > since the theoreticians apparently didn't seem to know that their own   
   > possibility theory was incorrect, and relied on fallacious   
   > methodology.   
      
   I do not know why do you think that the possibility theory is wrong. But   
   here is how it deals with partial conditions:   
      
   Pos(Price) >= Pos(Price|A) >= Pos(Price|A&B) ...   
   Nec(Price) <= Nec(Price|A) <= Nec(Price|A&B) ...   
      
   See, you can add new knowledge (conditions) *without* breaking estimations.   
   So everything remains correct if the rules are correct.   
      
   > Computer science is not of much help here; in its present   
   > state it is Boolean, and doesn't deal with truth values.   
      
   Having fuzzy computers... (:-))   
      
   > The   
   > interesting thing is that Earl and I independently arrived at similar   
   > conclusions without help from the theoreticians   
      
   >> > It is a major failing of our schools that not only don't they teach the   
   >> > true epistemological and methodological properties of fuzzy logic and   
   >> > fuzzy systems, but even when they do address fuzzy logic, they almost   
   >> > never teach HOW to build and implement a real fuzzy system. This is   
   >> > basically because so few academics have any real work exposure to fuzzy   
   >> > models.   
   >>   
   >> As a professional software architect I would say that it is not their   
   >> fault, because IMO, it is not their job. The issue (of dealing with   
   >> dependent data in an asynchronous system) arise in practically every   
   >> application area of software development. It is a fundamental software   
   >> design problem. So it should be taught there. This problem is nasty,   
   >> sometimes extremely, but doable. And yes, one should avoid it, if one   
   >> can.   
   >   
      
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