Dear prof. Zadeh, > If u is a real number, than the grade of membership of u in the fuzzy > set "approximately a" is simply the subjective degree to which u fits > your intended meaning of "approximately a."
But we don't know how our brain work and because of that it is very difficult to justify fuzzy logic operations. > "Approximately a" could be interpreted--much less simply--in terms of > random sets or, equivalently, in terms of a convex combination of > intervals, but the results would be the same as those obtained through > the use of the machinery of fuzzy sets. There are many different mathematical tools that led finally, to the semantic of fuzzy sets. Some of them are listed in the following paper: http://citeseer.nj.nec.com/bilgic95measurement.html The problem with random set interpretation and similar technique is that: 1) they works only under very strong assumptions which are simply no practical (for example in the case of random set the family of the interval measurements have to be nested). 2) they are actually the branch of probability theory (and cannot be apply to non-probabilistic types of uncertainty). Some other uncertain problems can be described using uncertain probability. The statements "the bottle is half empty" (i.e. m(bottle|Empty)=0.5) describes the subjective degree to which fits your intended meaning of that statements and in this situation the phenomena is really non-random. Then cannot be described using probability theory, then also using random sets. In that framework it is vary difficult justify the following relations m(bottle|Empty and Full) = min{ m(bottle|Empty), m(bottle|Full)} m(bottle|Empty or Full) = max{ m(bottle|Empty), m(bottle|Full)} or another t-norms. However, there are some useful heuristics which supports this way of thinking. > If you interpret "approximately a" as a probability distribution, > then various problems > arise, among which is the problem of the meaning of conjunction. Probability theory has a lot of week points (Elishakoff I., 2000, Possible limitations of probabilistic methods in engineering. Applied Mechanics Reviews, Vol.53, No.2, s.19-25) but why fuzzy logic is a solution of the probability theory problems? Fuzzy logic has a lot of very successful applications because this is a good approximation tools for complicated nonlinear functions. (http://citeseer.nj.nec.com/337091.html Universal Approximation Theorem for Uninorm-Based Fuzzy Systems Modeling Ronald R. Yager, Vladik Kreinovich, 2000). However if we neglect tuning then it is very difficult to apply fuzzy logic because in existing fuzzy logic application (mostly in control theory) meaning of fuzzy membership function is actually completely not important (the algorithm is convergent for any choice of t-norm). Fuzzy logic was a topic of my Ph.D. dissertation and according to my experience discussion about relation between probability theory and fuzzy logic are repeated after each 6 months. In these discussions I discover that: 1) definition of fuzzy set doesn't exist, 2) there is no person in this word who really understand what is fuzzy set (otherwise all that discussion will be quite short), 3) "fuzzy logic has many successful applications" this is the only one real argument that support the fuzzy logic development (i.e. theoretical background is not very important - - except approximations theorems). Additionally there are absolutely no conclusions from that discussion. Here are some examples: http://www.mat.univie.ac.at/~andrzej/papers/fuzzy22.htm http://www.mat.univie.ac.at/~andrzej/papers/thomas.htm http://groups.yahoo.com/group/Behavioral-Finance/messages/1040 Much more such discussion can be found here: http://zeus.polsl.gliwice.pl/~pownuk/fuzzy.htm In my opinion this topic is very interesting topic but till now there are very little sources which give some objective information but there are many of them which are based on "believe theory" (i.e. one author believe another author and so on http://www.labs.agilent.com/personal/Danny_Abramovitch/pubs/fuzzy_acc_ta lk_2e.pdf). You and other fuzzy researchers did really greater job in control theory. Fuzzy logic is probably quite good approximation method of nonlinear system but before applying fuzzy logic in another areas some extra work has to be done. Regards, Andrzej Pownuk --------------------------------- Ph.D., research associate at: Chair of Theoretical Mechanics Faculty of Civil Engineering Silesian University of Technology URL: http://zeus.polsl.gliwice.pl/~pownuk --------------------------------- P.S. There are some papers about imprecise defined mean and variance EXACT BOUNDS ON SAMPLE VARIANCE OF INTERVAL DATA(2002) Scott Ferson, Lev Ginzburg, Vladik Kreinovich, and Monica Aviles http://www.cs.utep.edu/vladik/2002/tr02-13c.pdf ABSOLUTE BOUNDS ON THE MEAN OF SUM, PRODUCT, MAX, AND MIN: A PROBABILISTIC EXTENSION OF INTERVAL ARITHMETIC(2002) Scott Ferson, Lev Ginzburg, Vladik Kreinovich, and Jorge Lopez http://www.cs.utep.edu/vladik/2002/tr02-12a.pdf COMPUTING HIGHER CENTRAL MOMENTS FOR INTERVAL DATA(2003) Vladik Kreinovich, Luc Longpre, Scott Ferson, and Lev Ginzburg http://www.cs.utep.edu/vladik/2003/tr03-14.pdf Some other information about probabilistic properties of inexact data can be found here: http://www.c3.lanl.gov/~joslyn/papers.html
