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Section - What about Fuzzy Logic?

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Fuzzy logic is an area of research based on the work of L.A. Zadeh. It is a
departure from classical two-valued sets and logic, that uses "soft"
linguistic (e.g. large, hot, tall) system variables and a continuous range
of truth values in the interval [0,1], rather than strict binary (True or
False) decisions and assignments.

Fuzzy logic is used where a system is difficult to model exactly (but an
inexact model is available), is controlled by a human operator or expert, or
where ambiguity or vagueness is common. A typical fuzzy system consists of a
rule base, membership functions, and an inference procedure.

Most fuzzy logic discussion takes place in the newsgroup
(where there is a fuzzy logic FAQ) but there is also some work (and
discussion) about combining fuzzy logic with neural network approaches in

Early work combining neural nets and fuzzy methods used competitive networks
to generate rules for fuzzy systems (Kosko 1992). This approach is sort of a
crude version of bidirectional counterpropagation (Hecht-Nielsen 1990) and
suffers from the same deficiencies. More recent work (Brown and Harris 1994;
Kosko 1997) has been based on the realization that a fuzzy system is a
nonlinear mapping from an input space to an output space that can be
parameterized in various ways and therefore can be adapted to data using the
usual neural training methods (see "What is backprop?") or conventional
numerical optimization algorithms (see "What are conjugate gradients,
Levenberg-Marquardt, etc.?"). 

A neural net can incorporate fuzziness in various ways: 

 o The inputs can be fuzzy. Any garden-variety backprop net is fuzzy in this
   sense, and it seems rather silly to call a net "fuzzy" solely on this
   basis, although Fuzzy ART (Carpenter and Grossberg 1996) has no other
   fuzzy characteristics. 
 o The outputs can be fuzzy. Again, any garden-variety backprop net is fuzzy
   in this sense. But competitive learning nets ordinarily produce crisp
   outputs, so for competitive learning methods, having fuzzy output is a
   meaningful distinction. For example, fuzzy c-means clustering (Bezdek
   1981) is meaningfully different from (crisp) k-means. Fuzzy ART does not
   have fuzzy outputs. 
 o The net can be interpretable as an adaptive fuzzy system. For example,
   Gaussian RBF nets and B-spline regression models (Dierckx 1995, van
   Rijckevorsal 1988) are fuzzy systems with adaptive weights (Brown and
   Harris 1994) and can legitimately be called neurofuzzy systems. 
 o The net can be a conventional NN architecture that operates on fuzzy
   numbers instead of real numbers (Lippe, Feuring and Mischke 1995). 
 o Fuzzy constraints can provide external knowledge (Lampinen and Selonen

More information on neurofuzzy systems is available online: 

 o The Fuzzy Logic and Neurofuzzy Resources page of the Image, Speech and
   Intelligent Systems (ISIS) research group at the University of
   Southampton, Southampton, Hampshire, UK: 
 o The Neuro-Fuzzy Systems Research Group's web page at Tampere University
   of Technology, Tampere, Finland: and 
 o Marcello Chiaberge's Neuro-Fuzzy page at 
 o The homepage of the research group on Neural Networks and Fuzzy Systems
   at the Institute of Knowledge Processing and Language Engineering,
   Faculty of Computer Science, University of Magdeburg, Germany, at 
 o Jyh-Shing Roger Jang's home page at with
   information on ANFIS (Adaptive Neuro-Fuzzy Inference Systems), articles
   on neuro-fuzzy systems, and more links. 
 o Andrew Gray's Hybrid Systems FAQ at the University of Otago at


   Bezdek, J.C. (1981), Pattern Recognition with Fuzzy Objective Function
   Algorithms, New York: Plenum Press. 

   Bezdek, J.C. & Pal, S.K., eds. (1992), Fuzzy Models for Pattern
   Recognition, New York: IEEE Press. 

   Brown, M., and Harris, C. (1994), Neurofuzzy Adaptive Modelling and
   Control, NY: Prentice Hall. 

   Carpenter, G.A. and Grossberg, S. (1996), "Learning, Categorization, Rule
   Formation, and Prediction by Fuzzy Neural Networks," in Chen, C.H.
   (1996), pp. 1.3-1.45. 

   Chen, C.H., ed. (1996) Fuzzy Logic and Neural Network Handbook, NY:
   McGraw-Hill, ISBN 0-07-011189-8. 

   Dierckx, P. (1995), Curve and Surface Fitting with Splines, Oxford:
   Clarendon Press. 

   Hecht-Nielsen, R. (1990), Neurocomputing, Reading, MA: Addison-Wesley. 

   Klir, G.J. and Folger, T.A.(1988), Fuzzy Sets, Uncertainty, and
   Information, Englewood Cliffs, N.J.: Prentice-Hall. 

   Kosko, B.(1992), Neural Networks and Fuzzy Systems, Englewood Cliffs,
   N.J.: Prentice-Hall. 

   Kosko, B. (1997), Fuzzy Engineering, NY: Prentice Hall. 

   Lampinen, J and Selonen, A. (1996), "Using Background Knowledge for
   Regularization of Multilayer Perceptron Learning", Submitted to
   International Conference on Artificial Neural Networks, ICANN'96, Bochum,

   Lippe, W.-M., Feuring, Th. and Mischke, L. (1995), "Supervised learning
   in fuzzy neural networks," Institutsbericht Angewandte Mathematik und
   Informatik, WWU Muenster, I-12, 

   Nauck, D., Klawonn, F., and Kruse, R. (1997), Foundations of
   Neuro-Fuzzy Systems, Chichester: Wiley, ISBN 0-471-97151-0. 

   van Rijckevorsal, J.L.A. (1988), "Fuzzy coding and B-splines," in van
   Rijckevorsal, J.L.A., and de Leeuw, J., eds., Component and
   Correspondence Analysis, Chichester: John Wiley & Sons, pp. 33-54. 

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