Subject: What about Fuzzy Logic?
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 comp.ai.fuzzy
(where there is a fuzzy logic FAQ) but there is also some work (and
discussion) about combining fuzzy logic with neural network approaches
in comp.ai.neural-nets.
Early work combining neural nets and fuzzy methods used competitive
networks to generate rules for fuzzy systems (Kosko 1992). This approach
is essentially the same thing as bidirectional counterpropagation
(Hecht-Nielsen 1990) and suffers from the same deficiencies. More recent
work (Brown and Harris 1994) 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:
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.
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.
The net can be interpretable as an adaptive fuzzy system.
For example, Gaussian RBF nets and B-spline regression models
are fuzzy systems with adaptive weights (Brown and Harris 1994)
and so can legitimately be called neurofuzzy systems.
References:
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.
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.
--Warren S. Sarle SAS Institute Inc. The opinions expressed here saswss@unx.sas.com SAS Campus Drive are mine and not necessarily (919) 677-8000 Cary, NC 27513, USA those of SAS Institute.