Fuzzy sets were introduced by Zadeh (1965) as a means of representing and manipulating data that was not precise, but rather fuzzy. Fuzzy logic pro­ vides an inference morphology that enables approximate human reasoning capabilities to be applied to knowledge-based systems. The theory of fuzzy logic provides a mathematical strength to capture the uncertainties associ­ ated with human cognitive processes, such as thinking and reasoning. The conventional approaches to knowledge representation lack the means for rep­ resentating the meaning of fuzzy concepts. As a consequence, the approaches based on first order logic and classical probablity theory do not provide an appropriate conceptual framework for dealing with the representation of com­ monsense knowledge, since such knowledge is by its nature both lexically imprecise and noncategorical. The developement of fuzzy logic was motivated in large measure by the need for a conceptual framework which can address the issue of uncertainty and lexical imprecision. Some of the essential characteristics of fuzzy logic relate to the following [242]. . In fuzzy logic, exact reasoning is viewed as a limiting case of ap­ proximate reasoning. . In fuzzy logic, everything is a matter of degree. . In fuzzy logic, knowledge is interpreted a collection of elastic or, equivalently, fuzzy constraint on a collection of variables. . Inference is viewed as a process of propagation of elastic con­ straints. . Any logical system can be fuzzified. There are two main characteristics of fuzzy systems that give them better performance für specific applications.

Fuzzy Systems: An introduction to fuzzy logic; Operations on fuzzy sets; Fuzzy relations; The extension principle; The extension principle for n-place functions; Metrics for fuzzy numbers; Measures of possibility and necessity; Fuzzy implications; Linguistic variables; The theory of approximate reasoning; An introduction to fuzzy logic controllers; Defuzzification methods; Inference mechanisms; Construction of data base and rule base of FLC; The ball and beam problem; Aggregation in fuzzy system modeling; Averaging operators; Fuzzy screening systems; Applications of fuzzy systems.- Artificial Neural Networks: The perceptron learning rule; The delta learning rule; The delta learning rule with semilinear activation function; The generalized delta learning rule; Effectivity of neural networks; Winner-take-all-learning; Applications of artificial neural networks.- Fuzzy Neural Networks: Integration of fuzzy logic and neural networks; Fuzzy neurons; Hybrid neural nets; Computation of fuzzy logic inferences by hybrid neural net; Trainable neural nets for fuzzy IF-THEN rules; Implementation of fuzzy rules by regular FNN of Type 2; Implementation of fuzzy rules by regular FNN of Type 3; Tuning fuzzy control parameters by neural nets; Fuzzy rule extraction from numerical data; Neuro-fuzzy classifiers; FULLINS; Applications of fuzzy neural systems.- Appendix: Case study: A portfolio problem; Exercises.