Fuzzy logic provides a wide variety of concepts and techniques for representing and processing knowledge that is imprecise, uncertain, or unreliable. As humans, we deal with fuzzy information every day. When we consider whether to walk or run to our car when it is raining, we may partition the concept of "raining" into, "drizzling", "normal rain", or "pouring". Each of these is a fuzzy concept covering some range of measurement in inches-per-hour, perhaps. A conventional logical system would put hard limits on specific values of rainfall. For example, a limit of 0.5 inches-per-hour would mean that at 0.4999 inches-per-hour you would walk and at 0.5001 inches-per-hour you would run. A fuzzy logical system would consider each of these concepts, "drizzling", "normal rain", "pouring", and even "walking" and "running" as ranges of values. Different degrees of walking and running, a range of speeds, would result from different degrees of raining.

Recently, fuzzy logic has been finding a rapidly growing number of applications in fields ranging from consumer electronics and photography to medical-diagnosis systems and securities-management funds. What is exploited in most of these applications is fuzzy logic's tolerance for imprecision. In effect, the operative principle of fuzzy logic is this: precision is costly; minimize the precision needed to perform a task.

In contrast to classical logical systems, fuzzy logic considers modes of reasoning that are approximate rather than exact. This part of fuzzy logic is referred to as fuzzy inference, which forms the foundation of fuzzy expert systems. Knowledge can be represented in the form of a system of rules in which the antecedents, consequences, or both are fuzzy rather than crisp. As an example, the relationship between interest rate, unemployment, and inflation may be expressed as a collection of fuzzy rules exemplified by:

The three linguistic values of inflation (low, moderate, and high) are fuzzy sets. The meaning of such values is defined by their membership functions (See Figure).

The importance of the calculus of fuzzy rules stems from the fact that much of human knowledge lends itself to representation in the form of a hierarchy of fuzzy rules. Furthermore, the inference mechanisms in the calculus of fuzzy rules are relatively simple and in harmony with the modes of human reasoning, which are approximate rather than exact. As a consequence, the expression of knowledge in the form of fuzzy rules is relatively natural and simple to understand.

What is of central importance in practical applications is that the fuzziness of the antecedents eliminates the need for a precise match with the input. As a consequence, in a fuzzy rule-based system, each rule is fired to a degree that is a function of the degree of match between its antecedent and the input. The mechanism of imprecise matching provides a basis for interpolation. Interpolation, in turn, serves to minimize the number of fuzzy rules needed to describe an input-output relation.

In summary, the calculus of fuzzy rules provides a systematic way of handling systems in which it is either necessary or advantageous to describe the input-output relations in the form of fuzzy rules. The calculus of fuzzy rules is straightforward and close to intuition. Furthermore, it is largely self-contained and does not require an extensive familiarity with fuzzy logic. The bottom line is that fuzzy rule-based systems are simpler, cheaper, and more robust than their conventional counterparts.

Neural Networks

Evolutionary Programming and Genetic Algorithms

Case Based Reasoning