Читать книгу Introduction to Fuzzy Logic - James K. Peckol - Страница 62

The birth of modern fuzzy logic is usually traced to the seminal paper Fuzzy Sets published in 1965 by Lotfi A. Zadeh. In his paper, Zadeh described the mathematics of fuzzy subsets and, by extension, the mathematics of fuzzy logic. The concept of the fuzzy event was introduced by Zadeh (1968) and has been used in various ways since early attempts to model inexact concepts were prevalent in human reasoning. The initial work led to the development of the branch of mathematics called fuzzy logic. This logic, actually a superset of classical binary‐valued logic, does not restrict set membership to absolutes (Yes or No) but tolerates varying degrees of membership.

Using these criteria, an element is assigned a grade of membership in a parent set. The domain of this attribute is the closed interval [0, 1]. If the grade of membership values is restricted to the two extrema, then fuzzy logic reduces to two‐valued or crisp logic.

In his work, Zadeh proposed that people often base their thinking and decisions on imprecise or nonnumerical information. He further believed that the membership of an element in a set need not be restricted to the values 0 and 1 (corresponding to FALSE and TRUE) but could easily be extended to include all real numbers in the interval 0.0–1.0 including the endpoints. He further felt that such a concept should not be considered in isolation but rather viewed as a methodology that moves from a discrete world to a continuous one. To augment such thinking, he proposed a collection of operations supporting his new logic.

Zadeh introduced his ideas as a new way of representing the vagueness common in everyday thinking and language. His fuzzy sets are a natural generalization or superset of classical sets or Boolean logic that are one of the basic structures underlying contemporary mathematics. Under Boolean algebra, a proposition takes a narrow view that a value is either completely true or completely false. In contrast, fuzzy logic introduces the concept of partial truth under which values are expressed anywhere within, and including, the two extremes of TRUE and FALSE.

Based on the idea of the fuzzy variable, Zadeh (1979) further proposed a theory of approximate reasoning. This theory postulates the notion of a possibility distribution on a linguistic variable. Using this concept, he was able to reason using vague concepts such as young, old, tall, or short. Zadeh also introduced the ideas of semantic equivalence and semantic entailment on the possibility distributions of linguistic propositions. Using these concepts, he was able to determine that a statement and its double negative are equivalent and that very small is more restrictive than small. Such conclusions derive from either the equality or containment of corresponding distributions.

Zadeh's theory is generally effective in reconciling ambiguous natural language expressions. The scope of the work was initially limited to laboratory sentences comparing hair color, age, or height between various people. Zadeh's work provided a good tool for future efforts, particularly in combination with or to enhance other forms of reasoning.

As often follows the introduction of a new concept or idea, questions arise: Why does that thing do this? Why doesn't it do that? Can you make it do another thing? An early criticism of Zadeh's fuzzy sets was: “Why can't your fuzzy set members have an uncertainty associated with them?” Zadeh eventually dealt with the issue by proposing more sophisticated kinds of fuzzy sets. New criteria evolved the original concept into numbered types of fuzzy subsets. His initial work became type‐1 fuzzy sets. Additional concepts grew from type‐2 fuzzy sets to ultimately type‐n in a 1976 paper to incorporate greater uncertainty into set membership. Naturally, if a type‐2 or higher set has no uncertainty in its members, it reduces to a type‐1. In this text, we will work primarily with type‐1 fuzzy subsets.