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A first course in fuzzy logic. (English) Zbl 0856.03019

Boca Raton, FL: CRC Press. 266 p. (1997).
In many situations, experts use statements in which they are not totally sure, in particular, statements that use imprecise terms (such as “medium”). As a result, when we describe, for a computer, the expert’s knowledge, we must, at least, indicate the expert’s degree of belief in different statements. Since in a computer, “true” is usually described as 1 and “false” as 0, it is natural to describe the resulting intermediate degrees of belief by “intermediate” numbers, i.e., by numbers from the interval \([0, 1]\) (more complicated descriptions are also used). Methods of processing these degrees are often called fuzzy logic. Many successful expert systems (starting from the first successful expert system MYCIN), and many intelligent control systems use fuzzy logic, i.e., use numbers from the interval \([0, 1]\) (or more complicated objects) to describe the expert’s degree of confidence in different statements. Because of these successes, there is an interest in learning fuzzy logic methods. Several textbooks in fuzzy logic have been written, but these textbooks are mostly written for engineers by engineers, often with gaps or imprecise parts that make these texts unsuitable for mathematicians. The fact that most introductory texts are of this type adds to the belief that many mathematicians share that there is something wrong with “fuzzy mathematics”. There are mathematically precise books on fuzzy logic, but mainly on the level of advanced research monographs, not introductions.
This book, to the best of my knowledge, is the first introductory text in fuzzy logic that is specifically addressed to mathematicians (or to engineers who are interested in mathematical rigor). This one-semester textbook is written by two mathematicians and grew out of their experience of successfully teaching this material, both in the USA (which is still the main center of fuzzy logic research) and in Japan (which is, nowadays, the main center of fuzzy logic applications). The book starts with the basic motivations for using fuzzy logic, and goes via the description of fuzzy analogues of “and” and “or” operations (usually called t-norms and t-conorms) to fuzzy analogues of probability (including possibility measure and interval-valued probabilities), of measure, and of integral. Along the way, it covers several important research topics, many of which were largely developed by the authors themselves. These topics include the universal approximation property of fuzzy systems, logical aspects of “fuzzy logic” (in particular, a decision algorithm is presented), mathematical foundations of interval-valued modifications of fuzzy logic, and the question of sensitivity of different fuzzy operations. Also covered are the properties of fuzzy relations, fuzzy modeling, and decision making under fuzzy uncertainty. The material covered in every chapter is supplemented by carefully selected exercises. Solutions to some of these exercises are presented at the end of the book; in a pedagogically wise manner, the solution to other (similar) exercises is left to the reader.
This book can definitely be recommended as a textbook for a first course in fuzzy logic.

MSC:

03B52 Fuzzy logic; logic of vagueness
03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations
68T27 Logic in artificial intelligence
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