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**Foundations of fuzzy systems.**
*(English)*
Zbl 0843.68109

Chichester: Wiley. xii, 265 p. (1994).

The monograph “Foundations of Fuzzy Systems” provides an unitary appropriate framework for theoretical developments in fuzzy theory. Emerging from the classical set theory, the fuzzy theory founded by L. Zadeh in the early ‘60, has proved a continually increasing usefulness especially in knowledge-based industrial applications as control tasks and expert systems. The aim of the book is to supply a methodically ordered introductory course in fuzzy systems, the involved material being structured into three chapters. Each chapter is divided into several subject-specific sections, including historical remarks and a series of additional references guiding the reader in deepening the acquired knowledge beyond the content of the book.

The fundamental concepts of the fuzzy set theory and its relations with fuzzy logic are extensively presented in the second chapter of the book. A series of illustrative examples are given, helping the reader to get a better understanding, the meaning and the corresponding usefulness of the various concepts defined here. The possibilistic reasoning viewed as an alternative way in modelling some aspects of vagueness and uncertainty existing in knowledge-based systems, allowing and justifying the execution of reasoning mechanism on the basis of imperfect knowledge, is investigated in the third chapter of the book. The special concept of possibilistic introduced here differs from other approaches, the possibility distributions being viewed as information-compressed representations random sets. Next, and outline of an expert system performing approximate reasoning is given as an application in the field of artificial intelligence, followed by an argument concerning the derivation of the underlying possibility distributions as an interpretation of the possibilistic inference rules.

The logic based inference mechanisms are introduced mainly as an alternative approach to possibilistic reasoning and a series of comments with respect to other approaches to possibility theory are also formulated here.

The third chapter of the book is devoted to issues in fuzzy control. Following to some brief description of the basic differences between the classical control approaches and the fuzzy control, the standard methods of fuzzy control are described here. Next, the development of a fuzzy controller based on the idea of interpolation in the framework of equality relations is presented, leading to the conclusion that the semantic background could allow the design of intuitively justified fuzzy controllers but with a formal foundation as well. It is argued that the relational equations could offer an appropriate framework in approaching the fuzzy control modelling.

The monograph is distinguished by the precision, clarity, soundness and mathematical rigour the results are presented here. The book could prove extremely useful for developing teaching activities and as a starting point in developing fundamental research in this area as well.

The fundamental concepts of the fuzzy set theory and its relations with fuzzy logic are extensively presented in the second chapter of the book. A series of illustrative examples are given, helping the reader to get a better understanding, the meaning and the corresponding usefulness of the various concepts defined here. The possibilistic reasoning viewed as an alternative way in modelling some aspects of vagueness and uncertainty existing in knowledge-based systems, allowing and justifying the execution of reasoning mechanism on the basis of imperfect knowledge, is investigated in the third chapter of the book. The special concept of possibilistic introduced here differs from other approaches, the possibility distributions being viewed as information-compressed representations random sets. Next, and outline of an expert system performing approximate reasoning is given as an application in the field of artificial intelligence, followed by an argument concerning the derivation of the underlying possibility distributions as an interpretation of the possibilistic inference rules.

The logic based inference mechanisms are introduced mainly as an alternative approach to possibilistic reasoning and a series of comments with respect to other approaches to possibility theory are also formulated here.

The third chapter of the book is devoted to issues in fuzzy control. Following to some brief description of the basic differences between the classical control approaches and the fuzzy control, the standard methods of fuzzy control are described here. Next, the development of a fuzzy controller based on the idea of interpolation in the framework of equality relations is presented, leading to the conclusion that the semantic background could allow the design of intuitively justified fuzzy controllers but with a formal foundation as well. It is argued that the relational equations could offer an appropriate framework in approaching the fuzzy control modelling.

The monograph is distinguished by the precision, clarity, soundness and mathematical rigour the results are presented here. The book could prove extremely useful for developing teaching activities and as a starting point in developing fundamental research in this area as well.

Reviewer: L.State (Bucureşti)