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**Fuzzy relational calculus. Theory, applications and software.**
*(English)*
Zbl 1083.03048

This book deals with the theory of fuzzy relation equations and is divided into four parts. The first part consists of 6 chapters and the second one has 3. The third part is formed only by Chapter 10. The fourth part includes three Appendices: Appendix A contains solved samples listed in Table 10.1, Appendix B is a list of symbols used in the book, and Appendix C is a long bibliography. A CD containing a toolbox tutorial is enclosed in the book. Each chapter is concluded with proper, detailed and annotated references.

Chapter 1 points out the most important basic concepts of fuzzy relations and related applications to fuzzy finite machines and fuzzy syntactic methods. Chapter 2 gives the fundamental notions of the various possible compositions of fuzzy relations. Chapter 3 deals with fuzzy relation equations defined on bounded chains, when the max-min composition is considered. The authors give all the direct methods for finding the whole set of solutions of such fuzzy relation equations. Chapter 4 contains fuzzy linear systems of inequalities and the resolution of the related inverse problem on bounded chains. Applications to fuzzy linear programming are also presented. In Chapter 5 the authors study fuzzy relation equations on bounded chains with min-max composition by determining the whole solution set. Chapter 6 contains results about the direct and inverse problems of intuitionistic fuzzy relation equations. Chapter 7, which follows the ideas of previous papers of the first author [see, e.g., Fuzzy Sets Syst. 141, 415–437 (2004; Zbl 1059.68066)], develops a theory for behaviour, reduction and minimization of fuzzy finite machines over [0,1] by using methods and algorithms for the direct and inverse problems given in the first part of the book. A section dedicated to intuitionistic fuzzy finite machines is also presented. Chapter 8 describes the usage of fuzzy languages for syntactic pattern recognition and classification of distorted images, following essentially the papers of the first author [see, e.g., Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 12, 89–104 (2004; Zbl 1074.68032)]. Chapter 9 contains applications to artificial intelligent systems whose inference engine is based on fuzzy relational calculus. Examples from textile and chemical engineering are given.

The final Chapter 10 contains the software, implemented in MATLAB, for all the compositions defined in Chapters 2, 3, 5 and 6. Further, Section 10.5 is dedicated to intuitionistic fuzzy relational calculus and Section 10.6 deals with working examples from the textile industry and finite fuzzy machines. The CD contains many examples of fuzzy equations with the related input data, the whole solution set and the computation time.

Chapter 1 points out the most important basic concepts of fuzzy relations and related applications to fuzzy finite machines and fuzzy syntactic methods. Chapter 2 gives the fundamental notions of the various possible compositions of fuzzy relations. Chapter 3 deals with fuzzy relation equations defined on bounded chains, when the max-min composition is considered. The authors give all the direct methods for finding the whole set of solutions of such fuzzy relation equations. Chapter 4 contains fuzzy linear systems of inequalities and the resolution of the related inverse problem on bounded chains. Applications to fuzzy linear programming are also presented. In Chapter 5 the authors study fuzzy relation equations on bounded chains with min-max composition by determining the whole solution set. Chapter 6 contains results about the direct and inverse problems of intuitionistic fuzzy relation equations. Chapter 7, which follows the ideas of previous papers of the first author [see, e.g., Fuzzy Sets Syst. 141, 415–437 (2004; Zbl 1059.68066)], develops a theory for behaviour, reduction and minimization of fuzzy finite machines over [0,1] by using methods and algorithms for the direct and inverse problems given in the first part of the book. A section dedicated to intuitionistic fuzzy finite machines is also presented. Chapter 8 describes the usage of fuzzy languages for syntactic pattern recognition and classification of distorted images, following essentially the papers of the first author [see, e.g., Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 12, 89–104 (2004; Zbl 1074.68032)]. Chapter 9 contains applications to artificial intelligent systems whose inference engine is based on fuzzy relational calculus. Examples from textile and chemical engineering are given.

The final Chapter 10 contains the software, implemented in MATLAB, for all the compositions defined in Chapters 2, 3, 5 and 6. Further, Section 10.5 is dedicated to intuitionistic fuzzy relational calculus and Section 10.6 deals with working examples from the textile industry and finite fuzzy machines. The CD contains many examples of fuzzy equations with the related input data, the whole solution set and the computation time.

Reviewer: Salvatore Sessa (Napoli)

### MSC:

03E72 | Theory of fuzzy sets, etc. |

68Q45 | Formal languages and automata |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

68-02 | Research exposition (monographs, survey articles) pertaining to computer science |

68T10 | Pattern recognition, speech recognition |