Advertisement

Decomposition and Resolution of Fuzzy Relation Equations (II) Based on Boolean-Type Implications

  • Yanbin Luo
  • Chunjie Yang
  • Yongming Li
  • Daoying Pi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3809)

Abstract

The problem of solving fuzzy relation equations (II) based on Boolean-type implications is studied in the present paper. Decomposition of fuzzy relation equations (II) based on Boolean-type implications is first presented in a finite case. Then, the solution existence of fuzzy relation equations (II) based on Boolean-type implications is discussed, and for nice Boolean-type implications, some new solvability criteria based upon the notion of ”solution matrices” are given. It is also shown that for each solution a of a fuzzy relation equation (II) based on Boolean-type implication, there exists a minimal solution a * of this equation, such that a * is less than or equal to a, whenever the solution set of this equation is nonempty. The complete solution set of fuzzy relation equation (II) based on Boolean-type implication can be determined by all minimal solutions of this equation. Finally, an effective method to solve fuzzy relation equations (II) based on Boolean-type implications is proposed.

Keywords

Minimal Solution Resolution Problem Fuzzy Relation Great Solution Minimal Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    De Basets, B.: Analytical solution methods for fuzzy relational equations. In: Dubois, D., Prade, H. (eds.) Fundamentals of fuzzy sets. The handbooks of fuzzy sets series, vol. 1, pp. 291–340. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar
  2. 2.
    Dubois, D., Prade, H.: Fuzzy sets in approximate reasoning, Part 1: Inference with possibility distributions. Fuzzy Sets Syst. 100(suppl.), 73–132 (1999)CrossRefGoogle Scholar
  3. 3.
    Luo, Y., Li, Y.: Decomposition and resolution of θ-Fuzzy relation equations based on R-implications. Fuzzy Syst. Math. 4, 81–87 (2003) (in Chinese)Google Scholar
  4. 4.
    Luo, Y., Li, Y.: Decomposition and resolution of min-implication fuzzy relation equations based on S-implications. Fuzzy Sets Syst. 148, 305–317 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Li, Y., Shi, Z., Li, Z.: Approximation theory of fuzzy systems based upon genuine many valued implications-MIMO cases. Fuzzy Sets Syst. 130, 159–174 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Miyakoshi, M., Shimbo, M.: Solutions of composite fuzzy relational equations with triangular norms. Fuzzy Sets Syst. 16, 53–63 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Sanchez, E.: Resolution of composite fuzzy relation equations. Inform. Contr. 30, 38–48 (1976)zbMATHCrossRefGoogle Scholar
  8. 8.
    Stamou, G.B., Tzafestas, S.G.: Fuzzy relation equations and fuzzy inference systems: an inside approach. IEEE Trans. Syst., Man, Cybern.-Part B 29, 694–702 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Yanbin Luo
    • 1
  • Chunjie Yang
    • 1
  • Yongming Li
    • 2
  • Daoying Pi
    • 1
  1. 1.National Laboratory of Industrial Control TechnologyZhejiang UniversityHangzhouChina
  2. 2.Institute of Fuzzy Systems, College of Mathematics and Information SciencesShaanxi Normal UniversityXi’anChina

Personalised recommendations