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Novel Neural Algorithms for Solving Fuzzy Relation Equations

  • Xiaozhong Li
  • Da Ruan

Abstract

Due to the difficulty in a given system of using neural networks to solve fuzzy relation equations, the best learning rate sometimes cannot be decided easily and strict theoretical analyses on convergence of algorithms are not given. To overcome these problems, we present in this chapter some novel neural algorithms based on fuzzy δ rules. We first describe such algorithms for max-min operator networks, then we demonstrate that these algorithms can also be extended to max-times operator network. Important results include some improved fuzzy δ rules, a convergence theorem, and an equivalence theorem which reflects that fuzzy theory and neural networks can reach the same goal by different routes. We also discuss the fuzzy bidirectional associative memory network and its training algorithms, and prove all important theorems with additional simulation and comparison results.

Furthermore, we propose a more powerful algorithm for solving many types of fuzzy relation equations. The algorithm is based on a specially designed fuzzy neuron. This fuzzy neuron is found by replacing the operators of the traditional neuron by a pair of abstract fuzzy operators such as \(\left( {\hat{ + },\hat{ \bullet }} \right)\), which we call fuzzy neuron operators. Afterwards, we discuss the relationship between the fuzzy neuron operators and the t-norm and t-conorm, and point out that fuzzy neuron operators are based on the t-norm, but are much wider than the t-norm. And finally, we report some simulation results of some pairs of typical compositional operators.

Keywords

Fuzzy System Stable Point Fuzzy Neural Network Maximum Solution Fuzzy Relation Equation 
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.

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References

  1. [1]
    A. Blanco, M. Delgado, and I. Requena, “Identification of fuzzy relational equations by fuzzy neural networks,” Fuzzy Sets and Systems 71 (1995), 215–226.CrossRefGoogle Scholar
  2. [2]
    J. J. Buckley and Y. Hayashi, “Fuzzy neural networks: A survey,” Fuzzy Sets and Systems 66 (1994), 1–13.MathSciNetCrossRefGoogle Scholar
  3. [3]
    A. Di Nola, W. Pedrycz, S. Sessa, and P. Z. Wang, “Fuzzy relation equations under a class of triangular norms: a survey and new results,” Stochastica 8 (1984), 89–145.Google Scholar
  4. [4]
    A. Di Nola, W. Pedrycz, and S. Sessa, “Some theoretical aspects of fuzzy relation equations describing fuzzy systems,” Inform. Sci. 34 (1984), 241–264.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    S. Gottwald, “Approximately solving fuzzy relation equations: Some mathematical results and some heuristic proposals,” Fuzzy Sets and Systems 66 (1994), 175–193.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    M. M. Gupta and D. H. Rao, “On the principles of fuzzy neural networks,” Fuzzy Sets and Systems 61 (1994), 1–18.MathSciNetCrossRefGoogle Scholar
  7. [7]
    G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic, Theory and Applications (Prentice Hall PTR, 1995).Google Scholar
  8. [8]
    B. Kosko, Neural Networks and Fuzzy Systems (Prentice Hall, 1990).Google Scholar
  9. [9]
    X. Li, Researches on a self-learning and adaptive fuzzy systems and fuzzy neural network, PhD thesis, Beijing University of Post and Telecommunications, Beijing, 1994.Google Scholar
  10. [10]
    X. Li, P. Z. Wang, and C. Luo, Fuzzy Neural Network, Guizhou Science and Technology Press, 1994 (in Chinese).Google Scholar
  11. [11]
    X. Li and Sh. Bai, “A neural algorithm to solve some fuzzy relation equations,” Proc. Int. Conf. on Neural Information Processing, 417–420, Beijing 1995.Google Scholar
  12. [12]
    X. Li, “A fuzzy perceptron and its convergence theorem,” Proc. Fourth Int. Conf. for Young Computer Scientists, 506–512, Beijing, 1995.Google Scholar
  13. [13]
    X. Li and D. Ruan, “Novel neural algorithms based on fuzzy δ rules for solving fuzzy relation equations: Part I,” accepted by Fuzzy Sets and Systems, 1996.Google Scholar
  14. [14]
    X. Li and D. Ruan, “Novel neural algorithms based on fuzzy δ rules for solving fuzzy relation equations: Part II”, accepted by Fuzzy Sets and Systems, 1997.Google Scholar
  15. [15]
    C. Luo, An Introduction to Fuzzy Sets, Beijing Normal University Press, 1989 (in Chinese).Google Scholar
  16. [16]
    M. Miyakoshi and M Shimobo, “Solutions of fuzzy relational equations with triangular norms,” Fuzzy Sets and Systems 16 (1985), 53–63.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    W. Pedrycz, “Relational Structures in fuzzy sets and neurocomputation,” Proc. Int. Conf. on Fuzzy Logic and Neural Networks, lizuka (1990), 235–238.Google Scholar
  18. [18]
    E. Sanchez, “Resolution of composite fuzzy relation equations,” Inform. and Control 30 (1976), 38–48.zbMATHCrossRefGoogle Scholar
  19. [19]
    S. Sessa, “Some results in the setting of fuzzy relation equation theory,” Fuzzy Sets and Systems 14 (1984), 237–248.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Xiaozhong Li
    • 1
  • Da Ruan
    • 1
  1. 1.Belgian Nuclear Research Centre (SCK•CEN)MolBelgium

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