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Fundamentals of Fuzzy Relation Calculus

  • Siegfried Gottwald
Chapter
Part of the International Series in Intelligent Technologies book series (ISIT, volume 7)

Abstract

This chapter introduces the basic notions of the fuzzy relation calculus, also in their t-norm based form. Typical properties of fuzzy relations are discussed in connection with particular types of relations like fuzzy equivalence or fuzzy ordering relations. Finally there is a discussion of fuzzy relational equations, their relationship to fuzzy control, and their solvability resp. approximate solvability.

Keywords

Membership Function Relational Equation Fuzzy Controller Relational Product Membership Degree 
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]
    Dinola, A.: An algorithm of calculation of lower solutions of fuzzy relation equation. Stochastica 3, 33–40 (1984).MathSciNetGoogle Scholar
  2. [2]
    Dinola, A.: Relational equations in totally ordered lattices and their complete resolution. J. Math. Anal. Appl. 107, 148–155 (1985).MathSciNetCrossRefGoogle Scholar
  3. [3]
    Dinola, A.; Sessa, S.; Pedrycz, W.; Sanchez, E.: Fuzzy Relation Equations and Their Applications to Knowledge Engineering. Theory and Decision Libr., ser. D, Kluwer Academic Publ., Dordrecht 1989.Google Scholar
  4. [4]
    Gottwald, S.: On the existence of solutions of systems of fuzzy equations. Fuzzy Sets and Systems 12, 301–302 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Gottwald, S.: Criteria for non-interactivity of fuzzy logic controller rules. In: Large Scale Systems: Theory and Applications 1983 (A. Straczak, ed.), Pergamon Press, Oxford 1984 229–233.Google Scholar
  6. [6]
    Gottwald, S.: Characterizations of the solvability of fuzzy equations. Elektron. Informationsverarb. Kybernetik 22, 67–91 (1986).MathSciNetzbMATHGoogle Scholar
  7. [7]
    Gottwald, S.: Fuzzy Sets and Fuzzy Logic. Foundation of Application — from a Mathematical Point of View. Vieweg, Braunschweig/Wiesbaden and Teknea, Toulouse 1993.zbMATHGoogle Scholar
  8. [8]
    Gottwald, S.; Pedrycz, W.: Analysis and synthesis of fuzzy controller. Problems Control Inform. Theory 14, 33–45 (1985).MathSciNetzbMATHGoogle Scholar
  9. [9]
    Gottwald, S.; Pedrycz, W.: On the suitability of fuzzy models: an evaluation through fuzzy integrals. Internat. J. Man-Machine Studies 24, 141–151 (1986).zbMATHCrossRefGoogle Scholar
  10. [10]
    Gottwald, S.; Pedrycz, W.: On the methodology of solving fuzzy relational equations and its impact on fuzzy modelling. In: Fuzzy Logic in Knowledge-Based Systems, Decision and Control (M. M. Gupta, T. Yamakawa, eds.), North-Holland Publ. Comp., Amsterdam 1988, 197–210.Google Scholar
  11. [11]
    Klaua, D.: Über einen zweiten Ansatz zur mehrwertigen Mengenlehre. Monatsber. Deut. Akad. Wiss. Berlin 8, 161–177 (1966).MathSciNetzbMATHGoogle Scholar
  12. [12]
    Klaua, D.: Grundbegriffe einer mehrwertigen Mengenlehre. Monatsber. Deut. Akad. Wiss. Berlin 8, 781–802 (1966).Google Scholar
  13. [13]
    Klir, G.; Yuan, B.: Approximate solutions of systems of fuzzy relation equations. In: Proc. 3rd IEEE Internat. Conf. Fuzzy Systems, FUZZ-IEEE ‘94, Orlando/FL, IEEE Soc. 1994, 1452–1457.CrossRefGoogle Scholar
  14. [14]
    Nauck, D.; Klawonn, F.; Kruse, R.: Neuronale Netze und Fuzzy-Systeme. Vieweg, Braunschweig/Wiesbaden 1994.zbMATHGoogle Scholar
  15. [15]
    Sanchez, E.: Solutions in composite fuzzy relation equations: application to medical diagnosis in Brouwerian logic. In: Fuzzy Automata and Decision Processes (M. M. Gupta, G. N. Saridis, B. N. Gaines, eds.), North-Holland Publ. Comp., Amsterdam 1977, 221–234.Google Scholar
  16. [16]
    Sanchez, E.: Solution of fuzzy equations with extended operations. Fuzzy Sets and Systems 12, 237–248 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    Sessa, S.: Some results in the setting of fuzzy relation equations theory. Fuzzy Sets and Systems 14, 281–297 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Wagenknecht, M.; Hartmann, K.: On the solution of direct and inverse problems for fuzzy equation systems with tolerances. In: Fuzzy Sets Applications, Methodological Approaches, and Results (St. Bocklisch et al., eds.); Mathematical Research, 30, Akademie-Verlag, Berlin 1986, 37–44.Google Scholar
  19. [19]
    Wagenknecht, M.; Hartmann, K.: Fuzzy modelling with tolerances. Fuzzy Sets and Systems 20, 325–332 (1986).MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Wu, W.-M.: Fuzzy reasoning and fuzzy relational equations. Fuzzy Sets and Systems 20, 67–78 (1986).MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Zadeh, L.A.: Similarity relations and fuzzy orderings. Information Sci. 3, 159–176 (1971).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Siegfried Gottwald
    • 1
  1. 1.Institute of Logic and Philosophy of ScienceLeipzig UniversityLeipzigGermany

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