Advertisement

The Theory and Applications of Generalized Complex Fuzzy Propositional Logic

  • Dan E. TamirEmail author
  • Mark Last
  • Abraham Kandel
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 291)

Abstract

The current definition of complex fuzzy logic has two limitations. First, the derivation uses complex fuzzy relations; hence, it assumes the existence of complex fuzzy sets. Second, current theory is based on a restricted polar representation of complex fuzzy proposition, where only one component of a complex fuzzy proposition carries fuzzy information. In this chapter we present a novel form of complex fuzzy logic. The new theory, referred to as generalized complex fuzzy logic, overcomes the limitations of the current theory and provides several advantages. First, the derivation of the new theory is based on axiomatic approach and does not assume the existence of complex fuzzy sets or complex fuzzy classes. Second, the new form supports Cartesian and polar representation of complex logical propositions with two components of fuzzy information. Hence, the new form significantly improves the expressive power and inference capability of complex fuzzy logic. Finally, the new form is compatible with (yet independent of) the definition of complex fuzzy classes; thereby providing further improvement in the expressive power and inference capability. The chapter surveys the current state of complex fuzzy sets, complex fuzzy classes, and complex fuzzy logic; and provides a new and generalized complex fuzzy propositional logic theory. The new theory has potential for usage in advanced complex fuzzy logic systems and latent for extension into multidimensional fuzzy propositional and predicate logic. Moreover, it can be used for inference with type 2 (or higher) fuzzy sets. Furthermore, the introduction of complex logic can be used for analysis of periodic temporal fuzzy processes where the period is fuzzy.

Keywords

Fuzzy Logic Expressive Power Fuzzy Information Polar Representation Fuzzy Class 
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.
    Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Kandel, A.: Fuzzy Mathematical Techniques with Applications. Addison Wesley, New York (1987)Google Scholar
  3. 3.
    Zadeh, L.A.: Fuzzy Logic and Its Application to Approximate Reasoning. In: Proceedings ofthe IFIP Congress, pp. 591–594 (1974)Google Scholar
  4. 4.
    Drianko, D., Hellendorf, H., Reinfrank, M.: An introduction to fuzzy control. Springer (1993)Google Scholar
  5. 5.
    Halpern, J.Y.: Reasoning about uncertainty. MIT Press (2003)Google Scholar
  6. 6.
    Last, M., Klein, Y., Kandel, A.: Knowledge Discovery in Time Series Databases. IEEE Transactions on Systems, Man, and Cybernetics 31(Part B,1), 160–169 (2001)Google Scholar
  7. 7.
    Tamir, D.E., Kandel, A.: An axiomatic approach to fuzzy set theory. Information Sciences 52, 75–83 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Tamir, D.E., Kandel, A.: Fuzzy semantic analysis and formal specification of conceptual knowledge. Information Sciences, Intelligent systems 82(3-4), 181–196 (1995)zbMATHCrossRefGoogle Scholar
  9. 9.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning - Part I. Information Sciences 7, 199–249 (1975)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ronen, M., Shabtai, R., Guterman, H.: Hybrid model building methodology using unsupervised fuzzy clustering and supervised neural networks. Biotechnology and Bioengineering 77(4), 420–429 (2002)CrossRefGoogle Scholar
  11. 11.
    Agarwal, D., Tamir, D.E., Last, M., Kandel, A.: A comparative study of software testing using artificial neural networks and Info-Fuzzy networks. Accepted for publication in the IEEE, Transactions on Man Machine and Cybernetics, Part A (2012)Google Scholar
  12. 12.
    Tamir, D.E., Kandel, A.: The Pyramid Fuzzy C-means Algorithm. International Journal of Computational Intelligence in Control (December 2011)Google Scholar
  13. 13.
    De, S.P., Krishna, R.P.: A new approach to mining fuzzy databases using nearest neighbor classification by exploiting attribute hierarchies. International Journal of Intelligent Systems, 2004 19(12), 1277–1290 (2004)zbMATHCrossRefGoogle Scholar
  14. 14.
    Qiu, T., Chen, X., Liu, Q., Huang, H.: Granular computing approach to finding association rules in relational database. International Journal of Intelligent Systems 25(2), 165–179 (2010)zbMATHGoogle Scholar
  15. 15.
    Ramot, D., Milo, R., Friedman, M., Kandel, A.: Complex fuzzy sets. IEEE Transactions on Fuzzy Systems 2002 10(2), 171–186 (2002)CrossRefGoogle Scholar
  16. 16.
    Ramot, D., Friedman, M., Langholz, G., Kandel, A.: Complex fuzzy logic. IEEE Transactions on Fuzzy Systems 11(4), 450–461 (2003)CrossRefGoogle Scholar
  17. 17.
    Tamir, D.E., Lin, J., Kandel, A.: A New Interpretation of Complex Membership Grade. International Journal of Intelligent Systems (2011)Google Scholar
  18. 18.
    Tamir, D.E., Kandel, A.: Axiomatic Theory of Complex Fuzzy Logic and Complex Fuzzy Classes (invited paper). Int. J. of Computers, Communications & Control VI(3) (2011)Google Scholar
  19. 19.
    Dick, S.: Towards complex fuzzy logic. IEEE Transaction on Fuzzy Systems 13, 405–414 (2005)CrossRefGoogle Scholar
  20. 20.
    Cintula, P.: Weakly implicative fuzzy logics. Archive for Mathematical Logic 45(6), 673–704 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Baaz, M., Hajek, P., Montagna, F., Veith, H.: Complexity of t-tautologies. Annals of Pure and Applied Logic 113(1-3), 3–11 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Běhounek, L., Cintula, P.: Fuzzy class theory. Fuzzy Sets and Systems 154(1), 34–55 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Fraenkel, A.A., Bar-Hillel, Y., Levy, A.: Foundations of set theory, 2nd edn. Elsevier, Amsterdam (1973)zbMATHGoogle Scholar
  24. 24.
    Montagna, F.: On the predicate logics of continuous t-norm BL-algebras. Archives of Mathematical Logic 44, 97–114 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Moses, D., Degani, O., Teodorescu, H., Friedman, M., Kandel, A.: Linguistic coordinate transformations for complex fuzzy sets. In: 1999 IEEE International Conference on Fuzzy Systems, Seoul, Korea (1999)Google Scholar
  26. 26.
    Zhang, G., Dillon, T.S., Cai, K., Ma, J., Lu, J.: Operation properties and delta equalities of complex fuzzy sets. International Journal on Approximate Reasoning 50(8), 1227–1249 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Nguyen, H.T., Kandel, A., Kreinovich, V.: Complex fuzzy sets: towards new foundations. In: Proceedings, 2000 IEEE International Conference on Fuzzy Systems, San Antonio, Texas (2000)Google Scholar
  28. 28.
    Deshmukh, A.Y., Bavaskar, A.B., Bajaj, P.R., Keskar, A.G.: Implementation of complex fuzzy logic modules with VLSI approach. International Journal on Computer Science and Network Security 8, 172–178 (2008)Google Scholar
  29. 29.
    Chen, Z., Aghakhani, S., Man, J., Dick, S.: ANCFIS: A Neuro-Fuzzy Architecture Employing Complex Fuzzy Sets. IEEE, Transactions on Man Machine and Cybernetics, Part A 19(2) (2011)Google Scholar
  30. 30.
    Man, J., Chen, Z., Dick, S.: Towards inductive learning of complex fuzzy inference systems. In: Proceedings of the International Conference of the North American Fuzzy Information Processing, San Diego, CA (2007)Google Scholar
  31. 31.
    Hirose, A.: Complex-valued neural networks. Springer (2006)Google Scholar
  32. 32.
    Michel, H.E., Awwal, A.A.S., Rancour, D.: Artificial neural networks using complex numbers and phase encoded weights-electronic and optical implementations. In: Proceedings of the International Joint Conference on Neural Networks, Vancouver, BC (2006)Google Scholar
  33. 33.
    Noest, A.J.: Discrete-state phasor neural nets. Physics Review A 38, 2196–2199 (1988)CrossRefGoogle Scholar
  34. 34.
    Leung, S.H., Hyakin, S.: The complex back propagation algorithm. IEEE Transactions on Signal Processing 39(9), 2101–2104 (1991)CrossRefGoogle Scholar
  35. 35.
    Iritani, N.T., Sakakibara, K.: Improvements of the traffic signal control by complex-valued Hopfield networks. In: Proceedings of the International Joint Conference on Neural Networks, Vancouver, BC,Google Scholar
  36. 36.
    Buckley, J.J.: Fuzzy complex numbers. Fuzzy Sets and Systems 33, 333–345 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Buckley, J.J., Qu, Y.: Fuzzy complex analysis I: differentiation. Fuzzy Sets and Systems 41, 269–284 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Buckley, J.J.: Fuzzy complex analysis II: integration. Fuzzy Sets and Systems 49, 171–179 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Buckley, J.J., Qu, Y.: Solving linear and quadratic fuzzy equations. Fuzzy Sets and Systems 38, 43–59 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Buckley, J.J., Qu, Y.: Solving fuzzy equations: a new solution concept. Fuzzy Sets and Systems 41, 291–301 (1991)CrossRefGoogle Scholar
  41. 41.
    Zhang, G.: Fuzzy limit theory of fuzzy complex numbers. Fuzzy Sets and Systems 46(2), 227–2352 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Wu, C., Qiu, J.: Some remarks for fuzzy complex analysis. Fuzzy Sets and Systems 106, 231–238 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Ma, S., Peng, D., Li, D.: Fuzzy complex value measure and fuzzy complex value measurable function. In: Proceedings of the Third Annual Conference on Fuzzy Information and Engineering, Haikou, China, pp. 187–192 (2008)Google Scholar
  44. 44.
    Mundici, D., Cignoli, R., D’Ottaviano, I.M.L.: Algebraic foundations of many-valued reasoning. Kluwer Academic Press (1999)Google Scholar
  45. 45.
    Hájek, P.: Fuzzy logic and arithmetical hierarchy. Fuzzy Sets and Systems 3(8), 359–363 (1995)CrossRefGoogle Scholar
  46. 46.
    Casasnovas, J., Rosselló, F.: Scalar and fuzzy cardinalities of crisp and fuzzy multisets. International Journal of Intelligent Systems 24(6), 587–623 (2009)zbMATHCrossRefGoogle Scholar
  47. 47.
    Cintula, P.: Advances in LΠ and LΠ1/2 logics. Archives of Mathematical Logic 42, 449–468 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.CS DepartmentTexas State UniversitySan MarcosUSA
  2. 2.Faculty of Engineering Sciences, Department of Information Systems EngineeringBen-Gurion University of the NegevBeer ShevaIsrael
  3. 3.CSE DepartmentUniversity of South FloridaTampaUSA

Personalised recommendations