Fifty Years of Fuzzy Logic and its Applications pp 661-681

Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 326) | Cite as

Complex Fuzzy Sets and Complex Fuzzy Logic an Overview of Theory and Applications

  • Dan E. Tamir
  • Naphtali D. Rishe
  • Abraham Kandel
Chapter

Abstract

Fuzzy Logic, introduced by Zadeh along with his introduction of fuzzy sets, is a continuous multi-valued logic system. Hence, it is a generalization of the classical logic and the classical discrete multi-valued logic (e.g. Łukasiewicz’ three/many-valued logic). Throughout the years Zadeh and other researches have introduced extensions to the theory of fuzzy setts and fuzzy logic. Notable extensions include linguistic variables, type-2 fuzzy sets, complex fuzzy numbers, and Z-numbers. Another important extension to the theory, namely the concepts of complex fuzzy logic and complex fuzzy sets, has been investigated by Kandel et al. This extension provides the basis for control and inference systems relating to complex phenomena that cannot be readily formalized via type-1 or type-2 fuzzy sets. Hence, in recent years, several researchers have used the new formalism, often in the context of hybrid neuro-fuzzy systems, to develop advanced complex fuzzy logic-based inference applications. In this chapter we reintroduce the concept of complex fuzzy sets and complex fuzzy logic and survey the current state of complex fuzzy logic, complex fuzzy sets theory, and related applications.

Keywords

Fuzzy set theory Fuzzy class theory Fuzzy logic Complex fuzzy sets Complex fuzzy classes Complex fuzzy logic Neuro-fuzzy systems 

References

  1. 1.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8(1), 338–353 (1965)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Zadeh, L.A.: Fuzzy algorithms. Inf. Control 12(2), 94–102 (1968)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning - part I. Inf. Sci. 7(1), 199–249 (1975)CrossRefGoogle Scholar
  4. 4.
    Zadeh, L. A.: From computing with numbers to computing with words - from manipulation of measurements to manipulation of perceptions. IEEE Trans. Circ. Syst. 45(1), 105–119 (1999)Google Scholar
  5. 5.
    Yager, R.R.: Fuzzy Sets and Applications: Selected Papers by L.A. Zadeh. Wiley, New York (1987)Google Scholar
  6. 6.
    Kandel, A.: Fuzzy Mathematical Techniques with Applications. Addison Wesley, Boston (1987)Google Scholar
  7. 7.
    Kosko, B.: Fuzzy logic. Sci. Am. 269(1), 76–81 (1993)Google Scholar
  8. 8.
    Běhounek, L., Cintula, P.: Fuzzy class theory. Fuzzy Sets Syst. 154(1), 34–55 (2005)CrossRefMATHGoogle Scholar
  9. 9.
    Tamir, D.E., Kandel, A.: An axiomatic approach to fuzzy set theory. Inf. Sci. 52(1), 75–83 (1990)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Tamir, D., Kandel, A.: Axiomatic theory of complex fuzzy logic and complex fuzzy classes. Int. J. Comput. Commun. Control 6(3), 508–522 (2011)Google Scholar
  11. 11.
    Drianko, D., Hellendorf, H., Reinfrank, M.: An Introduction to Fuzzy Control. Springer, London (1993)Google Scholar
  12. 12.
    Lee, C.C.: Fuzzy logic in control systems. IEEE Trans. Syst. Man Cybern. 20(2), 404–435 (1990)CrossRefMATHGoogle Scholar
  13. 13.
    De, S.P., Krishna, R.P.: A new approach to mining fuzzy databases using nearest neighbor classification by exploiting attribute hierarchies. Int. J. Intell. Syst. 19(12), 1277–1290 (2004)CrossRefMATHGoogle Scholar
  14. 14.
    Li, C., Chan, F.: Knowledge discovery by an intelligent approach using complex fuzzy sets. In: Pan, J., Chen, S., Nguyen, N.T. (eds) Intelligent Information and Database Systems, pp. 320–329. Springer, Berlin (2012)Google Scholar
  15. 15.
    Pedrycz, W., Gomide, F.: An Introduction to Fuzzy Sets Analysis and Design. MIT Press, Massachusetts (1998)Google Scholar
  16. 16.
    Halpern, J.Y.: Reasoning about Uncertainty. MIT Press, Massachusetts (2003)Google Scholar
  17. 17.
    Klir, G.J., Tina, A.: Fuzzy Sets, Uncertainty, and Information. Prentice Hall, Upper Saddle River (1988)Google Scholar
  18. 18.
    Lou, X., Hou, W., Li, Y., Wang, Z.: A fuzzy neural network model for predicting clothing thermal comfort. Comput. Math Appl. 53(12), 1840–1846 (2007)CrossRefGoogle Scholar
  19. 19.
    Constantin, V.: Fuzzy Logic and NeuroFuzzy Applications Explained. Prentice Hall, Upper Saddle River (1995)Google Scholar
  20. 20.
    Aaron, B., Tamir, D.E., Rishe, N.D., Kandel, A.: Dynamic incremental fuzzy C-means clustering. In Proceedings of The The Sixth International Conference on Pervasive Patterns and Applications, pp. 28–37. Venice, Italy (2014)Google Scholar
  21. 21.
    Höppner, F., Klawonn, F., Kruse, R., Runkler, K.: Fuzzy Cluster Analysis: Methods for Classification, Data Analysis and Image Recognition. Wiley, New York (1999)MATHGoogle Scholar
  22. 22.
    Tamir, D.E., Kandel, A.: The pyramid fuzzy C-means algorithm. Int. J. Comput. Intell. Control 2(2), 65–77 (2010)Google Scholar
  23. 23.
    Hu, D., Li, H., Yu, X.: The Information content of fuzzy relations and fuzzy rules. Comput. Math. Appl. 57, 202–216 (2009)Google Scholar
  24. 24.
    Kandel, A., Tamir, D.E., Rishe, N.D.: Fuzzy logic and data mining in disaster mitigation. In: Teodorescu, H.N., Kirschenbaum, A., Cojocaru, S., Bruderlein, C. (eds.) Improving Disaster Resilience and Mitigation - IT Means and Tools, pp. 167–186. Springer, Netherlands (2014)Google Scholar
  25. 25.
    Agarwal, D., Tamir, D.E., Last, M., Kandel, A.: A comparative study of software testing using artificial neural networks and info-fuzzy networks. IEEE Trans. Syst. Man Cybern. 42(5), 1183–1193 (2012)CrossRefGoogle Scholar
  26. 26.
    Last, M., Friedman, M., Kandel, A.: The data mining approach to automated software testing. In: Proceedings of The Proceedings of the Ninth ACM International Conference on Knowledge Discovery and Data Mining, pp. 388–396 (2003)Google Scholar
  27. 27.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning - part II. Inf. Sci. 7(1), 301–357 (1975)CrossRefGoogle Scholar
  28. 28.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning - part III. Inf. Sci. 9(1), 43–80 (1975)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Karnik, N.N., Mendel, J.M., Liang, Q.: Type-2 fuzzy logic systems. IEEE Trans. Fuzzy Syst. 7(6), 643–658 (1999)CrossRefGoogle Scholar
  30. 30.
    Qilian, L., Mendel, J.M.: Interval type-2 fuzzy logic systems. In: Proceedings of The The Ninth IEEE International Conference on Fuzzy Systems, pp. 328–333 (2000)Google Scholar
  31. 31.
    Buckley, J.J.: Fuzzy complex numbers. Fuzzy Sets Syst. 33(1), 333–345 (1989)CrossRefMATHGoogle Scholar
  32. 32.
    Yager, R.R.: On a view of zadeh Z-numbers, vol. 299, pp. 90–101 (2012)Google Scholar
  33. 33.
    Ramot, D., Milo, R., Friedman, M., Kandel, A.: Complex fuzzy sets. IEEE Trans. Fuzzy Syst. 10(2), 171–186 (2002)CrossRefGoogle Scholar
  34. 34.
    Ramot, D., Friedman, M., Langholz, G., Kandel, A.: Complex fuzzy logic. IEEE Trans. Fuzzy Syst. 11(4), 450–461 (2003)CrossRefGoogle Scholar
  35. 35.
    Moses, D., Degani, O., Teodorescu, H., Friedman, M., Kandel, A.: Linguistic coordinate transformations for complex fuzzy sets. In: Proceedings of The IEEE International Conference on Fuzzy Systems, pp. 1340–1345 (1999)Google Scholar
  36. 36.
    Tamir, D.E., Last, M., Kandel, A.: Complex fuzzy logic. In Seising, R., Trillas, E., Termini, S., Moraga, C. (eds.) On Fuzziness, pp. 665–672. Springer, London (2013)Google Scholar
  37. 37.
    Karnik, N.N., Mendel, J.M., Qilian, L.: Type-2 fuzzy logic systems. IEEE Trans Fuzzy Syst. 7(6), 643–658 (1999)CrossRefGoogle Scholar
  38. 38.
    Buckley, J.J., Qu, Y.: Solving fuzzy equations: a new solution concept. Fuzzy Sets Syst. 41(1), 291–301 (1991)CrossRefGoogle Scholar
  39. 39.
    Buckley, J.J., Qu, Y.: Solving linear and quadratic fuzzy equations. Fuzzy Sets Syst. 38(1), 43–59 (1990)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Buckley, J.J., Qu, Y.: Fuzzy complex analysis I: differentiation. Fuzzy Sets Syst. 41(1), 269–284 (1991)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Buckley, J.J.: Fuzzy complex analysis II: integration. Fuzzy Sets Syst. 49(1), 171–179 (1992)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Tamir, D.E., Kandel, A.: A new interpretation of complex membership grade. Int. J. Intell. Syst. 26(4), 285–312 (2011)CrossRefMATHGoogle Scholar
  43. 43.
    Tamir, D.E., Last, M., Kandel, A.: The theory and applications of generalized complex fuzzy propositional logic. In: Yager, R.R., Abbasov, A.M., Reformat, M.Z. Shahbazova, S.N. (eds.) Soft Computing: State of the Art Theory and Novel Applications Springer Series on Studies in Fuzziness and Soft Computing, pp. 177–192. Springer, Berlin (2013)Google Scholar
  44. 44.
    Zhang, G., Dillon, T.S., Cai, K., Ma, J., Lu, J.: Operation properties and delta equalities of complex fuzzy sets. Int. J. Approximate Reasoning 50(8), 1227–1249 (2009)CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Wu, C., Qiu, J.: Some remarks for fuzzy complex analysis. Fuzzy Sets Syst. 106(1), 231–238 (1999)CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    Ma, S., Peng, D., Li, D.: Fuzzy complex value measure and fuzzy complex value measurable function. In: Cao, B., Zhang, C., Li, T. (eds.) Fuzzy Information and Engineering, pp. 187–192 (2009)Google Scholar
  47. 47.
    Łukasiewicz, J.: On three-valued logic. In: Borkowski, L. (ed.) Selected Works by Jan Łukasiewicz (English Translation), pp. 87–88. North–Holland, Amsterdam (1970)Google Scholar
  48. 48.
    Guh, Y., Yang, M., Po, R., Lee, E.S.: Interval-valued fuzzy relation-based clustering with its application to performance evaluation. Comput. Math Appl. 57(5), 841–849 (2009)CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    Guosheng, C., Jianwei, Y.: Complex fuzzy reasoning schemes. In: Proceedings of The Third International Conference on Information and Computing, pp. 29–32 (2010)Google Scholar
  50. 50.
    Qiu, T., Chen, X., Liu, Q., Huang, H.: Granular computing approach to finding association rules in relational database. Int. J. Intell. Syst. 25(2), 165–179 (2010)MATHGoogle Scholar
  51. 51.
    Ronen, M., Shabtai, R., Guterman, H.: Hybrid model building methodology using unsupervised fuzzy clustering and supervised neural networks. Biotechnol. Bioeng. 77(4), 420–429 (2002)CrossRefGoogle Scholar
  52. 52.
    Tamir, D.E., Kandel, A.: Fuzzy semantic analysis and formal specification of conceptual knowledge. Inf. Sci. Intell. Syst. 82(3), 181–196 (1995)MATHGoogle Scholar
  53. 53.
    Zimmermann, H.: Fuzzy Set Theory and its Applications. Kluwer Academic Publishers, Massachusetts (2001)Google Scholar
  54. 54.
    Baaz, M., Hajek, P., Montagna, F., Veith, H.: Complexity of t-tautologies. Ann. Pure Appl. Logic 113(1), 3–11 (2002)MATHMathSciNetGoogle Scholar
  55. 55.
    Cintula, P.: Weakly implicative fuzzy logics. Arch. Math. Logic 45(6), 673–704 (2006)CrossRefMATHMathSciNetGoogle Scholar
  56. 56.
    Cintula, P.: Advances in LΠ and LΠ1/2 logics. Arch. Math. Logic 42(1), 449–468 (2003)CrossRefMATHMathSciNetGoogle Scholar
  57. 57.
    Hajek, P.: Arithmetical complexity of fuzzy logic - a survey. Soft. Comput. 9(1), 935–941 (2005)CrossRefMATHGoogle Scholar
  58. 58.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Massachusetts (19980Google Scholar
  59. 59.
    Hájek, P.: Fuzzy logic and arithmetical hierarchy. Fuzzy Sets Syst. 3(8), 359–363 (1995)CrossRefGoogle Scholar
  60. 60.
    Montagna, F.: On the predicate logics of continuous t-norm BL-algebras. Arch. Math. Logic 44(1), 97–114 (2005)CrossRefMATHMathSciNetGoogle Scholar
  61. 61.
    Montagna, F.: Three complexity problems in quantified fuzzy logic. Stud. Logica. 68(1), 143–152 (2001)CrossRefMATHMathSciNetGoogle Scholar
  62. 62.
    Mundici, D., Cignoli, R., D’Ottaviano, I.M.L.: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic Press, Massachusetts (1999)Google Scholar
  63. 63.
    She, Y., Wang, G.: An axiomatic approach of fuzzy rough sets based on residuated lattices. Comput. Math Appl. 58(1), 189–201 (2009)CrossRefMATHMathSciNetGoogle Scholar
  64. 64.
    Fraenkel, A.A., Bar-Hillel, Y., Levy, A.: Foundations of Set Theory, 2nd edn. Elsevier, Pennsylvania (1973)Google Scholar
  65. 65.
    Nguyen, H.T., Kandel, A., Kreinovich, V.: Complex fuzzy sets: towards new foundations. In: Proceedings of The Proceedings of the IEEE International Conference on Fuzzy Systems, pp. 1045–1048 (2000)Google Scholar
  66. 66.
    Tamir, D.E., Last, M., Teodorescu, N.H., Kandel, A.: Discrete complex fuzzy logic. In: Proceedings of The Proceedings of the North American Fuzzy Information Processing Society, pp. 1–6. California, USA (2012)Google Scholar
  67. 67.
    Dick, S.: Towards complex fuzzy logic. IEEE Trans. Fuzzy Syst. 13(1), 405–414 (2005)CrossRefMathSciNetGoogle Scholar
  68. 68.
    Yager, R.R., Abbasov, A.M.: Pythagorean membership grades, complex numbers, and decision making. Int. J. Intell. Syst. 28(5), 436–452 (2013)CrossRefGoogle Scholar
  69. 69.
    Greenfield, S., Chiclana, F.: Fuzzy in 3-D: contrasting complex fuzzy sets with type-2 fuzzy sets. In: Proceedings of The Joint Annual Meeting IFSA World Congress and NAFIPS, pp. 1237–1242 (2013)Google Scholar
  70. 70.
    Apolloni, B., Pedrycz, W., Bassis, S., Malchiodi, D.: Granular constructs. In: Apolloni, B., Pedrycz, W., Bassis, S. Malchiodi, D. (eds.) The Puzzle of Granular Computing, pp. 343–384. Springer, Berlin (2008)Google Scholar
  71. 71.
    Guangquan, Z., Dillon, T.S., Kai-Yuan, C., Jun, M., Jie, L.: Delta-equalities of complex fuzzy relations. In: Proceedings of The IEEE International 24th Conference on Advanced Information Networking and Applications, pp. 1218–1224 (2010)Google Scholar
  72. 72.
    Chen, Z., Aghakhani, S., Man, J., Dick, S.: ANCFIS: a neuro-fuzzy architecture employing complex fuzzy sets. IEEE Trans. Fuzzy Syst. 19(2), 305–322 (2009)CrossRefGoogle Scholar
  73. 73.
    Man, J.Y., Chen, Z., Dick, S.: Towards inductive learning of complex fuzzy inference systems. In: Proceedings of The Annual Meeting of the North American Fuzzy Information Processing Society, pp. 415–420 (2007)Google Scholar
  74. 74.
    Zhifei, C., Aghakhani, S., Man, J., Dick, S.: ANCFIS: a neurofuzzy architecture employing complex fuzzy sets. IEEE Int. Conf. Fuzzy Syst. 19(2), 305–322 (2011)CrossRefGoogle Scholar
  75. 75.
    Aghakhani, S., Dick, S.: An on-line learning algorithm for complex fuzzy logic. In: Proceedings of The The IEEE International Conference on Fuzzy Systems, pp. 1–7 (2010)Google Scholar
  76. 76.
    Yazdanbaksh, O., Krahn, A., Dick, S.: Predicting solar power output using complex fuzzy logic. In: Proceedings of The Joint IFSA World Congress and NAFIPS Annual Meeting, pp. 1243–1248 (2013)Google Scholar
  77. 77.
    Yazdanbakhsh, O., Dick, S.: Time-series forecasting via complex fuzzy logic, pp. 147–165 (2015)Google Scholar
  78. 78.
    Li, Y., Jang, T.Y.: Complex adaptive fuzzy inference systems. In: Proceedings of The Proceedings of the Asian Conference on Soft Computing in Intelligent Systems and Information Processing, pp. 551–556 (1996)Google Scholar
  79. 79.
    Li, C., Chiang, T.: Complex neurofuzzy ARIMA forecasting—a new approach using complex fuzzy sets. IEEE Trans. Fuzzy Syst. 21(3), 567–584 (2013)Google Scholar
  80. 80.
    Li, C., Chiang, T.: Function approximation with complex neuro-fuzzy system using complex fuzzy sets A new approach. New Gener. Comput. 29(3), 261–276 (2011)CrossRefGoogle Scholar
  81. 81.
    Li, C., Chan, F.: Complex-fuzzy adaptive image restoration an artificial-bee-colony-based learning approach. In: Nguyen, N.T., Kim, C., Janiak, A. (eds.) Intelligent Information and Database Systems, pp. 90–99. Springer, Berlin (2011)Google Scholar
  82. 82.
    Tamir, D.E., Mueller, C.J., Kandel, A.: Complex fuzzy logic reasoning based methodologies for quantitative software engineering. In: Pedrycz, W., Succi, G., Sillitti, A. (eds.) Computational Intelligence and Quantitative Software Engineering. Springer, Berlin (2015)Google Scholar
  83. 83.
    Tamir, D.E., Rishe, N.D., Last, M., Kandel, A.: Soft computing based epidemical crisis prediction. In: Yager, R.R., Reformat, M.Z., Alajlan, N. (eds.) Intelligent Methods for Cyberwarfare, pp. 43–76. Springer, Berlin (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Dan E. Tamir
    • 1
  • Naphtali D. Rishe
    • 2
  • Abraham Kandel
    • 2
  1. 1.Department of Computer Science, Texas State University San MarcosTexasUSA
  2. 2.School of Computing and Information Sciences, Florida International UniversityMiamiUSA

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