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.
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Notes
- 1.
The first documented reference by Zadeh to the concepts of Fuzzy Mathematics appeared in a 1962 paper.
References
Zadeh, L.A.: Fuzzy sets. Inf. Control 8(1), 338–353 (1965)
Zadeh, L.A.: Fuzzy algorithms. Inf. Control 12(2), 94–102 (1968)
Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning - part I. Inf. Sci. 7(1), 199–249 (1975)
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)
Yager, R.R.: Fuzzy Sets and Applications: Selected Papers by L.A. Zadeh. Wiley, New York (1987)
Kandel, A.: Fuzzy Mathematical Techniques with Applications. Addison Wesley, Boston (1987)
Kosko, B.: Fuzzy logic. Sci. Am. 269(1), 76–81 (1993)
Běhounek, L., Cintula, P.: Fuzzy class theory. Fuzzy Sets Syst. 154(1), 34–55 (2005)
Tamir, D.E., Kandel, A.: An axiomatic approach to fuzzy set theory. Inf. Sci. 52(1), 75–83 (1990)
Tamir, D., Kandel, A.: Axiomatic theory of complex fuzzy logic and complex fuzzy classes. Int. J. Comput. Commun. Control 6(3), 508–522 (2011)
Drianko, D., Hellendorf, H., Reinfrank, M.: An Introduction to Fuzzy Control. Springer, London (1993)
Lee, C.C.: Fuzzy logic in control systems. IEEE Trans. Syst. Man Cybern. 20(2), 404–435 (1990)
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)
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)
Pedrycz, W., Gomide, F.: An Introduction to Fuzzy Sets Analysis and Design. MIT Press, Massachusetts (1998)
Halpern, J.Y.: Reasoning about Uncertainty. MIT Press, Massachusetts (2003)
Klir, G.J., Tina, A.: Fuzzy Sets, Uncertainty, and Information. Prentice Hall, Upper Saddle River (1988)
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)
Constantin, V.: Fuzzy Logic and NeuroFuzzy Applications Explained. Prentice Hall, Upper Saddle River (1995)
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)
Höppner, F., Klawonn, F., Kruse, R., Runkler, K.: Fuzzy Cluster Analysis: Methods for Classification, Data Analysis and Image Recognition. Wiley, New York (1999)
Tamir, D.E., Kandel, A.: The pyramid fuzzy C-means algorithm. Int. J. Comput. Intell. Control 2(2), 65–77 (2010)
Hu, D., Li, H., Yu, X.: The Information content of fuzzy relations and fuzzy rules. Comput. Math. Appl. 57, 202–216 (2009)
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)
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)
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)
Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning - part II. Inf. Sci. 7(1), 301–357 (1975)
Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning - part III. Inf. Sci. 9(1), 43–80 (1975)
Karnik, N.N., Mendel, J.M., Liang, Q.: Type-2 fuzzy logic systems. IEEE Trans. Fuzzy Syst. 7(6), 643–658 (1999)
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)
Buckley, J.J.: Fuzzy complex numbers. Fuzzy Sets Syst. 33(1), 333–345 (1989)
Yager, R.R.: On a view of zadeh Z-numbers, vol. 299, pp. 90–101 (2012)
Ramot, D., Milo, R., Friedman, M., Kandel, A.: Complex fuzzy sets. IEEE Trans. Fuzzy Syst. 10(2), 171–186 (2002)
Ramot, D., Friedman, M., Langholz, G., Kandel, A.: Complex fuzzy logic. IEEE Trans. Fuzzy Syst. 11(4), 450–461 (2003)
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)
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)
Karnik, N.N., Mendel, J.M., Qilian, L.: Type-2 fuzzy logic systems. IEEE Trans Fuzzy Syst. 7(6), 643–658 (1999)
Buckley, J.J., Qu, Y.: Solving fuzzy equations: a new solution concept. Fuzzy Sets Syst. 41(1), 291–301 (1991)
Buckley, J.J., Qu, Y.: Solving linear and quadratic fuzzy equations. Fuzzy Sets Syst. 38(1), 43–59 (1990)
Buckley, J.J., Qu, Y.: Fuzzy complex analysis I: differentiation. Fuzzy Sets Syst. 41(1), 269–284 (1991)
Buckley, J.J.: Fuzzy complex analysis II: integration. Fuzzy Sets Syst. 49(1), 171–179 (1992)
Tamir, D.E., Kandel, A.: A new interpretation of complex membership grade. Int. J. Intell. Syst. 26(4), 285–312 (2011)
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)
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)
Wu, C., Qiu, J.: Some remarks for fuzzy complex analysis. Fuzzy Sets Syst. 106(1), 231–238 (1999)
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)
Łukasiewicz, J.: On three-valued logic. In: Borkowski, L. (ed.) Selected Works by Jan Łukasiewicz (English Translation), pp. 87–88. North–Holland, Amsterdam (1970)
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)
Guosheng, C., Jianwei, Y.: Complex fuzzy reasoning schemes. In: Proceedings of The Third International Conference on Information and Computing, pp. 29–32 (2010)
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)
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)
Tamir, D.E., Kandel, A.: Fuzzy semantic analysis and formal specification of conceptual knowledge. Inf. Sci. Intell. Syst. 82(3), 181–196 (1995)
Zimmermann, H.: Fuzzy Set Theory and its Applications. Kluwer Academic Publishers, Massachusetts (2001)
Baaz, M., Hajek, P., Montagna, F., Veith, H.: Complexity of t-tautologies. Ann. Pure Appl. Logic 113(1), 3–11 (2002)
Cintula, P.: Weakly implicative fuzzy logics. Arch. Math. Logic 45(6), 673–704 (2006)
Cintula, P.: Advances in LΠ and LΠ1/2 logics. Arch. Math. Logic 42(1), 449–468 (2003)
Hajek, P.: Arithmetical complexity of fuzzy logic - a survey. Soft. Comput. 9(1), 935–941 (2005)
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Massachusetts (19980
Hájek, P.: Fuzzy logic and arithmetical hierarchy. Fuzzy Sets Syst. 3(8), 359–363 (1995)
Montagna, F.: On the predicate logics of continuous t-norm BL-algebras. Arch. Math. Logic 44(1), 97–114 (2005)
Montagna, F.: Three complexity problems in quantified fuzzy logic. Stud. Logica. 68(1), 143–152 (2001)
Mundici, D., Cignoli, R., D’Ottaviano, I.M.L.: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic Press, Massachusetts (1999)
She, Y., Wang, G.: An axiomatic approach of fuzzy rough sets based on residuated lattices. Comput. Math Appl. 58(1), 189–201 (2009)
Fraenkel, A.A., Bar-Hillel, Y., Levy, A.: Foundations of Set Theory, 2nd edn. Elsevier, Pennsylvania (1973)
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)
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)
Dick, S.: Towards complex fuzzy logic. IEEE Trans. Fuzzy Syst. 13(1), 405–414 (2005)
Yager, R.R., Abbasov, A.M.: Pythagorean membership grades, complex numbers, and decision making. Int. J. Intell. Syst. 28(5), 436–452 (2013)
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)
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)
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)
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)
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)
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)
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)
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)
Yazdanbakhsh, O., Dick, S.: Time-series forecasting via complex fuzzy logic, pp. 147–165 (2015)
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)
Li, C., Chiang, T.: Complex neurofuzzy ARIMA forecasting—a new approach using complex fuzzy sets. IEEE Trans. Fuzzy Syst. 21(3), 567–584 (2013)
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)
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)
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)
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)
Acknowledgment
This work is based in part upon work supported by the National Science Foundation under grants I/UCRC IIP-1338922, AIR IIP-1237818, SBIR IIP-1330943, III-Large IIS-1213026, MRI (CNS-1429345, CNS-0821345, CNS-1126619), and CREST HRD-0833093 and by DHS S&T at TerraFly (http://terrafly.com) and the NSF CAKE Center (http://cake.fiu.edu).
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Tamir, D.E., Rishe, N.D., Kandel, A. (2015). Complex Fuzzy Sets and Complex Fuzzy Logic an Overview of Theory and Applications. In: Tamir, D., Rishe, N., Kandel, A. (eds) Fifty Years of Fuzzy Logic and its Applications. Studies in Fuzziness and Soft Computing, vol 326. Springer, Cham. https://doi.org/10.1007/978-3-319-19683-1_31
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