×

A systematic review of complex fuzzy sets and logic. (English) Zbl 1397.03093

Summary: Complex fuzzy sets and logic are an extension of type-1 fuzzy sets wherein memberships may be complex-valued. This has been an area of growing research focus in the fuzzy systems community for over a decade, with successful applications in time series forecasting and other areas. We conduct a systematic review of this topic to provide a framework to position new research in the field, consolidate the available theoretical results, catalogue the current applications of complex fuzzy sets and logic, identify the key open questions facing researchers in this area, and suggest possible future directions for research in this field.

MSC:

03E72 Theory of fuzzy sets, etc.
03B52 Fuzzy logic; logic of vagueness
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations

Software:

ANFIS; ANCFIS
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. Aghakhani, S. Dick, An on-line learning algorithm for complex fuzzy logic, in: FUZZ-IEEE, Barcelona, Spain, 2010, 7 pp.; S. Aghakhani, S. Dick, An on-line learning algorithm for complex fuzzy logic, in: FUZZ-IEEE, Barcelona, Spain, 2010, 7 pp.
[2] Al-Husban, A.; Salleh, A. R., Complex fuzzy hyperring based on complex fuzzy spaces, AIP Conf. Proc., 1691, (2015)
[3] A. Al-Husban, A.R. Salleh, Complex fuzzy ring, presented at the Int. C. Research and Education in Mathematics, Kuala Lumpur, Malaysia, 2015.; A. Al-Husban, A.R. Salleh, Complex fuzzy ring, presented at the Int. C. Research and Education in Mathematics, Kuala Lumpur, Malaysia, 2015.
[4] Al-Husban, A.; Salleh, A. R.; Hassan, N., Complex fuzzy normal subgroup, AIP Conf. Proc., 1678, (2015)
[5] Al-Husban, R.; Salleh, A. R., Complex vague relation, AIP Conf. Proc., 1691, (2015)
[6] Alkouri, A. S.; Salleh, A. R., Complex Atanassov’s intuitionistic fuzzy relation, Abstr. Appl. Anal., 2013, (2013), 18 pp · Zbl 1448.03040
[7] Alkouri, A. S.; Salleh, A. R., Complex intuitionistic fuzzy sets, AIP Conf. Proc., 1482, (2012)
[8] Alkouri, A. S.; Salleh, A. R., Linguistic variables, hedges and several distances on complex fuzzy sets, J. Intell. Fuzzy Syst., 26, 2527-2535, (2014) · Zbl 1305.03043
[9] Alkouri, A. S.; Salleh, A. R., Some operations on complex Atanassov’s intuitionistic fuzzy sets, AIP Conf. Proc., 1571, 987-993, (2013)
[10] Atanassov, K. T., Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20, 87-96, (1986) · Zbl 0631.03040
[11] Běhounek, L.; Cintula, P., Fuzzy class theory, Fuzzy Sets Syst., 154, 34-55, (2005) · Zbl 1086.03043
[12] Bellman, R.; Giertz, M., On the analytic formalism of the theory of fuzzy sets, Inf. Sci., 5, 149-156, (1973) · Zbl 0251.02059
[13] Buckley, J. J., Fuzzy complex numbers, Fuzzy Sets Syst., 33, 333-345, (1989) · Zbl 0739.30038
[14] Çağman, N.; Çıtak, F.; Enginoğlu, S., Fuzzy parameterized fuzzy soft set theory and its applications, Turk. J. Fuzzy Syst., 1, 21-35, (2010)
[15] Chen, Z.; Aghakhani, S.; Man, J.; Dick, S., ANCFIS: a neurofuzzy architecture employing complex fuzzy sets, IEEE Trans. Fuzzy Syst., 19, 305-322, (2011)
[16] Cordón, O.; Herrera, F.; Villar, P., Generating the knowledge base of a fuzzy rule-based system by the genetic learning of the data base, IEEE Trans. Fuzzy Syst., 9, 667-674, (2001)
[17] Das, S. K.; Panda, D. C.; Sethi, N.; Gantayat, S. S., Inductive learning of complex fuzzy relation, Int. J. Comput. Inf. Sci. Eng. Inf. Technol., 1, 29-38, (2011)
[18] Deshmukh, A. Y.; Bavaskar, A. B.; Bajaj, P. R.; Keskar, A. G., Implementation of complex fuzzy logic modules with VLSI approach, Int. J. Comput. Sci. Netw. Secur., 8, 172-178, (2008)
[19] Di Zenzo, S., A many-valued logic for approximate reasoning, IBM J. Res. Dev., 32, 552-565, (1988)
[20] Dick, S., Toward complex fuzzy logic, IEEE Trans. Fuzzy Syst., 13, 405-414, (2005)
[21] Dick, S.; Yager, R. R.; Yazdanbakhsh, O., On Pythagorean and complex fuzzy set operations, IEEE Trans. Fuzzy Syst., 24, 1009-1021, (2016)
[22] Edwards, C., The logic of Boolean matrices, Comput. J., 15, 247-253, (1972) · Zbl 0253.94020
[23] Goguen, J. A., L-fuzzy sets, J. Math. Anal. Appl., 18, 145-174, (1967) · Zbl 0145.24404
[24] Greenfield, S.; Chiclana, F., Fuzzy in 3-D: contrasting complex fuzzy sets with type-2 fuzzy sets, (IFSA/NAFIPS, Edmonton, AB, Canada, (2013)), 1237-1242
[25] S. Greenfield, F. Chiclana, S. Dick, Interval-Valued Complex Fuzzy Logic, presented at the FUZZ-IEEE, Vancouver, BC, Canada, 2016.; S. Greenfield, F. Chiclana, S. Dick, Interval-Valued Complex Fuzzy Logic, presented at the FUZZ-IEEE, Vancouver, BC, Canada, 2016.
[26] S. Greenfield, F. Chiclana, S. Dick, Join and Meet Operations for Interval-Valued Complex Fuzzy Logic, presented at the NAFIPS, El Paso, TX, USA, 2016.; S. Greenfield, F. Chiclana, S. Dick, Join and Meet Operations for Interval-Valued Complex Fuzzy Logic, presented at the NAFIPS, El Paso, TX, USA, 2016.
[27] Hata, R.; Murase, K., Generation of fuzzy rules by a complex valued neuro-fuzzy learning algorithm, J. Jpn. Soc. Fuzzy Theory Intell. Inform., 27, 533-548, (2015)
[28] Herrmann, C. S., A hybrid fuzzy-neural expert system for diagnosis, (IJCAI, (1995)), 494-501
[29] Hirose, A., Complex-valued neural networks, (2006), Spinger-Verlag Berlin, Germany · Zbl 1126.68067
[30] Hirose, A., Complex-valued neural networks: theories and applications, (2003), World Scientific Publishing Co. Singapore · Zbl 1058.68096
[31] Jang, J. S.R., ANFIS: adaptive-network-based fuzzy inference system, IEEE Trans. Syst. Man Cybern., 23, 665-685, (1993)
[32] Jiang, Y.; Tang, Y.; Liu, H.; Chen, Z., Entropy on intuitionistic fuzzy soft sets and on interval-valued fuzzy soft sets, Inf. Sci., 240, 95-114, (2013) · Zbl 1320.68190
[33] Kandel, A., Fuzzy expert systems, (1991), CRC Press Boca Raton, FL, USA
[34] Kandel, A.; Tamir, D.; Rishe, N. D., Fuzzy logic and data mining in disaster mitigation, (Teodorescu, H.-N.; etal., Improving Disaster Resilience and Mitigation - IT Means and Tools, (2014), Springer Dordrecht, The Netherlands), 167-186
[35] Karacay, T., Harmonic analysis in fuzzy systems, Int. J. Sci. Res., 3, 1300-1306, (2014)
[36] Karpenko, D.; Van Gorder, R. A.; Kandel, A., The Cauchy problem for complex fuzzy differential equations, Fuzzy Sets Syst., 245, 18-29, (2014) · Zbl 1315.35227
[37] Kasuba, T., Simplified fuzzy ARTMAP, AI Expert, 18-25, (Nov. 1993)
[38] Kaufman, A.; Gupta, M. M., Introduction to fuzzy arithmetic, (1985), Van Nostrand Reinhold Co. New York, NY · Zbl 0588.94023
[39] B. Kitchenham, Procedures for Performing Systematic Reviews, Keele University, Keele2004.; B. Kitchenham, Procedures for Performing Systematic Reviews, Keele University, Keele2004.
[40] Klir, G.; Yuan, B., Fuzzy sets and fuzzy logic, (1995), Prentice Hall Upper Saddle River, NJ, USA · Zbl 0915.03001
[41] Kosheleva, O.; Kreinovich, V., Approximate nature of traditional fuzzy methodology naturally leads to complex-valued fuzzy degrees, (FUZZ-IEEE, Beijing, China, (2014)), 1475-1479
[42] Kosheleva, O.; Kreinovich, V.; Ngamsantivong, T., Why complex-valued fuzzy? why complex values in general? A computational explanation, (IFSA/NAFIPS, Edmonton, AB, Canada, (2013)), 1233-1236
[43] Kumar, T.; Bajaj, R. K., On complex intuitionistic fuzzy soft sets with distance measures and entropies, J. Math., 2014, (2014), 12 pp · Zbl 1489.03021
[44] Lee, C. C., Fuzzy logic in control systems: fuzzy logic controller, part II, IEEE Trans. Syst. Man Cybern., 20, 419-435, (1990) · Zbl 0707.93037
[45] Li, C., Adaptive image restoration by a novel neuro-fuzzy approach using complex fuzzy sets, Int. J. Intell. Inf. Database Syst., 7, 479-495, (2013)
[46] Li, C.; Chan, F.-T., Knowledge discovery by an intelligent approach using complex fuzzy sets, Lect. Notes Comput. Sci., 7196, 320-329, (2012)
[47] Li, C.; Chan, F., Complex-fuzzy adaptive image restoration - an artificial-bee-colony-based learning approach, Lect. Notes Comput. Sci., vol. 6592, 90-99, (2011)
[48] Li, C.; Chiang, T.-W., Complex fuzzy computing to time series prediction A multi-swarm PSO learning approach, Lect. Notes Comput. Sci., vol. 6592, 242-251, (2011)
[49] Li, C.; Chiang, T.-W., Complex neuro-fuzzy self-learning approach to function approximation, Lect. Notes Comput. Sci., vol. 5991, 289-299, (2010)
[50] Li, C.; Chiang, T.-W., Function approximation with complex neuro-fuzzy system using complex fuzzy sets - a new approach, New Gener. Comput., 29, 261-276, (2011) · Zbl 1251.68245
[51] Li, C.; Chiang, T.-W., Intelligent financial time series forecasting: a complex neuro-fuzzy approach with multi-swarm intelligence, Int. J. Appl. Math. Comput. Sci., 22, 787-800, (2012) · Zbl 1286.91149
[52] Li, C.; Chiang, T.-W.; Yeh, L.-C., A novel self-organizing complex neuro-fuzzy approach to the problem of time series forecasting, Neurocomputing, 99, 467-476, (2012)
[53] Li, C.; Chiang, T., Complex neuro-fuzzy ARIMA forecasting - a new approach using complex fuzzy sets, IEEE Trans. Fuzzy Syst., 21, 567-584, (2011)
[54] Li, C.; Wu, T.; Chan, F.-T., Self-learning complex neuro-fuzzy system with complex fuzzy sets and its application to adaptive image noise canceling, Neurocomputing, 94, 121-139, (2012)
[55] Loo, C. K.; Memariani, A.; Liew, W. S., A novel complex-valued fuzzy ARTMAP for sparse dictionary learning, (ICONIP, (2013)), 360-368
[56] J. Ma, R. Wickramasuriya, P. Perez, M. Safadi, Data-driven forecasts of regional demand for infrastructure services, presented at the Int. Symp. Next Gen. Infrastructure, Wollongong, Australia, 2013.; J. Ma, R. Wickramasuriya, P. Perez, M. Safadi, Data-driven forecasts of regional demand for infrastructure services, presented at the Int. Symp. Next Gen. Infrastructure, Wollongong, Australia, 2013.
[57] Ma, J.; Zhang, G.; Lu, J., A method for multiple periodic factor prediction problems using complex fuzzy sets, IEEE Trans. Fuzzy Syst., 20, 32-45, (2012)
[58] Maji, P. K., More on intuitionistic fuzzy soft sets, Lect. Notes Comput. Sci., 5908, 231-240, (2009)
[59] Man, J. Y.; Chen, Z.; Dick, S., Towards inductive learning of complex fuzzy inference systems, (NAFIPS, San Diego, CA, USA, (2007)), 415-420
[60] Mizraji, E., The operators of vector logic, Math. Log. Q., 42, 27-40, (1996) · Zbl 0836.03007
[61] Mizraji, E., Vector logics: the matrix-vector representation of logical calculus, Fuzzy Sets Syst., 50, 179-185, (1992)
[62] Moses, D.; Degani, O.; Teodorescu, H.-N.; Friedman, M.; Kandel, A., Linguistic coordinate transformations for complex fuzzy sets, (FUZZ-IEEE, (1999)), 1340-1345
[63] Moses, D.; Teodorescu, H.-N.; Friedman, M.; Kandel, A., Complex membership grades with an application to the design of adaptive filters, Comput. Sci. J. Mold., 7, 253-283, (1999) · Zbl 1045.03522
[64] Nguyen, H. T.; Kandel, A.; Kreinovich, V., Complex fuzzy sets: towards new foundations, (FUZZ-IEEE, (2000)), 1045-1048
[65] Nguyen, H. T.; Kreinovich, V.; Shekhter, V., On the possibility of using complex values in fuzzy logic for representing inconsistencies, Int. J. Intell. Syst., 13, 683-714, (1998)
[66] Pedrycz, W., Fuzzy control and fuzzy systems, (1993), John Wiley & Sons New York, NY, USA · Zbl 0839.93006
[67] Ramot, D.; Friedman, M.; Langholz, G.; Kandel, A., Complex fuzzy logic, IEEE Trans. Fuzzy Syst., 11, 450-461, (2003)
[68] Ramot, D.; Milo, R.; Friedman, M.; Kandel, A., Complex fuzzy sets, IEEE Trans. Fuzzy Syst., 10, 171-186, (2002)
[69] Rengarajulu, S., Parameterized soft complex fuzzy sets, J. Prog. Res. Math., 4, 303-308, (2015)
[70] Seki, H.; Nakashima, T., Complex-valued SIRMs connected fuzzy inference model, (IEEE-GrC, (2014)), 250-253
[71] C. Servin, V. Kreinovich, O. Kosheleva, From 1-D to 2-D fuzzy: a proof that interval-valued and complex-valued are the only distributive options, presented at the NAFIPS, 2015.; C. Servin, V. Kreinovich, O. Kosheleva, From 1-D to 2-D fuzzy: a proof that interval-valued and complex-valued are the only distributive options, presented at the NAFIPS, 2015.
[72] Sgurev, V., Features of disjunction and conjunction in the complex propositional S-logic, C. R. Acad. Bulgare Sci., 1491-1502, (2014) · Zbl 1324.03008
[73] Shapiro, S., Classical logic, (Zalta, E. N., The Stanford Encyclopedia of Philosophy, (2013), Metaphysics Research Lab, Stanford University Stanford, CA, USA)
[74] Shende, V. V.; Bullock, S. S.; Markov, I. L., Synthesis of quantum-logic circuits, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 25, 1000-1010, (2006)
[75] Shoorangiz, R.; Marhaban, M. H., Complex neuro-fuzzy system for function approximation, Int. J. Appl. Electron. Phys. Robot., 1, 5-9, (2013)
[76] Sun, H.; Wang, S.; Jiang, Q., FCM-based model selection algorithms for determining the number of clusters, Pattern Recognit., 37, 2027-2037, (2004) · Zbl 1056.68583
[77] Tamir, D. E.; Jin, L.; Kandel, A., A new interpretation of complex membership grade, Int. J. Intell. Syst., 26, 285-312, (2011) · Zbl 1219.03061
[78] Tamir, D. E.; Kandel, A., Axiomatic theory of complex fuzzy logic and complex fuzzy classes, Int. J. Comput. Commun. Control, 6, 562-576, (2011)
[79] Tamir, D. E.; Last, M.; Kandel, A., The theory and applications of generalized complex fuzzy propositional logic, (Yager, R. R.; etal., Soft Computing: State of the Art Theory and Novel Applications, (2013), Springer Berlin, Germany), 177-192 · Zbl 1283.03053
[80] Tamir, D. E.; Rishe, N. D.; Kandel, A., Complex fuzzy sets and complex fuzzy logic: an overview of theory and applications, (Tamir, D. E.; etal., Fifty Years of Fuzzy Logic and its Applications, (2015), Springer International Publishing Cham, Switzerland), 661-681 · Zbl 1359.03039
[81] Tamir, D. E.; Rishe, N. D.; Last, M.; Kandel, A., Soft computing based epidemical crisis prediction, (Yager, R. R.; etal., Intelligent Methods for Cyber Warfare, (2015), Springer International Publishing Cham, Switzerland), 43-67
[82] Tamir, D. E.; Teodorescu, H.-N.; Last, M.; Kandel, A., Discrete complex fuzzy logic, (NAFIPS, (2012)), 6 pp
[83] Thirunavukarasu, P.; Suresh, R.; Thamilmani, P., Complex neuro fuzzy system using complex fuzzy sets and update the parameters by PSO-GA and RLSE method, Int. J. Eng. Innov. Technol., 3, 117-122, (2013)
[84] Turunen, E., Algebraic structures in fuzzy logic, Fuzzy Sets Syst., 52, 181-188, (1992) · Zbl 0791.03010
[85] Vakil-Baghmisheh, M.-T.; Pavešić, N., A fast simplified fuzzy ARTMAP network, Neural Process. Lett., 17, 273-316, (2003)
[86] Voxman, W.; Goetschel, R., A note on the characterization of the MAX and MIN operators, Inf. Sci., 30, 5-10, (1983) · Zbl 0597.04002
[87] T. Whalen, Real and imaginary truth in complex fuzzy implication, in: NAFIPS, Redmond, Washington, 2015.; T. Whalen, Real and imaginary truth in complex fuzzy implication, in: NAFIPS, Redmond, Washington, 2015.
[88] Yager, R., Pythagorean membership grades in multi-criteria decision making, IEEE Trans. Fuzzy Syst., 22, 958-965, (2014)
[89] Yager, R. R.; Abbasov, A. M., Pythagorean membership grades, complex numbers, and decision making, Int. J. Intell. Syst., 28, 436-452, (2013)
[90] O. Yazdanbakhsh, S. Dick, Multivariate Time Series Forecasting using Complex Fuzzy Logic, presented at the NAFIPS, Redmond, WA, USA, 2015.; O. Yazdanbakhsh, S. Dick, Multivariate Time Series Forecasting using Complex Fuzzy Logic, presented at the NAFIPS, Redmond, WA, USA, 2015.
[91] Yazdanbakhsh, O.; Dick, S., Time-series forecasting via complex fuzzy logic, (Sadeghian, A.; Tahayori, H., Frontiers of Higher Order Fuzzy Sets, (2015), Springer New York, NY, USA), 147-165 · Zbl 1437.62356
[92] Yazdanbaksh, O.; Krahn, A.; Dick, S., Predicting solar power output using complex fuzzy logic, (IFSA/NAFIPS, Edmonton, AB, Canada, (2013)), 1243-1248
[93] Yubazaki, N.; Yi, J.; Hirota, K., SIRMs (single input rule modules) connected fuzzy inference model, J. Adv. Comput. Intell. Intell. Inform., 1, 23-30, (1997)
[94] Zadeh, L. A., The concept of a linguistic variable and its application to approximate reasoning - I, Inf. Sci., 8, 199-249, (1975) · Zbl 0397.68071
[95] Zadeh, L. A., Probability theory and fuzzy logic, (2002, 23 December 2016), available:
[96] Zhang, G.; Dillon, T. S.; Cai, K.-Y.; Ma, J.; Lu, J., Delta-equalities of complex fuzzy relations, (IEEE International Conference on Advanced Information Networking and Applications, (2010)), 1218-1224
[97] Zhang, G.; Dillon, T. S.; Cai, K.-Y.; Ma, J.; Lu, J., Operation properties and δ-equalities of complex fuzzy sets, Int. J. Approx. Reason., 50, 1227-1249, (2009) · Zbl 1196.03077
[98] Zhao, Z.-Q.; Ma, S.-Q., Complex fuzzy matrix and its convergence problem research, (Cao, B.-Y.; etal., Fuzzy Systems & Operations Research and Management, (2016), Springer International Cham, Switzerland), 157-162 · Zbl 1371.15030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.