The fuzzy XNOR connective, as the dual construction of the fuzzy symmetric difference operator, has been defined and discussed by Li, Qin and He [18]. The aim of this paper is to give a systematic investigation of properties and structures of fuzzy XNOR connectives. The main results are: (1) It is proved that the biresiduation of a t-norm is indeed a fuzzy XNOR connective. (2) The structures and properties for biresiduations and two canonical constructions of fuzzy XNOR connectives given in [18] are examined. The associativity of fuzzy XNOR connectives is specially investigated. (3) Three additional ways of constructing fuzzy XNOR connectives are proposed.
AlsinaC. and TrillasE., On the symmetric difference of fuzzy sets, Fuzzy Sets and Systems153 (2005), 181–194.
2.
AlsinaC. and TrillasE., On the difference of fuzzy sets, International Journal of Intelligent Systems23 (2008), 355–363.
3.
BaczyńskiM., Residual implications revisited. Notes on the Smets-Magrez theorem, Fuzzy Sets and Systems145 (2004), 267–277.
4.
BedregalB.C., SantiagoR.H.N., DimuroG.P. and ReiserR.H.S., Interval valued R-implications and automorphisms, Pre-Proceedings of the 2nd Work on Logical and Semantic Frameworks, with Applications, Ouro Preto, (2007), pp. 82–97.
5.
BedregalB.C., ReiserR.H.S. and DimuroG.P., Xor-implications and E-implications: Classes of fuzzy implications based on fuzzy Xor, Electronic Notes in Theoretical Computer Science247 (2009), 5–18.
6.
BedregalB.C., ReiserR.H.S. and DimuroG.P., Revising XORimplications: Classes of fuzzy (co)implications based on fuzzy XOR (xnor) connectives, International Journal of Uncertainty, Fuzziness and Knowlegde-Based Systems14 (2013), 1–29.
7.
BerrettiniG., SimiA., MalacarneA., BogoniA. and PotiL., Ultrafast integrable and reconfigurable XNOR, AND, NOR and NOT photonic logic gate, IEEE Photonics Technology Letters18(8) (2006), 917–919.
8.
BustinceH., BurilloP. and SoriaF., Automorphisms, negations and implication operators, Fuzzy Sets and Systems134(2) (2003), 209–229.
9.
CalvoT., KolesárováA., KomorníkováM. and MesiarR., Aggregation operators: Properties, classes and construction methods, in: CalvoT., MayorG., MesiarR., (Eds.), Aggregation Operators New Trends and Applications, hysica-Verlag, Heidelberg(2002), pp. 3–104.
10.
ChakradharS.T., BushnellM.L. and AgrawalV.D., Toward massively parallel automatic test generation, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems9(9) (1990), 981–994.
11.
DombiJ., On the equivalence of two symmetric differences, Proceedings of IEEE 8th International Symposium on Intelligent Signal Processing (2013), pp. 32–36.
12.
EndertonH.B., A Mathematical Introduction to Logic, Academic Press Publisher, Boston, 2001.
13.
FodorJ.C. and RoubensM., Fuzzy preference modeling and multicriteria decision supportKluwer Academic Publishers, Dordrecht, 1994.
14.
FujitaM., KomatsuS. and SaitoH., Formal verification techniques for digital systems, in: DiabHassan B. and ZomayaAlbert Y.: Deendable Computing Systems, John Wiley & Sons, Inc., 2005.
15.
HeX.X., LiY.F. and QinK.Y., On the associative property of fuzzy Xor connectives, Journal of Intelligent and Fuzzy Systems25(1) (2013), 1–7.
KumarS. and WillnerA.E., Simultaneous four-wave mixing and cross-gain modulation for implementing an all-optical XNOR logic gate using a single SOA, Optics Express14(12) (2006), 5092–5097.
18.
LiY.F., QinK.Y. and HeX.X., Fuzzy XNOR connectives in fuzzy logic, Soft Computing15 (2011), 2457–2465.
19.
LiY.F., QinK.Y. and HeX.X., Some new approaches to constructing similarity measures, Fuzzy Sets and Systems234 (2014), 46–60.
20.
LiY.F., QinK.Y., HeX.X. and MengD., Robustness of fuzzy reasoning with respect to general divergence measures, Fuzzy Sets and Systems294 (2016), 63–78.
21.
OvchinnikovS. and RoubensM., On strict preference relations, Fuzzy Sets and Systems43(3) (1991), 319–326.
22.
RaseenM., PrasadP.W.C. and SenanayakeS.M.N.A., XOR/XNOR functional behavior on ROBDD representation, in: Proceedings of the 14th IASTED International Conference on Applied Simulation and Modeling, (2005), pp. 115–119.
SchweizerB. and SklarA., Probabilistic Metric Spaces, Dover Publications Inc, New York, 1983.
25.
ShenZ.Q. and ZhangD.X., Symmetric difference operators on fuzzy sets. doi: 10.1016/j.fss.2015.12.005.
26.
TrillasE., Sobre funciones de negación en la teoría de conjuntos difusos, Stochastica3(1) (1979), 47–60.
27.
WangX.Z., RuanD. and KerreE.E., Mathematics of Fuzziness-Basic Issues, Springer, Heidelberg, 2009.
28.
ZanotelliR.M., ReiserR.H.S., da Costa CavalheiroS.A. and FossL., Sensitivity and dual constructions on the fuzzy f-Xor class, 2013 2nd Workshop-School on Theoretical Computer Science (2013), 105–110.
29.
ZengS.Z. and XiaoY., TOPSIS method for intuitionistic fuzzy multiple-criteria decision making and its application to investment selection, Kybernetes45(2) (2016), 282–296.
30.
ZengS.Z., ChenJ.P. and LiX.S., A hybrid method for pythagorean fuzzy multiple-criteria decision making, International Journal of Information Technology & Decision Making15(2) (2016), 403–422.
31.
ZengS.Z., SuW.H. and ZhangC.H., Intuitionistic fuzzy generalized probabilistic ordered weighted averaging operator and its application to group decision making, Technological and Economic Development of Economy22(2) (2016), 177–193.
