Login Register Cart 0
Generic placeholder image

Journal of Fuzzy Logic and Modeling in Engineering

Editor-in-Chief

ISSN (Print): 2666-2949
ISSN (Online): 2666-2957

Back
Journal Home About Journal Editorial Board Current Issue Volumes/Issues
Subscribe
Research Article

A Novel Construction Method of (OP) Polynomial and Rational Fuzzy Implications

Author(s): Maria N. Rapti and Basil K. Papadopoulos*

Volume 1, Issue 2, 2022

Published on: 23 September, 2022

Article ID: e100622205847 Pages: 11

DOI: 10.2174/2666294901666220610143613

Price: $65

Abstract

In this article, we develop new constructed methods with specific conditions. The first method is a generalization of convex combination using n fuzzy implications. The second method is a parameterization of Lukasiewicz implication in an Ordering Property (OP) fuzzy implication form. The innovation in this work is the presentation of three new constructed methods of (OP) polynomial and (OP) rational fuzzy implications. We investigate some families of Ordering Property (OP) and Ordering Property (OP) Rational fuzzy implications. To these methods, we give some coefficient conditions in order to satisfy basic properties like ordering property (OP), identity property (IP) and contrapositive symmetry (CP).

Background: Fuzzy implication functions are one of the main operations in fuzzy logic. They generalize the classical implication, which takes values in the set {0, 1}, to fuzzy logic, where the truth values belong to the unit interval [0, 1]. The study of this class of operations has been extensively developed in the literature in the last 30 years from both theoretical and applicational points of view.

Introduction: In this paper, we develop five new methods for constructing fuzzy implications with specific properties. The paper starts by presenting the first fuzzy implication construction machine that uses n fuzzy implications with specific conditions. Next, we parameterize Lukasiewicz implication and create new families of (OP) polynomial and (OP) rational implications. For each method we investigate which conditions are satisfied and we give some examples.

Methods: The first constructed method uses n fuzzy implications in a linear product representation. The second method is an (OP) polynomial implication a parameterized Lukasiewicz implication. The third method is a rational implication with five parameters. In the fourth method we give a general form in the previous method by changing variables x and y with increasing functions. Finally, the last method is another (OP) rational implication with three parameters.

Results: In each method we present the properties that are satisfied. We generalize the (OP) polynomial and rational by replacing the variables with monotonic functions or add powers on them. Finally, we generalize and we give examples of new produced fuzzy implications.

Conclusion: As a future work, we can create new families of rational implications by changing the polynomials of the numerator and denominator so that they satisfy more properties. Finally, the new methods we presented can contribute in the construction of uninorms and copulas under certain conditions.

Keywords: (OP)-polynomial fuzzy implication, (OP)-rational fuzzy implications, ordering property, identity principle, natural negation, contrapositive symmetry.

Graphical Abstract

[1]
M. Baczyński, and B. Jayaram, "On the characterizations of -implications", Fuzzy Sets Syst., vol. 158, no. 15, pp. 1713-1727, 2007.
[http://dx.doi.org/10.1016/j.fss.2007.02.010]
[2]
M. Baczynski, and B. Jayaram, "(S, N) - and R-implications; a state-of-the- art survey", Fuzzy Sets Syst., vol. 159, no. 14, pp. 1836-1859, 2008.
[http://dx.doi.org/10.1016/j.fss.2007.11.015]
[3]
M. Baczynski, and B. Jayaram, Fuzzy Implications., Springer-Verlang: Berlin, Heidelberg, Germany, 2008.
[http://dx.doi.org/10.1007/978-3-540-69082-5]
[4]
H. Bustince, V. Mohedano, E. Barrenechea, and M. Pagola, "Definition and construction of fuzzy DI-subsethood measures", Inf. Sci., vol. 176, no. 21, pp. 3190-3231, 2006.
[http://dx.doi.org/10.1016/j.ins.2005.06.006]
[5]
S. Massanet, J. Vicente, R. Clapes, and D.R. Aguilera, "On fuzzypolynomials implications In:", Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2014. Communications in Computer and Information Science..
A. Lauret, O. Strauss, B. Bouchon-Meunier, R.R. Yager, Eds. Springer, Cham, 2014. [http://dx.doi.org/10.1007/978-3-319-08795-5_15]
[6]
A. Kolesova, S. Massanet, R. Mesiar, J.V. Pierra, and J. Torrens, "Polynomial constructions of fuzzy implications functions: The quadratic case", Inf. Sci., vol. 494, pp. 60-79, 2019.
[http://dx.doi.org/10.1016/j.ins.2019.04.040]
[7]
S. Massanet, and J. Torrens, "An Overview of Construction Methods of Fuzzy Implications", Stud. Fuzziness Soft Comput., vol. 300, pp. 1-30, 2013.
[http://dx.doi.org/10.1007/978-3-642-35677-3_1]
[8]
S. Massanet, and J. Torrens, "Eds. A new method of generating fuzzy implications from given ones", Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (EUSFLATLFA 2011), Aug 2011, Aix-Les-Bains, France, Atlantis Press: Amsterdam, Noord-Holland, The Netherlands..
[9]
S. Massanet, and J. Torrens, "Threshold generation method of construction of a new implication from two given ones", Fuzzy Sets Syst., vol. 205, no. 16, pp. 50-75, 2012.
[http://dx.doi.org/10.1016/j.fss.2012.01.013]
[10]
D. Hlinena, M. Kalina, and P. Kral, Implication Functions Generated Using Functions of One Variable., Springer-Verlag Berlin Heidelberg, 2013.
[http://dx.doi.org/10.1007/978-3-642-35677-3_6]
[11]
E. Sainio, E. Turunen, and R. Mesiar, "A characterization of fuzzy implications generated by generalized quantifiers", Fuzzy Sets Syst., vol. 159, pp. 491-499, 2008.
[http://dx.doi.org/10.1016/j.fss.2007.09.018]
[12]
J. Drewniak, and J. Sobera, "Compositions of fuzzy implications In:", Advances in Fuzzy Implications Functions.
Studies in Fuzziness and Soft Computing, M. Baczyński, G. Beliakov, H. Bustince Sola, A. Pradera, Eds. Springer: Berlin, Heidelberg, pp. 155-176, 2013. [http://dx.doi.org/10.1007/978-3-642-35677-3_7]
[13]
F. Durante, E.P. Klement, R. Meriar, and C. Sempi, "Conjunctors and their residual implicators: Characterizations and construction methods", Mediterr. J. Math., vol. 4, no. 3, pp. 343-356, 2007.
[http://dx.doi.org/10.1007/s00009-007-0122-1]
[14]
E. Esmi, B. Laiate, F. Santo Pedro, and L.C. Barros, "Calculus for fuzzy functions with strongly linearly independent fuzzy coefficients", Fuzzy Sets Syst., vol. 436, pp. 1-31, 2022.
[http://dx.doi.org/10.1016/j.fss.2021.10.006]
[15]
M. Baczynski, B. Jayaram, S. Massanet, and J. Torrens, "Fuzzy implications: Past, present, and future", In:", Springer Handbook of Computational Intelligence. J. Kacprzyk, W. Pedrycz, Eds. Springer: Berlin, Heidelberg, 2015, pp. 183-202.
[16]
M. Baczyński, and B. Jayaram, "QL-implications: Some properties and intersections", Fuzzy Sets Syst., vol. 161, no. 2, pp. 158-188, 2010.
[http://dx.doi.org/10.1016/j.fss.2008.09.021]
[17]
E.E. Kerre, C. Huang, and D. Ruan, Fuzzy Set Theory and Approximate Reasoning.. Wu Han University Press: Wu Chang, 2004.
[18]
M. Mas, M. Monserrat, J. Torrens, and E. Trillas, "A survey on fuzzy implication functions", IEEE Trans. Fuzzy Syst., vol. 15, no. 6, pp. 1107-1121, 2007.
[http://dx.doi.org/10.1109/TFUZZ.2007.896304]
[19]
H. Bustince, M. Pagola, and E. Barrenechea, "Construction of fuzzy indices from fuzzy DI-subsethood measures: Application to the global comparison of images", Inf. Sci., vol. 177, no. 3, pp. 906-929, 2007.
[http://dx.doi.org/10.1016/j.ins.2006.07.021]
[20]
P. Yan, and G. Chen, "Discovering a cover set of ARsi with hierarchy from quantitative databases", Inf. Sci., vol. 173, no. 4, pp. 319-336, 2005.
[http://dx.doi.org/10.1016/j.ins.2005.03.003]
[21]
A.C. Bogiatzis, and B.K. Papadopoulos, "Local thresholding of degraded or unevenly illuminated documents using fuzzy inclusion and entropy measures", Evol. Syst., vol. 10, no. 4, pp. 593-619, 2019.
[http://dx.doi.org/10.1007/s12530-018-09262-5]
[22]
A. Bogiatzis, and B. Papadopoulos, "Global image thresholding adaptive neuro-fuzzy inference system trained with fuzzy inclusion and entropy measures", Symmetry (Basel), vol. 11, no. 2, p. 286, 2019.
[http://dx.doi.org/10.3390/sym11020286]
[23]
S. Massanet, J.V. Riera, and D. Ruiz-Aguilera, "On (OP)- polynomial implications", 16th World Congress of the International Fuzzy Systems Association (IPSA), 9th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT), Jun 30-July 3 2015, Gijon, Asturias, Spain, Atlantis Press: Amsterdam, The Netherlands..
[24]
B. Jayaram, and R. Mesiar, "I-Fuzzy equivalence relations and I-fuzzy partitions", Inf. Sci., vol. 179, no. 9, pp. 1278-1297, 2009.
[http://dx.doi.org/10.1016/j.ins.2008.12.027]
[25]
A.C. Bogiatzis, and B.K. Papadopoulos, "Producing fuzzy inclusion and entropy measures and their application on global image thresholding", Evol. Syst., vol. 9, no. 4, pp. 331-353, 2018.
[http://dx.doi.org/10.1007/s12530-017-9200-1]
[26]
B. Jayaram, "On the law of importation $(x y) z (x (y z))$ in fuzzy logic", IEEE Trans. Fuzzy Syst., vol. 16, no. 1, pp. 130-144, 2008.
[http://dx.doi.org/10.1109/TFUZZ.2007.895969]
[27]
M. Baczyński, and B. Jayaram, "(U,N)-implications and their characterizations", Fuzzy Sets Syst., vol. 160, no. 14, pp. 2049-2062, 2009.
[http://dx.doi.org/10.1016/j.fss.2008.11.001]
[28]
J. Balasubramaniam, "Yager’s new class of implications Jf and some classical tautologies", Inf. Sci., vol. 177, no. 3, pp. 930-946, 2007.
[http://dx.doi.org/10.1016/j.ins.2006.08.006]
[29]
S. Massanet, and J. Torrens, "On a new class of fuzzy implications: h-Implications and generalizations", Inf. Sci., vol. 181, no. 11, pp. 2111-2127, 2011.
[http://dx.doi.org/10.1016/j.ins.2011.01.030]
[30]
B. Jayaram, and R. Mesiar, "On special fuzzy implications", Fuzzy Sets Syst., vol. 160, no. 14, pp. 2063-2085, 2009.
[http://dx.doi.org/10.1016/j.fss.2008.11.004]
[31]
P. Grzerorzewski, "Probabilistic implications", Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-LFA 2011), Aug 2011, Aix-Les-Bains, France, Atlantis Press: Amsterdam, The Netherlands..
[http://dx.doi.org/10.2991/eusflat.2011.67]
[32]
M. Mas, D.R. Aguilera, and J. Torrens, "Uninorms based residual implications satisfying the Modus Ponens property with respect to a uninorm", Fuzzy Sets Syst., vol. 359, pp. 22-41, 2019.
[http://dx.doi.org/10.1016/j.fss.2018.09.014]
[33]
R. Mesiar, and A. Kolesárová, "Copulas and fuzzy implications", Int. J. Approx. Reason., vol. 117, pp. 52-59, 2020.
[http://dx.doi.org/10.1016/j.ijar.2019.11.006]
[34]
F. Durante, and C. Sempi, Principles of Copula Theory., Chapman and Hall/CRC: New York, 2015.
[http://dx.doi.org/10.1201/b18674]
[35]
R.B. Nelsen, An Introduction to Copulas.. 2nd Ed. Springer: New York, 2006.
[36]
M. Baczyński, and B. Jayaram, Fuzzy Implications, Studies in Fuzziness and Soft Computing., vol. 231. Springer: Berlin, Heidelberg, 2008.

Rights & Permissions Print Cite
Article Metrics
1
© 2024 Bentham Science Publishers | Privacy Policy