Skip to content
2000
  1. Home
  2. A-Z Publications
  3. Journal of Fuzzy Logic and Modeling in Engineering
  4. Volume 1, Issue 2
  5. Article
Volume 1, Issue 2
  • ISSN: 2666-2949
  • E-ISSN: 2666-2957

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).

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.

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.

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.

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.

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.

© 2022 Bentham Science Publishers
Loading

Article metrics loading...

/content/journals/flme/10.2174/2666294901666220610143613
2022-09-01
2024-11-22

  • From This Site

    /content/journals/flme/10.2174/2666294901666220610143613
    dcterms_title,dcterms_subject,pub_keyword
    -contentType:Contributor -contentType:Concept -contentType:Institution
    10
    5
Loading full text...

Full text loading...

References

  1. BaczyńskiM. JayaramB. On the characterizations of -implications.Fuzzy Sets Syst.2007158151713172710.1016/j.fss.2007.02.010
     [Google Scholar]
  2. BaczynskiM. JayaramB. (S, N) - and R-implications; a state-of-the- art survey.Fuzzy Sets Syst.2008159141836185910.1016/j.fss.2007.11.015
     [Google Scholar]
  3. BaczynskiM. JayaramB. Fuzzy Implications.Berlin, Heidelberg, GermanySpringer-Verlang200810.1007/978‑3‑540‑69082‑5
     [Google Scholar]
  4. BustinceH. MohedanoV. BarrenecheaE. PagolaM. Definition and construction of fuzzy DI-subsethood measures.Inf. Sci.2006176213190323110.1016/j.ins.2005.06.006
     [Google Scholar]
  5. MassanetS. VicenteJ. ClapesR. AguileraD.R. 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, 2014.10.1007/978‑3‑319‑08795‑5_15
     [Google Scholar]
  6. KolesovaA. MassanetS. MesiarR. PierraJ.V. TorrensJ. Polynomial constructions of fuzzy implications functions: The quadratic case.Inf. Sci.2019494607910.1016/j.ins.2019.04.040
     [Google Scholar]
  7. MassanetS. TorrensJ. An Overview of Construction Methods of Fuzzy Implications.Stud. Fuzziness Soft Comput.201330013010.1007/978‑3‑642‑35677‑3_1
     [Google Scholar]
  8. MassanetS. TorrensJ. Eds. A new method of generating fuzzy implications from given onesProceedings 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.
     [Google Scholar]
  9. MassanetS. TorrensJ. Threshold generation method of construction of a new implication from two given ones.Fuzzy Sets Syst.201220516507510.1016/j.fss.2012.01.013
     [Google Scholar]
  10. HlinenaD. KalinaM. KralP. Implication Functions Generated Using Functions of One Variable.Springer-Verlag Berlin Heidelberg201310.1007/978‑3‑642‑35677‑3_6
     [Google Scholar]
  11. SainioE. TurunenE. MesiarR. A characterization of fuzzy implications generated by generalized quantifiers.Fuzzy Sets Syst.200815949149910.1016/j.fss.2007.09.018
     [Google Scholar]
  12. DrewniakJ. SoberaJ. Compositions of fuzzy implications In:Advances in Fuzzy Implications FunctionsStudies in Fuzziness and Soft Computing, M. Baczyński, G. Beliakov, H. Bustince Sola, A. Pradera, Eds. Springer: Berlin, Heidelberg, pp. 155-176, 2013.10.1007/978‑3‑642‑35677‑3_7
     [Google Scholar]
  13. DuranteF. KlementE.P. MeriarR. SempiC. Conjunctors and their residual implicators: Characterizations and construction methods.Mediterr. J. Math.20074334335610.1007/s00009‑007‑0122‑1
     [Google Scholar]
  14. EsmiE. LaiateB. Santo PedroF. BarrosL.C. Calculus for fuzzy functions with strongly linearly independent fuzzy coefficients.Fuzzy Sets Syst.202243613110.1016/j.fss.2021.10.006
     [Google Scholar]
  15. BaczynskiM. JayaramB. MassanetS. TorrensJ. Fuzzy implications: Past, present, and future", In:Springer Handbook of Computational IntelligenceJ. Kacprzyk, W. Pedrycz, Eds. Springer: Berlin, Heidelberg, 2015, pp. 183-202.
     [Google Scholar]
  16. BaczyńskiM. JayaramB. QL-implications: Some properties and intersections.Fuzzy Sets Syst.2010161215818810.1016/j.fss.2008.09.021
     [Google Scholar]
  17. KerreE.E. HuangC. RuanD. Fuzzy Set Theory and Approximate Reasoning.Wu Han University Press: Wu Chang, 2004.
     [Google Scholar]
  18. MasM. MonserratM. TorrensJ. TrillasE. A survey on fuzzy implication functions.IEEE Trans. Fuzzy Syst.20071561107112110.1109/TFUZZ.2007.896304
     [Google Scholar]
  19. BustinceH. PagolaM. BarrenecheaE. Construction of fuzzy indices from fuzzy DI-subsethood measures: Application to the global comparison of images.Inf. Sci.2007177390692910.1016/j.ins.2006.07.021
     [Google Scholar]
  20. YanP. ChenG. Discovering a cover set of ARsi with hierarchy from quantitative databases.Inf. Sci.2005173431933610.1016/j.ins.2005.03.003
     [Google Scholar]
  21. BogiatzisA.C. PapadopoulosB.K. Local thresholding of degraded or unevenly illuminated documents using fuzzy inclusion and entropy measures.Evol. Syst.201910459361910.1007/s12530‑018‑09262‑5
     [Google Scholar]
  22. BogiatzisA. PapadopoulosB. Global image thresholding adaptive neuro-fuzzy inference system trained with fuzzy inclusion and entropy measures.Symmetry (Basel)201911228610.3390/sym11020286
     [Google Scholar]
  23. MassanetS. RieraJ.V. Ruiz-AguileraD. On (OP)- polynomial implications16th 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.
     [Google Scholar]
  24. JayaramB. MesiarR. I-Fuzzy equivalence relations and I-fuzzy partitions.Inf. Sci.200917991278129710.1016/j.ins.2008.12.027
     [Google Scholar]
  25. BogiatzisA.C. PapadopoulosB.K. Producing fuzzy inclusion and entropy measures and their application on global image thresholding.Evol. Syst.20189433135310.1007/s12530‑017‑9200‑1
     [Google Scholar]
  26. JayaramB. On the law of importation $(x y) z (x (y z))$ in fuzzy logic.IEEE Trans. Fuzzy Syst.200816113014410.1109/TFUZZ.2007.895969
     [Google Scholar]
  27. BaczyńskiM. JayaramB. (U,N)-implications and their characterizations.Fuzzy Sets Syst.2009160142049206210.1016/j.fss.2008.11.001
     [Google Scholar]
  28. BalasubramaniamJ. Yager’s new class of implications Jf and some classical tautologies.Inf. Sci.2007177393094610.1016/j.ins.2006.08.006
     [Google Scholar]
  29. MassanetS. TorrensJ. On a new class of fuzzy implications: h-Implications and generalizations.Inf. Sci.2011181112111212710.1016/j.ins.2011.01.030
     [Google Scholar]
  30. JayaramB. MesiarR. On special fuzzy implications.Fuzzy Sets Syst.2009160142063208510.1016/j.fss.2008.11.004
     [Google Scholar]
  31. GrzerorzewskiP. Probabilistic implicationsProceedings 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.10.2991/eusflat.2011.67
     [Google Scholar]
  32. MasM. AguileraD.R. TorrensJ. Uninorms based residual implications satisfying the Modus Ponens property with respect to a uninorm.Fuzzy Sets Syst.2019359224110.1016/j.fss.2018.09.014
     [Google Scholar]
  33. MesiarR. KolesárováA. Copulas and fuzzy implications.Int. J. Approx. Reason.2020117525910.1016/j.ijar.2019.11.006
     [Google Scholar]
  34. DuranteF. SempiC. Principles of Copula Theory.New YorkChapman and Hall/CRC201510.1201/b18674
     [Google Scholar]
  35. NelsenR.B. An Introduction to Copulas.2nd Ed. Springer: New York, 2006.
     [Google Scholar]
  36. BaczyńskiM. JayaramB. Fuzzy Implications, Studies in Fuzziness and Soft Computing.Berlin, HeidelbergSpringer2008231
     [Google Scholar]
/content/journals/flme/10.2174/2666294901666220610143613
Loading

  • Article Type:     Research Article
Keyword(s): (OP) rational fuzzy implications; (OP)-polynomial fuzzy implication; contrapositive symmetry; identity principle; natural negation; ordering property

Most Cited Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error
Please enter a valid_number test