Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Introduction Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

12. Fuzzy Implications: Past, Present, and Future

verfasst von : Michał Baczynski, Balasubramaniam Jayaram, Sebastia Massanet, Joan Torrens

Erschienen in: Springer Handbook of Computational Intelligence

Verlag: Springer Berlin Heidelberg

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Fuzzy implications are a generalization of the classical two-valued implication to the multi-valued setting. They play a very important role both in the theory and applications, as can be seen from their use in, among others, multivalued mathematical logic, approximate reasoning, fuzzy control, image processing, and data analysis. The goal of this chapter is to present the evolution of fuzzy implications from their beginnings to the current days. From the theoretical point of view, we present the basic facts, as well as the main topics and lines of research around fuzzy implications. We also devote a specific section to state and recall a list of main application fields where fuzzy implications are employed, as well as another one to the main open problems on the topic.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Fuzzy Relations: Past, Present, and Future
Nächstes Kapitel Fuzzy Rule-Based Systems
Literatur
[12.1]
M. Baczyński, B. Jayaram: $(S,N)$- and $R$-implications: A state-of-the-art survey, Fuzzy Sets Syst. 159(14), 1836–1859 (2008)MathSciNetCrossRefMATH
[12.2]
M. Mas, M. Monserrat, J. Torrens, E. Trillas: A survey on fuzzy implication functions, IEEE Trans. Fuzzy Syst. 15(6), 1107–1121 (2007)CrossRef
[12.3]
M. Baczyński, B. Jayaram: Fuzzy Implications, Studies in Fuzziness and Soft Computing, Vol. 231 (Springer, Berlin, Heidelberg 2008)MATH
[12.4]
M. Baczyński, G. Beliakov, H. Bustince, A. Pradera (Eds.): Advances in Fuzzy Implication Functions, Studies in Fuzziness and Soft Computing, Vol. 300 (Springer, Berlin, Heidelberg 2013)MATH
[12.5]
[12.6]
M. Mas, M. Monserrat, J. Torrens: Two types of implications derived from uninorms, Fuzzy Sets Syst. 158(23), 2612–2626 (2007)MathSciNetCrossRefMATH
[12.7]
[12.8]
I. Aguiló, J. Suñer, J. Torrens: A characterization of residual implications derived from left-continuous uninorms, Inf. Sci. 180(20), 3992–4005 (2010)MathSciNetCrossRefMATH
[12.9]
E. Trillas, C. Alsina: On the law $[(p\wedge q)\to r]=[(p\to r)\vee(q\to r)]$ in fuzzy logic, IEEE Trans. Fuzzy Syst. 10(1), 84–88 (2002)CrossRef
[12.10]
J. Balasubramaniam, C.J.M. Rao: On the distributivity of implication operators over T and S norms, IEEE Trans. Fuzzy Syst. 12(2), 194–198 (2004)CrossRef
[12.11]
D. Ruiz-Aguilera, J. Torrens: Distributivity of strong implications over conjunctive and disjunctive uninorms, Kybernetika 42(3), 319–336 (2006)MathSciNetMATH
[12.12]
D. Ruiz-Aguilera, J. Torrens: Distributivity of residual implications over conjunctive and disjunctive uninorms, Fuzzy Sets Syst. 158(1), 23–37 (2007)MathSciNetCrossRefMATH
[12.13]
[12.14]
M. Baczyński: On the distributivity of fuzzy implications over continuous and Archimedean triangular conorms, Fuzzy Sets Syst. 161(10), 1406–1419 (2010)MathSciNetCrossRefMATH
[12.15]
M. Baczyński: On the distributivity of fuzzy implications over representable uninorms, Fuzzy Sets Syst. 161(17), 2256–2275 (2010)MathSciNetCrossRefMATH
[12.16]
M. Baczyński, B. Jayaram: On the distributivity of fuzzy implications over nilpotent or strict triangular conorms, IEEE Trans. Fuzzy Syst. 17(3), 590–603 (2009)CrossRef
[12.17]
F. Qin, M. Baczyński, A. Xie: Distributive equations of implications based on continuous triangular norms (I), IEEE Trans. Fuzzy Syst. 20(1), 153–167 (2012)CrossRef
[12.18]
M. Baczyński, F. Qin: Some remarks on the distributive equation of fuzzy implication and the contrapositive symmetry for continuous, Archimedean t-norms, Int. J. Approx. Reason. 54(2), 290–296 (2012)MathSciNetCrossRefMATH
[12.19]
B. Jayaram: On the law of importation $(x\wedge y)\to z\equiv(x\to(y\to z))$ in fuzzy logic, IEEE Trans. Fuzzy Syst. 16(1), 130–144 (2008)CrossRef
[12.20]
M. Štěpnička, B. Jayaram: On the suitability of the Bandler--Kohout subproduct as an inference mechanism, IEEE Trans. Fuzzy Syst. 18(2), 285–298 (2010)CrossRef
[12.21]
S. Massanet, J. Torrens: The law of importation versus the exchange principle on fuzzy implications, Fuzzy Sets Syst. 168(1), 47–69 (2011)MathSciNetCrossRefMATH
[12.22]
H. Bustince, J. Fernandez, J. Sanz, M. Baczyński, R. Mesiar: Construction of strong equality index from implication operators, Fuzzy Sets Syst. 211(16), 15–33 (2013)MathSciNetCrossRefMATH
[12.23]
P. Hájek, L. Kohout: Fuzzy implications and generalized quantifiers, Int. J. Uncertain. Fuzziness Knowl. Syst. 4(3), 225–233 (1996)MathSciNetCrossRefMATH
[12.24]
P. Hájek, M.H. Chytil: The GUHA method of automatic hypotheses determination, Computing 1(4), 293–308 (1966)CrossRefMATH
[12.25]
P. Hájek, T. Havránek: The GUHA method-its aims and techniques, Int. J. Man-Mach. Stud. 10(1), 3–22 (1977)CrossRefMATH
[12.26]
P. Hájek, T. Havránek: Mechanizing Hypothesis Formation: Mathematical Foundations for a General Theory (Springer, Heidelberg 1978)CrossRefMATH
[12.27]
[12.28]
F. Durante, E. Klement, R. Mesiar, C. Sempi: Conjunctors and their residual implicators: Characterizations and construction methods, Mediterr. J. Math. 4(3), 343–356 (2007)MathSciNetCrossRefMATH
[12.29]
M. Carbonell, J. Torrens: Continuous R-implications generated from representable aggregation functions, Fuzzy Sets Syst. 161(17), 2276–2289 (2010)MathSciNetCrossRefMATH
[12.30]
[12.31]
V. Biba, D. Hliněná: Generated fuzzy implications and known classes of implications, Acta Univ. M. Belii Ser. Math. 16, 25–34 (2010)MathSciNetMATH
[12.32]
H. Bustince, J. Fernandez, A. Pradera, G. Beliakov: On $({TS},{N})$-fuzzy implications, Proc. AGOP 2011, Benevento, ed. by B. De Baets, R. Mesiar, L. Troiano (2011) pp. 93–98
[12.33]
I. Aguiló, M. Carbonell, J. Suñer, J. Torrens: Dual representable aggregation functions and their derived S-implications, Lect. Notes Comput. Sci. 6178, 408–417 (2010)CrossRef
[12.34]
S. Massanet, J. Torrens: An overview of construction methods of fuzzy implications. In: Advances in Fuzzy Implication Functions, Studies in Fuzziness and Soft Computing, Vol. 300, ed. by M. Baczyński, G. Beliakov, H. Bustince, A. Pradera (Springer, Berlin, Heidelberg 2013) pp. 1–30CrossRef
[12.35]
J. Balasubramaniam: Yager's new class of implications ${J}_{f}$ and some classical tautologies, Inf. Sci. 177(3), 930–946 (2007)MathSciNetCrossRefMATH
[12.36]
M. Baczyński, B. Jayaram: Yager's classes of fuzzy implications: Some properties and intersections, Kybernetika 43(2), 157–182 (2007)MathSciNetMATH
[12.37]
[12.38]
S. Massanet, J. Torrens: On a generalization of Yager's implications, Commun. Comput. Inf. Sci. Ser. 298, 315–324 (2012)CrossRefMATH
[12.39]
S. Massanet, J. Torrens: On a new class of fuzzy implications: h-implications and generalizations, Inf. Sci. 181(11), 2111–2127 (2011)MathSciNetCrossRefMATH
[12.40]
[12.41]
S. Jenei: New family of triangular norms via contrapositive symmetrization of residuated implications, Fuzzy Sets Syst. 110(2), 157–174 (2000)MathSciNetCrossRefMATH
[12.42]
[12.43]
Y. Shi, B.V. Gasse, D. Ruan, E.E. Kerre: On dependencies and independencies of fuzzy implication axioms, Fuzzy Sets Syst. 161(10), 1388–1405 (2010)MathSciNetCrossRefMATH
[12.44]
S. Massanet, J. Torrens: Threshold generation method of construction of a new implication from two given ones, Fuzzy Sets Syst. 205, 50–75 (2012)MathSciNetCrossRefMATH
[12.45]
S. Massanet, J. Torrens: On the vertical threshold generation method of fuzzy implication and its properties, Fuzzy Sets Syst. 226, 32–52 (2013)MathSciNetCrossRefMATH
[12.46]
N.R. Vemuri, B. Jayaram: Fuzzy implications: Novel generation process and the consequent algebras, Commun. Comput. Inf. Sci. Ser. 298, 365–374 (2012)CrossRefMATH
[12.47]
P. Grzegorzewski: Probabilistic implications, Proc. EUSFLAT-LFA 2011, ed. by S. Galichet, J. Montero, G. Mauris (Aix-les-Bains, France 2011) pp. 254–258
[12.48]
[12.49]
P. Grzegorzewski: On the properties of probabilistic implications. In: Eurofuse 2011, Advances in Intelligent and Soft Computing, Vol. 107, ed. by P. Melo-Pinto, P. Couto, C. Serôdio, J. Fodor, B. De Baets (Springer, Berlin, Heidelberg 2012) pp. 67–78
[12.50]
A. Dolati, J. Fernández Sánchez, M. Úbeda-Flores: A copula-based family of fuzzy implication operators, Fuzzy Sets Syst. 211(16), 55–61 (2013)MathSciNetCrossRefMATH
[12.51]
P. Grzegorzewski: Survival implications, Commun. Comput. Inf. Sci. Ser. 298, 335–344 (2012)CrossRefMATH
[12.52]
G. Deschrijver, E. Kerre: On the relation between some extensions of fuzzy set theory, Fuzzy Sets Syst. 133(2), 227–235 (2003)MathSciNetCrossRefMATH
[12.53]
G. Deschrijver, E. Kerre: Triangular norms and related operators in ${L}^{{*}}$-fuzzy set theory. In: Logical, Algebraic, Analytic, Probabilistic Aspects of Triangular Norms, ed. by E. Klement, R. Mesiar (Elsevier, Amsterdam 2005) pp. 231–259CrossRef
[12.54]
G. Deschrijver: Implication functions in interval-valued fuzzy set theory. In: Advances in Fuzzy Implication Functions , Studies in Fuzziness and Soft Computing, Vol. 300, ed. by M. Baczyński, G. Beliakov, H. Bustince, A. Pradera (Springer, Berlin, Heidelberg 2013) pp. 73–99CrossRef
[12.55]
C. Alcalde, A. Burusco, R. Fuentes-González: A constructive method for the definition of interval-valued fuzzy implication operators, Fuzzy Sets Syst. 153(2), 211–227 (2005)MathSciNetCrossRefMATH
[12.56]
H. Bustince, E. Barrenechea, V. Mohedano: Intuitionistic fuzzy implication operators-an expression and main properties, Int. J. Uncertain. Fuzziness Knowl. Syst. 12(3), 387–406 (2004)MathSciNetCrossRefMATH
[12.57]
G. Deschrijver, E. Kerre: Implicators based on binary aggregation operators in interval-valued fuzzy set theory, Fuzzy Sets Syst. 153(2), 229–248 (2005)MathSciNetCrossRefMATH
[12.58]
R. Reiser, B. Bedregal: K-operators: An approach to the generation of interval-valued fuzzy implications from fuzzy implications and vice versa, Inf. Sci. 257, 286–300 (2013)MathSciNetCrossRefMATH
[12.59]
G. Mayor, J. Torrens: Triangular norms in discrete settings. In: Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, ed. by E.P. Klement, R. Mesiar (Elsevier, Amsterdam 2005) pp. 189–230CrossRef
[12.60]
M. Mas, M. Monserrat, J. Torrens: S-implications and R-implications on a finite chain, Kybernetika 40(1), 3–20 (2004)MathSciNetMATH
[12.61]
[12.62]
J. Casasnovas, J. Riera: S-implications in the set of discrete fuzzy numbers, Proc. IEEE-WCCI 2010, Barcelona (2010), pp. 2741–2747
[12.63]
J.V. Riera, J. Torrens: Fuzzy implications defined on the set of discrete fuzzy numbers, Proc. EUSFLAT-LFA 2011 (2011) pp. 259–266
[12.64]
[12.65]
[12.66]
J.C. Bezdek, D. Dubois, H. Prade: Fuzzy Sets in Approximate Reasoning and Information Systems (Kluwer, Dordrecht 1999)CrossRefMATH
[12.67]
D. Driankov, H. Hellendoorn, M. Reinfrank: An Introduction to Fuzzy Control, 2nd edn. (Springer, London 1996)CrossRefMATH
[12.68]
D. Dubois, H. Prade: Fuzzy sets in approximate reasoning, Part 1: Inference with possibility distributions, Fuzzy Sets Syst. 40(1), 143–202 (1991)MathSciNetCrossRefMATH
[12.69]
G.J. Klir, B. Yuan: Fuzzy sets and fuzzy logic-theory and applications (Prentice Hall, Hoboken 1995)MATH
[12.70]
L.A. Zadeh: Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Syst. Man Cybern. 3(1), 28–44 (1973)MathSciNetCrossRefMATH
[12.71]
W. Bandler, L.J. Kohout: Semantics of implication operators and fuzzy relational products, Int. J. Man-Mach. Stud. 12(1), 89–116 (1980)MathSciNetCrossRefMATH
[12.72]
W.E. Combs, J.E. Andrews: Combinatorial rule explosion eliminated by a fuzzy rule configuration, IEEE Trans. Fuzzy Syst. 6(1), 1–11 (1998)CrossRef
[12.73]
B. Jayaram: Rule reduction for efficient inferencing in similarity based reasoning, Int. J. Approx. Reason. 48(1), 156–173 (2008)MathSciNetCrossRefMATH
[12.74]
[12.75]
[12.76]
L. Kitainik: Fuzzy inclusions and fuzzy dichotomous decision procedures. In: Optimization models using fuzzy sets and possibility theory, ed. by J. Kacprzyk, S. Orlovski (Reidel, Dordrecht 1987) pp. 154–170CrossRef
[12.77]
[12.78]
S. Mandal, B. Jayaram: Approximation capability of SISO SBR fuzzy systems based on fuzzy implications, Proc. AGOP 2011, ed. by B. De Baets, R. Mesiar, L. Troiano (University of Sannio, Benevento 2011) pp. 105–110
[12.79]
M. Štěpnička, B. De Baets: Monotonicity of implicative fuzzy models, Proc. FUZZ-IEEE, 2010 Barcelona (2010), pp. 1–7
[12.80]
D. Dubois, H. Prade, L. Ughetto: Checking the coherence and redundancy of fuzzy knowledge bases, IEEE Trans. Fuzzy Syst. 5(3), 398–417 (1997)CrossRef
[12.81]
B. De Baets: Idempotent closing and opening operations in fuzzy mathematical morphology, Proc. ISUMA-NAFIPS'95, Maryland (1995), pp. 228–233
[12.82]
B. De Baets: Fuzzy morphology: A logical approach. In: Uncertainty Analysis in Engineering and Science: Fuzzy Logic, Statistics, Neural Network Approach, ed. by B.M. Ayyub, M.M. Gupta (Kluwer, Dordrecht 1997) pp. 53–68
[12.83]
J. Serra: Image Analysis and Mathematical Morphology (Academic, London, New York 1988)
[12.84]
M. Nachtegael, E.E. Kerre: Connections between binary, gray-scale and fuzzy mathematical morphologies original, Fuzzy Sets Syst. 124(1), 73–85 (2001)MathSciNetCrossRefMATH
[12.85]
I. Bloch: Duality vs. adjunction for fuzzy mathematical morphology and general form of fuzzy erosions and dilations, Fuzzy Sets Syst. 160(13), 1858–1867 (2009)MathSciNetCrossRefMATH
[12.86]
M. González-Hidalgo, A. Mir Torres, D. Ruiz-Aguilera, J. Torrens Sastre: Edge-images using a uninorm-based fuzzy mathematical morphology: Opening and closing. In: Advances in Computational Vision and Medical Image Processing, Computational Methods in Applied Sciences, Vol. 13, ed. by J. Tavares, N. Jorge (Springer, Berlin, Heidelberg 2009) pp. 137–157CrossRef
[12.87]
M. González-Hidalgo, A. Mir Torres, J. Torrens Sastre: Noisy image edge detection using an uninorm fuzzy morphological gradient, Proc. ISDA 2009 (IEEE Computer Society, Los Alamitos 2009) pp. 1335–1340
[12.88]
M. Baczyński, B. Jayaram: Fuzzy implications: Some recently solved problems. In: Advances in Fuzzy Implication Functions, Studies in Fuzziness and Soft Computing, Vol. 300, ed. by M. Baczyński, G. Beliakov, H. Bustince, A. Pradera (Springer, Berlin, Heidelberg 2013) pp. 177–204CrossRef
[12.89]
B. Jayaram, M. Baczyński, R. Mesiar: R-implications and the exchange principle: The case of border continuous t-norms, Fuzzy Sets Syst. 224, 93–105 (2013)MathSciNetCrossRefMATH
Metadaten
Titel
Fuzzy Implications: Past, Present, and Future
verfasst von
Michał Baczynski
Balasubramaniam Jayaram
Sebastia Massanet
Joan Torrens
Copyright-Jahr
2015
Verlag
Springer Berlin Heidelberg
Buch
Springer Handbook of Computational Intelligence

Print ISBN: 978-3-662-43504-5

Electronic ISBN: 978-3-662-43505-2

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-662-43505-2_12