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Fuzzy Implications: Past, Present, and Future

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Springer Handbook of Computational Intelligence

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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.

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Abbreviations

AR:

approximate reasoning

BKS:

Bandler–Kohout subproduct

CP:

contrapositive symmetry

CRI:

compositional rule of inference

EP:

exchange property

FIM:

fuzzy inference mechanism

FMM:

fuzzy mathematical morphology

FRI:

fuzzy relational inference

GMP:

generalized modus ponens

IP:

identity principle

LI:

law of importation

MISO:

multiple inputs-single output

MM:

mathematical morphology

NP:

neutrality principle

OP:

ordering property

RAF:

representable aggregation function

SBR:

similarity based reasoning

SISO:

single input single output

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Authors and Affiliations

  1. Inst. Mathematics, University of Silesia, Bankowa 14, 40-007, Katowice, Poland

    Michał Baczynski

  2. Dep. Mathematics, Indian Institute of Technology Hyderabad, ODF Estate, Yeddumailaram, 502 205, Hyderabad, India

    Balasubramaniam Jayaram

  3. Dep. Mathematics and Computer Science, University of the Balearic Islands, Crta. Valldemossa km. 7,5, 07122, Palma de Mallorca, Spain

    Sebastia Massanet & Joan Torrens

Authors
  1. Michał Baczynski
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  2. Balasubramaniam Jayaram
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  3. Sebastia Massanet
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  4. Joan Torrens
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Correspondence to Michał Baczynski .

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Editors and Affiliations

  1. Systems Research Inst., Polish Academy of Sciences, ul. Newelska 6, 01-447, Warsaw, Poland

    Janusz Kacprzyk

  2. Dep. Electrical and Computer Engineering, University of Alberta, 116 Street 9107, T6J 2V4, Edmonton, Alberta, Canada

    Witold Pedrycz

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Baczynski, M., Jayaram, B., Massanet, S., Torrens, J. (2015). Fuzzy Implications: Past, Present, and Future. In: Kacprzyk, J., Pedrycz, W. (eds) Springer Handbook of Computational Intelligence. Springer Handbooks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43505-2_12

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