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Generators of Fuzzy Logical Operations

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Algebraic Techniques and Their Use in Describing and Processing Uncertainty

Part of the book series: Studies in Computational Intelligence ((SCI,volume 878))

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Abstract

We deal with fuzzy logical operations with values in the real unit interval. Many of them can be considered equivalent up to an isomorphism (i.e., increasing bijection) of the set of values. This is the case of all involutive fuzzy negations; an elegant proof was given by Nguyen and Walker (A first course in fuzzy logic, 2nd edn. Chapman & Hall/CRC, Boca Raton, 2000 [23]) . The situation is more tricky for binary operations, triangular norms, triangular conorms, and fuzzy implications. For the most common classes of these operations, the existence of their (additive or multiplicative) generators is known; however, their computation can be often unfeasible. We proved that a rather general subclass allows computing the generators from partial derivatives. Here we summarize preceding results in this direction (mostly with simplified proofs) and add several new ones.

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Notes

  1. 1.

    Although this is only a tiny piece of the work of Elbert Walker and his coauthors, we wish to recall this sample; one of many.

  2. 2.

    A different definition of the Archimedean property is used in general, see e.g. [13]. For continuous operations, this simpler condition is equivalent. Also, the definitions of generators which follow are simplified for the case of continuous Archimedean operations.

  3. 3.

    The operator notation might be considered unusual, but the use of Leibnitz’ or Newton’s notation for partial derivatives leads to formulations which are ambiguous or messy.

  4. 4.

    Only nilpotent t-norms may satisfy this condition.

  5. 5.

    More correctly, affine.

  6. 6.

    Only nilpotent t-norms may satisfy this condition.

  7. 7.

    More correctly, affine.

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Acknowledgements

The work was supported by the European Regional Development Fund, project “Center for Advanced Applied Science” (No. CZ.02.1.01/0.0/0.0/16_019/0000778).

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Authors and Affiliations

  1. Center for Machine Perception, Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University in Prague, Prague, Czech Republic

    Mirko Navara

  2. Department of Mathematics, Faculty of Engineering, Czech University of Life Sciences, Prague, Czech Republic

    Milan Petrík

Authors
  1. Mirko Navara
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  2. Milan Petrík
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Correspondence to Mirko Navara .

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Editors and Affiliations

  1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, USA

    Hung T. Nguyen

  2. Department of Computer Science, The University of Texas at El Paso, El Paso, TX, USA

    Vladik Kreinovich

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Navara, M., Petrík, M. (2020). Generators of Fuzzy Logical Operations. In: Nguyen, H., Kreinovich, V. (eds) Algebraic Techniques and Their Use in Describing and Processing Uncertainty. Studies in Computational Intelligence, vol 878. Springer, Cham. https://doi.org/10.1007/978-3-030-38565-1_8

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