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Towards Fuzzy Type Theory with Partial Functions

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Book cover Advances in Fuzzy Logic and Technology 2017 (EUSFLAT 2017, IWIFSGN 2017)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 643))

Abstract

This paper is a study of fuzzy type theory (FTT) with partial functions. Out of several possibilities we decided to introduce a special value “\(*\)” which represents “undefined”. In the interpretation of FTT, this value lays outside of the corresponding domain. In the syntax, it is naturally represented by the description operator acting on the empty (fuzzy) set which, of course, has no element and so, choosing an element from its kernel gives no result, i.e., it is undefined. We will demonstrate that our approach leads to reasonable characterization of the undefinedness. We will also show that any consistent theory of FTT has a model.

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Notes

  1. 1.

    Note that this “\(*\)” is a different element from “\(*\)” introduced for truth values.

  2. 2.

    Recall that the description operator represents, in fact, the defuzzification operation (cf. [14, Chapt. 3]).

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Acknowledgment

This paper was supported by the grant 16-19170S of GAČR, Czech Republic.

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Authors and Affiliations

  1. Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, NSC IT4Innovations, 30. dubna 22, 701 03, Ostrava 1, Czech Republic

    Vilém Novák

Authors
  1. Vilém Novák
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Correspondence to Vilém Novák .

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Editors and Affiliations

  1. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Janusz Kacprzyk

  2. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Eulalia Szmidt

  3. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Slawomir Zadrożny

  4. Institute of Biophysics and Biomedical Engineering, Department of Bioinformatics and Mathematical Modelling, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Krassimir T. Atanassov

  5. WIT - Warsaw School of Information Technology, Warsaw, Poland

    Maciej Krawczak

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Novák, V. (2018). Towards Fuzzy Type Theory with Partial Functions. In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K., Krawczak, M. (eds) Advances in Fuzzy Logic and Technology 2017. EUSFLAT IWIFSGN 2017 2017. Advances in Intelligent Systems and Computing, vol 643. Springer, Cham. https://doi.org/10.1007/978-3-319-66827-7_3

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  • DOI: https://doi.org/10.1007/978-3-319-66827-7_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66826-0

  • Online ISBN: 978-3-319-66827-7

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