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International Conference on Rough Sets and Knowledge Technology

RSKT 2015: Rough Sets and Knowledge Technology pp 347–356Cite as

Formalizing Lattice-Theoretical Aspects of Rough and Fuzzy Sets

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9436))

Abstract

Fuzzy sets and rough sets are well-known approaches to incomplete or imprecise data. In the paper we briefly report how these frameworks were successfully encoded with the help of one of the leading computer proof assistants in the world. Even though fuzzy sets are much closer to the set theory implemented within the Mizar library than rough sets, lattices as a basic viewpoint appeared a very feasible one. We focus on the lattice-theoretical aspects of rough and fuzzy sets to enable the application of external theorem provers like EQP or Prover9 as well as to translate them into TPTP format widely recognized in the world of automated proof search. The paper is illustrated with the examples taken just from one of the largest repositories of computer-checked mathematical knowledge – the Mizar Mathematical Library. Our formal development allows both for further generalizations, building on top of the existing knowledge, and even merging of these approaches.

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Notes

  1. 1.

    Most theorem provers are untyped, the Mizar language has types.

  2. 2.

    All the proofs are partially translated in the journal Formalized Mathematics and independently can be browsed at the homepage of the project.

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Author information

Authors and Affiliations

  1. Institute of Informatics, University of Białystok, Ciołkowskiego 1M, 15-245, Białystok, Poland

    Adam Grabowski

  2. University of Marketing and Distribution Sciences, Kobe, 3-1 Gakuen Nishimachi Nishi-ku, Kobe, 655-2188, Japan

    Takashi Mitsuishi

Authors
  1. Adam Grabowski
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  2. Takashi Mitsuishi
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Corresponding author

Correspondence to Adam Grabowski .

Editor information

Editors and Affiliations

  1. University of Milano-Bicocca, Milano, Italy

    Davide Ciucci

  2. Chongqing University of Posts and Telecommunications, Chongqing, China

    Guoyin Wang

  3. Indian Statistical Institute, Kolkata, India

    Sushmita Mitra

  4. Zhejiang Ocean University, Zhejiang, China

    Wei-Zhi Wu

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Grabowski, A., Mitsuishi, T. (2015). Formalizing Lattice-Theoretical Aspects of Rough and Fuzzy Sets. In: Ciucci, D., Wang, G., Mitra, S., Wu, WZ. (eds) Rough Sets and Knowledge Technology. RSKT 2015. Lecture Notes in Computer Science(), vol 9436. Springer, Cham. https://doi.org/10.1007/978-3-319-25754-9_31

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  • DOI: https://doi.org/10.1007/978-3-319-25754-9_31

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-25753-2

  • Online ISBN: 978-3-319-25754-9

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