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A Correspondence Between Credal Partitions and Fuzzy Orthopartitions

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Belief Functions: Theory and Applications (BELIEF 2022)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13506))

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Abstract

This work highlights the connections between fuzzy orthopartitions and credal partitions, which are both mathematical structures. It is shown that fuzzy orthopartitions are a more general way to represent partitions with uncertainty than credal partitions.

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Notes

  1. 1.

    We also assume here that \(m_i\) is normalized, namely \(m_i(\emptyset )=0\).

  2. 2.

    By an intuitionistic fuzzy set on a universe U, we mean a pair of functions \(\mu _i:U \rightarrow [0,1]\) and \(\nu _i:U \rightarrow [0,1]\) verifying the condition \(\mu _i(u)+\nu _i(u) \le 1\) for each \(u \in U\).

  3. 3.

    Of course, we need to suppose that \(2 \le n \le l\).

  4. 4.

    Let us recall that a fuzzy probabilistic partition \(\{m_1,\ldots ,m_l\}\) of U is a collection of Bayesian bbas, namely for each \(i \in \{1, \ldots , l\}\), \(\sum _{A \subseteq C}m_i(A)=1\) and \(m_i(A)=0\) for each \(A \subseteq C\) that is not a singleton.

  5. 5.

    We notice that the number of equations is less than or equal to the number of variables in \(S_j\), but this does not imply the existence of a solution. Indeed, if \(x=(x_1, \ldots , x_k)\) is a solution of \(S_j\), it must hold that \(x_1, \ldots , x_k \ge 0\) (i.e., \(x_i\) cannot be any real number).

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Author information

Authors and Affiliations

  1. Dipartimento di Informatica, Sistemistica e Comunicazione, Università degli Studi di Milano-Bicocca, Viale Sarca 336, 20126, Milano, Italy

    Stefania Boffa & Davide Ciucci

Authors
  1. Stefania Boffa
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  2. Davide Ciucci
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Corresponding author

Correspondence to Stefania Boffa .

Editor information

Editors and Affiliations

  1. University of Paris-Saclay, Gif sur Yvette, France

    Sylvie Le Hégarat-Mascle

  2. Sorbonne University, Paris, France

    Isabelle Bloch

  3. University of Paris-Saclay, Gif sur Yvette, France

    Emanuel Aldea

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Boffa, S., Ciucci, D. (2022). A Correspondence Between Credal Partitions and Fuzzy Orthopartitions. In: Le Hégarat-Mascle, S., Bloch, I., Aldea, E. (eds) Belief Functions: Theory and Applications. BELIEF 2022. Lecture Notes in Computer Science(), vol 13506. Springer, Cham. https://doi.org/10.1007/978-3-031-17801-6_24

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  • DOI: https://doi.org/10.1007/978-3-031-17801-6_24

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-17800-9

  • Online ISBN: 978-3-031-17801-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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