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Joint German/Austrian Conference on Artificial Intelligence (Künstliche Intelligenz)

KI 2017: KI 2017: Advances in Artificial Intelligence pp 58–71Cite as

A Thorough Formalization of Conceptual Spaces

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10505))

Abstract

The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points in a high-dimensional space and concepts are represented by convex regions in this space. After pointing out a problem with the convexity requirement, we propose a formalization of conceptual spaces based on fuzzy star-shaped sets. Our formalization uses a parametric definition of concepts and extends the original framework by adding means to represent correlations between different domains in a geometric way. Moreover, we define computationally efficient operations on concepts (intersection, union, and projection onto a subspace) and show that these operations can support both learning and reasoning processes.

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Notes

  1. 1.

    The weaker requirement of star-shapedness allows us to “cut out” some corners from the rectangle. This enables us to geometrically represent correlations.

  2. 2.

    We will drop the modifier “axis-parallel" from now on.

  3. 3.

    Note that if \(D_{S_1} \cap D_{S_2} = \emptyset \), then \(P_1 \cap P_2 \ne \emptyset \).

  4. 4.

    In some cases, the normalization constraint of the resulting domain weights might be violated. We can enforce this constraint by manually normalizing them.

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Authors and Affiliations

  1. Institute of Cognitive Science, Osnabrück University, Osnabrück, Germany

    Lucas Bechberger & Kai-Uwe Kühnberger

Authors
  1. Lucas Bechberger
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  2. Kai-Uwe Kühnberger
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Correspondence to Lucas Bechberger .

Editor information

Editors and Affiliations

  1. Fakultät für Informatik, Technische Universität Dortmund, Dortmund, Germany

    Gabriele Kern-Isberner

  2. FB Informatik, TU Darmstadt, Darmstadt, Hessen, Germany

    Johannes Fürnkranz

  3. FB Informatik, Universität Koblenz, Koblenz, Rheinland-Pfalz, Germany

    Matthias Thimm

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Bechberger, L., Kühnberger, KU. (2017). A Thorough Formalization of Conceptual Spaces. In: Kern-Isberner, G., Fürnkranz, J., Thimm, M. (eds) KI 2017: Advances in Artificial Intelligence. KI 2017. Lecture Notes in Computer Science(), vol 10505. Springer, Cham. https://doi.org/10.1007/978-3-319-67190-1_5

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  • DOI: https://doi.org/10.1007/978-3-319-67190-1_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67189-5

  • Online ISBN: 978-3-319-67190-1

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