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A Coq-Based Axiomatization of Tarski’s Mereogeometry

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Spatial Information Theory (COSIT 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9368))

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Abstract

During the last decade, the domain of Qualitative Spatial Reasoning, has known a renewal of interest for mereogeometry, a theory that has been initiated by Tarski. Mereogeometry relies on mereology, the Leśniewski’s theory of parts and wholes that is further extended with geometrical primitives and appropriate definitions. However, most approaches (i) depart from the original Leśniewski’s mereology which does not assume usual sets as a basis, (ii) restrict the logical power of mereology to a mere theory of part-whole relations and (iii) require the introduction of a connection relation. Moreover, the seminal paper of Tarki shows up unclear foundations and we argue that mereogeometry as it is introduced by Tarski, can be more suited to extend the whole theory of Leśniewski. For that purpose, we investigate a type-theoretical representation of space more closely related with the original ideas of Leśniewski and expressed with the Coq language. We show that (i) it can be given a more clear foundation, (ii) it can be based on three axioms instead of four and (iii) it can serve as a basis for spatial reasoning with full compliance with Leśniewski’s systems.

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Notes

  1. 1.

    Most descriptions suffer from the misunderstanding that mereology is formally nothing more than a particular elementary theory of partial ordering.

  2. 2.

    The two argument categories are on the right side and the resulting category on the left side.

  3. 3.

    This is partly due to the destruction of most of his work during the second world war and to the difficulty to assess protothetic.

  4. 4.

    Known as Girard–Reynolds polymorphic lambda calculus. It extends STT by the introduction of a mechanism of universal quantification over types (second-order) and is itself extended in CIC.

  5. 5.

    Also called sort.

  6. 6.

    Called spheres in Tarski’s paper.

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Authors and Affiliations

  1. LISTIC/Polytech Annecy-Chambéry, University of Savoie, Po. Box 80439, 74944, Annecy-le-vieux cedex, France

    Richard Dapoigny & Patrick Barlatier

Authors
  1. Richard Dapoigny
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  2. Patrick Barlatier
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Correspondence to Richard Dapoigny .

Editor information

Editors and Affiliations

  1. University of Zurich, Zurich, Switzerland

    Sara Irina Fabrikant

  2. ETH Zurich, Zurich, Switzerland

    Martin Raubal

  3. School of Computer Science and Informatics, University College Dublin, Dublin, Ireland

    Michela Bertolotto

  4. University of Winchester, Winchester, United Kingdom

    Clare Davies

  5. Dept. of Geography and Env. Studies, University of New Mexico, Albuquerque, New Mexico, USA

    Scott Freundschuh

  6. College of Arts and Science, University of Saskatchewan, Saskatoon, Canada

    Scott Bell

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Dapoigny, R., Barlatier, P. (2015). A Coq-Based Axiomatization of Tarski’s Mereogeometry. In: Fabrikant, S., Raubal, M., Bertolotto, M., Davies, C., Freundschuh, S., Bell, S. (eds) Spatial Information Theory. COSIT 2015. Lecture Notes in Computer Science(), vol 9368. Springer, Cham. https://doi.org/10.1007/978-3-319-23374-1_6

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  • DOI: https://doi.org/10.1007/978-3-319-23374-1_6

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