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Spatial Symmetry Driven Pruning Strategies for Efficient Declarative Spatial Reasoning

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Book cover Spatial Information Theory (COSIT 2015)

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

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Abstract

Declarative spatial reasoning denotes the ability to (declaratively) specify and solve real-world problems related to geometric and qualitative spatial representation and reasoning within standard knowledge representation and reasoning (KR) based methods (e.g., logic programming and derivatives). One approach for encoding the semantics of spatial relations within a declarative programming framework is by systems of polynomial constraints. However, solving such constraints is computationally intractable in general (i.e. the theory of real-closed fields).

We present a new algorithm, implemented within the declarative spatial reasoning system CLP(QS), that drastically improves the performance of deciding the consistency of spatial constraint graphs over conventional polynomial encodings. We develop pruning strategies founded on spatial symmetries that form equivalence classes (based on affine transformations) at the qualitative spatial level. Moreover, pruning strategies are themselves formalised as knowledge about the properties of space and spatial symmetries. We evaluate our algorithm using a range of benchmarks in the class of contact problems, and proofs in mereology and geometry. The empirical results show that CLP(QS) with knowledge-based spatial pruning outperforms conventional polynomial encodings by orders of magnitude, and can thus be applied to problems that are otherwise unsolvable in practice.

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Notes

  1. 1.

    Incompleteness refers to the inability of a spatial reasoning method to determine whether a given network of qualitative spatial constraints is consistent or inconsistent in general. Relation-algebraic spatial reasoning (i.e. using algebraic closure based on weak composition) has been shown to be incomplete for a number of spatial languages and cannot guarantee consistency in general, e.g. relative directions [23] and containment relations between linearly ordered intervals [22], Theorem 5.9.

  2. 2.

    We emphasise that this analytic geometry approach that we also adopt is not qualitative spatial reasoning in the relation algebraic sense; the foundations are similar (i.e. employing a finite language of spatial relations that are interpreted as infinite sets of configurations, determining consistency in the complete absence of numeric information, and so on) but the methods for determining consistency etc. come from different branches of spatial reasoning.

  3. 3.

    Standard geometric constraint languages of approaches including [12, 17, 27] consist of points, lines, circles, ellipses, and coincidence, tangency, perpendicularity, parallelism, and numerical dimension constraints; note the absence of e.g. mereotopology and “common-sense” relative orientation relations [35].

  4. 4.

    Spatial Reasoning (CLP(QS)). www.spatial-reasoning.com.

  5. 5.

    Important factors in determining the applicability of various analytic approaches are the degree of the polynomials (particularly the distinction between linear and nonlinear) and whether both equality and inequalities are permitted in the constraints.

  6. 6.

    That is, constructive methods may fail in building a consistent solution, and iterative root finding methods may fail to converge.

  7. 7.

    The properties of affine transformations and the geometric objects that they preserve are well understood; further information is readily available in introductory texts such as [26]. Our key contribution is formalising and exploiting this spatial knowledge as modular and extensible common-sense rules in intelligent knowledge-based spatial assistance systems.

  8. 8.

    All cases have been verified using Reduce as presented in Theorem 2.

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Authors and Affiliations

  1. Institute for Geoinformatics, University of Münster, Münster, Germany

    Carl Schultz

  2. Department of Computer Science, University of Bremen, Bremen, Germany

    Mehul Bhatt

  3. The DesignSpace Group, Bremen, Germany

    Carl Schultz & Mehul Bhatt

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  1. Carl Schultz
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  2. Mehul Bhatt
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Correspondence to Carl Schultz .

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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Schultz, C., Bhatt, M. (2015). Spatial Symmetry Driven Pruning Strategies for Efficient Declarative Spatial Reasoning. 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_16

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  • DOI: https://doi.org/10.1007/978-3-319-23374-1_16

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