On the decidability of connectedness constraints in 2D and 3D Euclidean spaces

international joint conference on artificial intelligence, Volume abs/1104.0219, 2011, Pages 957-962.

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connectedness constraintspatial constraint languageboolean operationinterior-connectedness predicatelow-dimensional euclidean spaceMore(6+)
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The moral is the same: the topological spaces of most interest for Qualitative Spatial Reasoning exhibit special characteristics which any topological constraint language able to express connectedness must take into account

Abstract:

We investigate (quantifier-free) spatial constraint languages with equality, contact and connectedness predicates, as well as Boolean operations on regions, interpreted over low-dimensional Euclidean spaces. We show that the complexity of reasoning varies dramatically depending on the dimension of the space and on the type of regions cons...More

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Introduction
  • A central task in Qualitative Spatial Reasoning is that of determining whether some described spatial configuration is geometrically realizable in 2D or 3D Euclidean space.
  • Such a description is given using a spatial logic—a formal language whose variables range over geometrical entities, and whose non-logical primitives represent geometrical relations and operations involving those entities.
  • An important extension of RCC8, known as BRCC8, features standard Boolean operations on regular closed sets [Wolter and Zakharyaschev, 2000]
Highlights
  • A central task in Qualitative Spatial Reasoning is that of determining whether some described spatial configuration is geometrically realizable in 2D or 3D Euclidean space
  • This paper investigated topological constraint languages featuring connectedness predicates and Boolean operations on regions
  • The obtained results rely on certain distinctive topological properties of Euclidean spaces
  • The moral is the same: the topological spaces of most interest for Qualitative Spatial Reasoning exhibit special characteristics which any topological constraint language able to express connectedness must take into account
Conclusion
  • This paper investigated topological constraint languages featuring connectedness predicates and Boolean operations on regions.
  • For example, the argument of Sec. 3 is based on the property of Lemma 1, while Sec. 4 relies on planarity considerations
  • In both cases, the moral is the same: the topological spaces of most interest for Qualitative Spatial Reasoning exhibit special characteristics which any topological constraint language able to express connectedness must take into account
Summary
  • Introduction:

    A central task in Qualitative Spatial Reasoning is that of determining whether some described spatial configuration is geometrically realizable in 2D or 3D Euclidean space.
  • Such a description is given using a spatial logic—a formal language whose variables range over geometrical entities, and whose non-logical primitives represent geometrical relations and operations involving those entities.
  • An important extension of RCC8, known as BRCC8, features standard Boolean operations on regular closed sets [Wolter and Zakharyaschev, 2000]
  • Conclusion:

    This paper investigated topological constraint languages featuring connectedness predicates and Boolean operations on regions.
  • For example, the argument of Sec. 3 is based on the property of Lemma 1, while Sec. 4 relies on planarity considerations
  • In both cases, the moral is the same: the topological spaces of most interest for Qualitative Spatial Reasoning exhibit special characteristics which any topological constraint language able to express connectedness must take into account
Funding
  • This work was partially supported by the U.K
  • EPSRC grants EP/E034942/1 and EP/E035248/1
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