Spatial Reasoning About Points in a Multidimensional Setting

  • Authors:

  • Philippe Balbiani;Jean-François Condotta

  • Affiliations:

  • Institut de Recherche en Informatique de Toulouse (IRIT), 118 route de Narbonne, 31062 Toulouse Cedex 4, France.;Institut de Recherche en Informatique de Toulouse (IRIT), 118 route de Narbonne, 31062 Toulouse Cedex 4, France. condotta@irit.fr

  • Venue:
  • Applied Intelligence
  • Year:
  • 2002

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Abstract

For in ≥ 1, we consider the possible relations between two points of the Euclidean space of dimension in. We define the in-point algebra on the pattern of the point algebra and the cardinal algebra. Generalizing the concept of convexity just as the one of preconvexity, we prove that the consistency problem of convex in-point networks is polynomial for in ≥ 1, whereas the consistency problem of preconvex in-point networks is NP-complete for in ≥ 3. We characterize a subset of the set of all preconvex relations: the set of all strongly preconvex relations, which contains the set of all convex relations. We demonstrate that the consistency problem of strongly preconvex in-point networks can be decided in polynomial time by means of the weak path-consistency method for all in ≥ 1. For in = 3 the set of all strongly preconvex relations is a maximal tractable subclass of the set of all in-point relations. Finally, we prove that the concept of strong preconvexity corresponds to the one of ORD-Horn representability.