Article

On the Decidability of Connectedness Constraints in 2D and 3D Euclidean Spaces

Computing Research Repository - CORR 04/2011; DOI: 10.5591/978-1-57735-516-8/IJCAI11-165
Source: arXiv

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 considered. For example, the logic with the
interior-connectedness predicate (and without contact) is undecidable over
polygons or regular closed sets in the Euclidean plane, NP-complete over
regular closed sets in three-dimensional Euclidean space, and ExpTime-complete
over polyhedra in three-dimensional Euclidean space.

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    ABSTRACT: We consider the quantifier-free languages, Bc and Bc0, obtained by augmenting the signature of Boolean algebras with a unary predicate representing, respectively, the property of being connected, and the property of having a connected interior. These languages are interpreted over the regular closed sets of n-dimensional Euclidean space (n greater than 1) and, additionally, over the regular closed polyhedral sets of n-dimensional Euclidean space. The resulting logics are examples of formalisms that have recently been proposed in the Artificial Intelligence literature under the rubric "Qualitative Spatial Reasoning." We prove that the satisfiability problem for Bc is undecidable over the regular closed polyhedra in all dimensions greater than 1, and that the satisfiability problem for both languages is undecidable over both the regular closed sets and the regular closed polyhedra in the Euclidean plane. However, we also prove that the satisfiability problem for Bc0 is NP-complete over the regular closed sets in all dimensions greater than 2, while the corresponding problem for the regular closed polyhedra is ExpTime-complete. Our results show, in particular, that spatial reasoning over Euclidean spaces is much harder than reasoning over arbitrary topological spaces.
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    ABSTRACT: Many formalisms discussed in the literature on qualitative spatial reasoning are designed for expressing static spatial constraints only. However, dynamic situations arise in virtually all applications of these formalisms, which makes it necessary to study variants and extensions dealing with change. This paper presents a study on the computational complexity of qualitative change. More precisely, we discuss the reasoning task of finding a solution to a temporal sequence of static reasoning problems where this sequence is subject to additional transition constraints. Our focus is primarily on smoothness and continuity constraints: we show how such transitions can be defined as relations and expressed within qualitative constraint formalisms. Our results demonstrate that for point-based constraint formalisms the interesting fragments are NP-complete in the presence of continuity constraints, even if the satisfiability problem of its static descriptions is tractable.
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