二维和三维欧氏空间中连通性约束的可判定性
On the Decidability of Connectedness Constraints in 2D and 3D Euclidean Spaces
摘要：
我们调查（无量词）空间约束语言的平等，接触和连通性谓词，以及布尔运算的区域，解释低维欧氏空间。我们表明，推理的复杂性显着变化取决于空间的尺寸和所考虑的区域的类型。例如，具有邻接连通性谓词（且无接触）的逻辑在欧氏平面中的多边形或正则闭集上是不可判定的，在三维欧氏空间中的正则闭集上是NP完全的，在三维欧氏空间中的多面体上是ExpTime完全的。
英文摘要：
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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