On Decomposition Operations in a Theory of Multidimensional Qualitative Space

Abstract

Mereotopological relations, such as contact, parthood and overlap, are central for representing spatial information qualitatively. While most existing mereotopological theories restrict models to entities of equal dimension (e.g., all are 2D regions), multidimensional mereotopologies are more flexible by allowing entities of different dimensions to co-exist. In many respects, they generalize traditional spatial data models based on geometric entities (points, simple lines, polylines, cells, polygon, and polyhedra) and algebraic topology that power much of the existing spatial information systems (e.g., GIS, CAD, and CAM). Geometric representations can typically be decomposed into atomic entities using set intersection and complementation operations, with non-atomic entities represented as sets of atomic ones. This paper accomplishes this for CODI, a first-order logic ontology of multidimensional mereotopology, by extending its axiomatization with the mereological closure operations intersection and difference that apply to pairs of regions regardless of their dimensions. We further prove that the extended theory satisfies important mereological principles and preserves many of the mathematical properties of set intersection and set difference. This decomposition addresses implementation concerns about the ontology CODI by offering a simple mechanism for determining the mereotopological relations between complex spatial entities, similar to the operations used in algebraic topological structures. It further underlines that CODI accommodates both quantitative/geometric and qualitative spatial knowledge.