Multidimensional Mereotopology

Antony Galton

To support commonsense reasoning about space, we require a qualitative calculus of spatial entities and their relations. One requirement for such a calculus, which has not so far been satisfactorily addressed in the mereotopological literature, is that it should be able to handle regions of different dimensions. Regions of the same dimension should admit Boolean sum and product operations, but regions of different dimensions should not. In this paper we propose a topological model for regions of different dimensions, based on the idea that a region of positive codimension is a regular closed subset of the boundary of a region of the next higher dimension. To satisfy the requirements of the commonsense theory, it is required that regions of the same dimension in the model can be summed, and we show that this is always the case. We conclude with a discussion of the possible applicability of the technical results to commonsense spatial reasoning.

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