A Foundation for Region-based Qualitative Geometry

We present a highly expressive logical language for describing qualitative configurations of spatial regions, based on Tarski's Geometry of Solids, in which the parthood relation and the concept of sphere are taken as primitive. We give a categorical axiom system, whose models can be interpreted classically in terms of Cartesian spaces over ℝ. We show that within this system the concept of sphere and the congruence relation are interdefinable. We investigate the 2nd-order character of the theory and prove incompleteness of some weaker 1st-order variants.

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