Semi - qualitative reasoning about distances : a preliminary

We introduce a family of languages intended for representing knowledge and reasoning about metric (and more general distance) spaces. While the simplest language can speak only about distances between individual objects and Boolean relations between sets, the more expressive ones are capable of capturing notions such as`somewhere in (or somewhere out of) the sphere of a certain radius', `everywhere in a certain ring', etc. The computational complexity of the satissability problem for formulas in our languages ranges from NP-completeness to undecidability and depends on the class of distance spaces in which they are interpreted. Besides the class of all metric spaces, we consider, for example, the spaces R R and N N with their natural metrics.