A modal logic framework for reasoning about comparative distances and topology

In 1944, McKinsey and Tarski proved that S4 is the logic of the topological interior and closure operators of any separable dense-in-itself metric space. Thus, the logic of topological interior and closure over arbitrary metric spaces coincides with the logic of the real line, the real plane, and any separable dense-in-itself metric space; it is finitely axiomatisable and PSpace-complete. Because of this result S4 has become a logic of prime importance in Qualitative Spatial Representation and Reasoning in Artificial Intelligence. And in Logic this result has triggered the investigation of a number of variants and extensions of S4 designed for reasoning about qualitative aspects of metric spaces. In parallel to this line of research (but without much interaction), Philosophical Logic and AI have suggested and investigated a variety of logics — such as conditional logics, certain non-monotonic logics, and logics of comparative similarity — which are naturally interpreted in metric (or more general distance) spaces and which contain a binary operator for comparing distances between points and sets in such spaces. The contribution of this paper is as follows: We suggest a uniform framework covering large parts of these two lines of research, thus enabling a comparison of the logics involved and a systematic investigation of their expressive power and computational complexity. This framework is obtained by decomposing the underlying modal-like operators into firstorder quantifier patterns. We then show that quite a powerful and natural fragment of the resulting first-order logic can be captured by one binary operator comparing distances between sets and one unary operator distinguishing between realised and limit distances (i.e., between minimum and infimum). Due to its greater expressive power, this logic turns out to behave quite differently from Tarski’s S4. We provide finite (Hilbert-style) axiomatisations and ExpTime-completeness proofs for the logics of various classes of distance spaces, in particular metric spaces. But we also show that the logic of the real line (and various other important metric spaces) is not recursively enumerable. This result is proved by an encoding of Diophantine equations.