From topology to metric: modal logic and quantification in metric spaces

We propose a framework for comparing the expressive power and computational behaviour of modal logics designed for reasoning about qualitative aspects of metric spaces. Within this framework we can compare such well-known logics as S4 (for the topology induced by the metric), wK4 (for the derivation operator of the topology), variants of conditional logic, as well as logics of comparative similarity. One of the main problems for the new family of logics is to delimit the borders between ‘decidable’ and ‘undecidable.’ As a first step in this direction, we consider the modal logic with the operator ‘closer to a set τ0 than to a set τ1’ interpreted in metric spaces. This logic contains S4 with the universal modality and corresponds to a very natural language within our framework. We prove that over arbitrary metric spaces this logic is ExpTime-complete. Recall that over R, Q, and Z, as well as their finite subspaces, this logic is undecidable.