Automated Reasoning About Metric and Topology

Abstract

In this paper we compare two approaches to automated reasoning about metric and topology in the framework of the logic \(\mathcal{MT}\) introduced in [10]. \(\mathcal{MT}\)-formulas are built from set variablesp ₁,p ₂,... (for arbitrary subsets of a metric space) using the Booleans ∧, ∨, →, and ¬, the distance operators∃ ^< a and ∃ ^≤ a , for \(a\in {\mathbb Q}^{> 0}\), and the topological interior and closure operators I and C. Intended models for this logic are of the form \(\mathfrak I=(\Delta,d,p_{1}^{\mathfrak I},p_{2}^{\mathfrak I},\dots)\) where (Δ,d) is a metric space and \(p_{i}^{\mathfrak I} \subseteq \Delta\). The extension \(\varphi^{\mathfrak I} \subseteq \Delta\) of an \(\mathcal{MT}\)-formula ϕ in \(\mathfrak I\) is defined inductively in the usual way, with I and C being interpreted as the interior and closure operators induced by the metric, and \((\exists^{<a}\varphi)^{\mathfrak I} = \{ x \in \Delta \mid \exists y\in \varphi^{\mathfrak I}\ d(x,y)<a \}\). In other words, \((\mathbf{I}\varphi)^{\mathfrak I}\) is the interior of \(\varphi^{\mathfrak I}\), \((\exists^{<a}\varphi)^{\mathfrak I}\) is the open a-neighbourhood of \(\varphi^{\mathfrak I}\), and \((\exists^{\le a}\varphi)^{\mathfrak I}\) is the closed one. A formula ϕ is satisfiable if there is a model \({\mathfrak I}\) such that \(\varphi^{\mathfrak I} \ne \emptyset\); ϕ is valid if ¬ϕ is not satisfiable.