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.