JELIA 2006: Logics in Artificial Intelligence pp 490-493

# Automated Reasoning About Metric and Topology

• Dmitry Tishkovsky
• Frank Wolter
• Michael Zakharyaschev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4160)

## 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.

## Keywords

Modal Logic Closure Operator Automate Reasoning Distance Operator Hybrid Logic
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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