JELIA 2006: Logics in Artificial Intelligence pp 490-493

# Automated Reasoning About Metric and Topology

• Ullrich Hustadt
• 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 variablesp1,p2,... (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 operatorsI 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.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 2006

## Authors and Affiliations

• Ullrich Hustadt
• 1
• Dmitry Tishkovsky
• 1
• Frank Wolter
• 1
• Michael Zakharyaschev
• 2
1. 1.Department of Computer ScienceUniversity of Liverpool
2. 2.School of Computer Science and Information SystemsBirkbeck College