Automated Reasoning About Metric and Topology

  • Ullrich Hustadt
  • Dmitry Tishkovsky
  • Frank Wolter
  • Michael Zakharyaschev
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 

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