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A modal logic framework for reasoning about comparative distances and topology

Sheremet, M. and Wolter, F. and Zakharyaschev, Michael (2010) A modal logic framework for reasoning about comparative distances and topology. Annals of Pure and Applied Logic 161 (4), pp. 534-559. ISSN 0168-0072.

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Official URL: http://dx.doi.org/10.1016/j.apal.2009.04.001

Abstract

We propose and investigate a uniform modal logic framework for reasoning about topology and relative distance in metric and more general distance spaces, thus enabling the comparison and combination of logics from distinct research traditions such as Tarski’s View the MathML source for topological closure and interior, conditional logics, and logics of comparative similarity. This framework is obtained by decomposing the underlying modal-like operators into first-order quantifier patterns. We then show that quite a powerful and natural fragment of the resulting first-order logic can be captured by one binary operator comparing distances between sets and one unary operator distinguishing between realised and limit distances (i.e., between minimum and infimum). Due to its greater expressive power, this logic turns out to behave quite differently from both View the MathML source and conditional logics. We provide finite (Hilbert-style) axiomatisations and ExpTime-completeness proofs for the logics of various classes of distance spaces, in particular metric spaces. But we also show that the logic of the real line (and various other important metric spaces) is not recursively enumerable. This result is proved by an encoding of Diophantine equations.

Item Type: Article
Keyword(s) / Subject(s): Modal logic, conditional logics, metric spaces, topology, diophantine equations
School or Research Centre: Birkbeck Schools and Research Centres > School of Business, Economics & Informatics > Computer Science and Information Systems
Depositing User: Administrator
Date Deposited: 01 Feb 2011 12:04
Last Modified: 17 Apr 2013 12:18
URI: http://eprints.bbk.ac.uk/id/eprint/1916

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