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A logic for metric and topology

Published online by Cambridge University Press:  12 March 2014

Frank Wolter  and
Michael Zakharyaschev
Frank Wolter
Affiliation:
Department of Computer Science, University of Liverpool, Liverpool L69 7ZF, UKE-mail:, frank@csc.liv.ac.uk URL: http://www.csc.liv.ac.uk/~frank
Michael Zakharyaschev
Affiliation:
Department of Computer Science, King's College London, Strand, London WC2R 2LS, UKE-mail:, mz@dcs.kcl.ac.uk URL: http://www.dcs.kcl.ac.uk/staff/mz
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Abstract

We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r’ including or excluding the boundary. We supply the logic with both the intended metric space semantics and a natural relational semantics, and show that the latter (i) provides finite partial representations of (in general) infinite metric models and (ii) reduces the standard ‘ε-definitions’ of closure and interior to simple constraints on relations. These features of the relational semantics suggest a finite axiomatisation of the logic and provide means to prove its EXPTIME-completeness (even if the rational numerical parameters are coded in binary). An extension with metric variables satisfying linear rational (in)equalities is proved to be decidable as well. Our logic can be regarded as a ‘well-behaved’ common denominator of logical systems constructed in temporal, spatial, and similarity-based quantitative and qualitative representation and reasoning. Interpreted on the real line (with its Euclidean metric), it is a natural fragment of decidable temporal logics for specification and verification of real-time systems. On the real plane, it is closely related to quantitative and qualitative formalisms for spatial representation and reasoning, but this time the logic becomes undecidable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 3 , September 2005 , pp. 795 - 828
Copyright
Copyright © Association for Symbolic Logic 2005

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