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Journal of Logic and Computation Advance Access originally published online on November 14, 2006
Journal of Logic and Computation 2006 16(6):891-916; doi:10.1093/logcom/exl020
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© The Author, 2006. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Articles

The Expressivity of Quantifying over Regions

Ernest Davis

Department of Computer Science, New York University, New York, USA.

E-mail: davise{at}cs.nyu.edu


  Abstract

We categorize in recursion-theoretic terms the expressivity of a number of first-order languages that allow quantification over regions in Euclidean space. Specifically we show the following:

(1) Let Formula be any class of closed regions in Euclidean space that includes all simple polygons. Let C(x, y) be the relation, ‘region x is connected to region y’ and let Convex(x) be the property, ‘region x is convex’. Then any relation over Formula that is analytical and invariant under affine transformations is first-order definable in the structure <Formula , C, Convex>.

(2) Let Formula be as in (1), and let Closer(x, y, z) be the relation ‘region x is closer to y than to z.’ Then any relation over Formula that is analytical and invariant under orthogonal transformations is first-order definable in the structure <Formula , Closer>.

(3) Let Formula be the class of finite unions of intervals in the real line. Then any relation over Formula that is analytical and invariant under linear transformations is first-order definable in the structure <Formula , Closer>.

(4) If the class of regions is restricted to be polygons with rational vertices, then results analogous to (1–3) hold, substituting ‘arithmetical relation’ for ‘analytical relation’.

Keywords: Expressivity; spatial representation; first-order definability; analytical relation


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