# The expressivity of quantifying over regions

Research output: Contribution to journalArticle

### 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 u 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 u that is analytical and invariant under affine transformations is first-order definable in the structure (u, C, Convex). (2) Let u be as in (1), and let Closer(x, y, z) be the relation 'region y is closer to y than to z.' Then any relation over u that is analytical and invariant under orthogonal transformations is first-order definable in the structure (u, Closer). (3) Let u be the class of finite unions of intervals in the real line. Then any relation over u that is analytical and invariant under linear transformations is first-order definable in the structure (u, 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'.

Original language English (US) Journal of Logic and Computation 16 6 https://doi.org/10.1093/logcom/exl020 Published - Dec 2006

### Fingerprint

Linear transformations
First-order
Invariant
Euclidean space
Closed region
Orthogonal Transformation
Simple Polygon
Linear transformation
Expressivity
Recursion
Real Line
Quantification
Polygon
Affine transformation
Union
Interval
Term
Class

### Keywords

• Analytical relation
• Expressivity
• First-order definability
• Spatial representation

### ASJC Scopus subject areas

• Computational Theory and Mathematics
• Safety, Risk, Reliability and Quality
• Logic
• Theoretical Computer Science

### Cite this

In: Journal of Logic and Computation, Vol. 16, No. 6, 12.2006.

Research output: Contribution to journalArticle

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