TY - JOUR

T1 - The expressivity of quantifying over regions

AU - Davis, Ernest

N1 - Funding Information:
My thanks to Martin Davis, who pointed me toward Hinman [12] and the concept of analytical relations, and to the reviewers, who made many helpful suggestions, especially pointing out the work of Schaefer and Stefanovic [19]. This research was supported in part by NSF grants #IIS-0097537 and #IIS-0534809.

PY - 2006/12

Y1 - 2006/12

N2 - 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'.

AB - 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'.

KW - Analytical relation

KW - Expressivity

KW - First-order definability

KW - Spatial representation

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U2 - 10.1093/logcom/exl020

DO - 10.1093/logcom/exl020

M3 - Article

AN - SCOPUS:33947637689

VL - 16

JO - Journal of Logic and Computation

JF - Journal of Logic and Computation

SN - 0955-792X

IS - 6

ER -