- Published 1969 in J. ACM

This paper studies the logical complexity of geometric properties in the plane. These properties are characterized according to the length of formulas necessary to express them. In this study a portion of the plane X will be considered as a collection of squares {x<subscrpt>1</subscrpt>, ..., x<subscrpt>n</subscrpt>} arranged into a fine grid. A <underline>pattern</underline> P is a subset of X and the grid is assumed to be fine enough so that approximations to geometrical figures can be obtained. P can be defined by mapping m: X &ran;{0,1} where for x ε X, m(x) &equil; 1 iff x ε P. A set of patterns or a geometrical <underline>property</underline> &rgr; can be expressed as a Boolean function f on the n variables x<subscrpt>1</subscrpt>, ..., x<subscrpt>n</subscrpt> such that the value of f is 1 under the assignment m iff P ε &rgr.

@article{Hodes1969TheLC,
title={The Logical Complexity of Geometric Properties in the Plane},
author={Louis Hodes},
journal={J. ACM},
year={1969},
volume={17},
pages={339-347}
}