Can one design a geometry engine? On the (un)decidability of affine Euclidean geometries

TL;DR

This paper surveys the decidability of consequence relations in various Euclidean geometries, highlighting a key undecidability result and establishing new undecidability results for certain geometries, while also identifying decidable universal theories.

Contribution

It demonstrates the undecidability of consequence relations in multiple geometries and shows that universal consequences of Pappian plane theories are decidable under certain conditions.

Findings

Undecidability of consequence relations in Hilbert, Wu's orthogonal, and Origami geometries.

Decidability of universal consequences for Pappian plane theories compatible with real analytic geometry.

Highlighting Ziegler's theorem as a key tool in proving undecidability.

Abstract

We survey the status of decidabilty of the consequence relation in various axiomatizations of Euclidean geometry. We draw attention to a widely overlooked result by Martin Ziegler from 1980, which proves Tarski's conjecture on the undecidability of finitely axiomatizable theories of fields. We elaborate on how to use Ziegler's theorem to show that the consequence relations for the first order theory of the Hilbert plane and the Euclidean plane are undecidable. As new results we add: (A) The first order consequence relations for Wu's orthogonal and metric geometries (Wen-Ts\"un Wu, 1984), and for the axiomatization of Origami geometry (J. Justin 1986, H. Huzita 1991)are undecidable. It was already known that the universal theory of Hilbert planes and Wu's orthogonal geometry is decidable. We show here using elementary model theoretic tools that (B) the universal first order consequences of any geometric theory $T$ of Pappian planes which is consistent with the analytic geometry of the reals is decidable.