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Can one design a geometry engine?

On the (un)decidability of certain affine Euclidean geometries

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Abstract

We survey the status of decidability of the first order consequences in various axiomatizations of Hilbert-style 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-Tsün 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.

The techniques used were all known to experts in mathematical logic and geometry in the past but no detailed proofs are easily accessible for practitioners of symbolic computation or automated theorem proving.

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Acknowledgments

This paper has its origin in my lecture notes on automated theorem proving [45], developed in the last 15 years. I was motivated to develop this material further, when I prepared a lecture on P. Bernays and the foundations of geometry, which I gave at the occasion of the unveiling in summer 2017 of a plaque at the house where P. Bernays used to live in Göttingen, before going into forced exile in 1933. P. Bernays edited Hilbert’s [35] from the 5th (1922) till the 10th edition (1967), see also [34, 36]. I am indebted to R. Kahle, who invited me to give this lecture. Without this invitation this paper would not have been written.

I was lucky enough to know P. Bernays personally, as well as some other pioneers of the modern foundations of geometry, among them R. Baer, H. Lenz, W. Rautenberg, W. Schwabhäuser, W. Szmielew and A. Tarski. I dedicate this paper to them, and to my wonderful teacher of descriptive geometry, M. Herter, at the Gymnasium Freudenberg during 1961-1967 in Zurich, Switzerland. Blessed be their memory.

I would also like to thank L. Kovacs and P. Schreck for their patience and flexibility concerning the deadline for submitting this paper to the special issue on Formalization of Geometry and Reasoning of the Annals of Mathematics and Artificial Intelligence. Special thanks are due to five anonymous referees and to J. Baldwin for critical remarks and suggestions, as well as for pointing out various imprecise statements, which I hope were all corrected.

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    Johann A. Makowsky

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Makowsky, J.A. Can one design a geometry engine?. Ann Math Artif Intell 85, 259–291 (2019). https://doi.org/10.1007/s10472-018-9610-1

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