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### Undecidable First-Order Theories of Affine Geometries

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### Abstract

Tarski initiated a logic-based approach to formal geometry that studies first-order structures with a ternary betweenness relation (\beta) and a quaternary equidistance relation (\equiv). Tarski established, inter alia, that the first-order (FO) theory of (R^2,\beta,\equiv) is decidable. Aiello and van Benthem (2002) conjectured that the FO-theory of expansions of (R^2,\beta) with unary predicates is decidable. We refute this conjecture by showing that for all n > 1, the FO-theory of monadic expansions of (R^n,\beta) is Pi^1_1-hard and therefore not even arithmetical. We also define a natural and comprehensive class C of geometric structures (T,\beta), where T is a subset of R^n, and show that for each structure (T,\beta) in C, the FO-theory of the class of monadic expansions of (T,\beta) is undecidable. We then consider classes of expansions of structures (T,\beta) with restricted unary predicates, for example finite predicates, and establish a variety of related undecidability results. In addition to decidability questions, we briefly study the expressivity of universal MSO and weak universal MSO over expansions of (R^n,\beta). While the logics are incomparable in general, over expansions of (R^n,\beta), formulae of weak universal MSO translate into equivalent formulae of universal MSO. An extended version of this article can be found on the ArXiv (arXiv:1208.4930v1).

### BibTeX - Entry

@InProceedings{kuusisto_et_al:LIPIcs:2012:3691,
author =	{Antti Kuusisto and Jeremy Meyers and Jonni Virtema},
title =	{{Undecidable First-Order Theories of Affine Geometries}},
booktitle =	{Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL},
pages =	{470--484},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-42-2},
ISSN =	{1868-8969},
year =	{2012},
volume =	{16},
editor =	{Patrick C{\'e}gielski and Arnaud Durand},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},