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Undecidable First-Order Theories of Affine Geometries (1310.8200v3)

Published 29 Oct 2013 in cs.LO and math.LO

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 (R2,\beta,\equiv) is decidable. Aiello and van Benthem (2002) conjectured that the FO-theory of expansions of (R2,\beta) with unary predicates is decidable. We refute this conjecture by showing that for all n>1, the FO-theory of the class of expansions of (R2,\beta) with just one unary predicate is not even arithmetical. We also define a natural and comprehensive class C of geometric structures (T,\beta), and show that for each structure (T,\beta) in C, the FO-theory of the class of expansions of (T,\beta) with a single unary predicate is undecidable. We then consider classes of expansions of structures (T,\beta) with a restricted unary predicate, for example a finite predicate, and establish a variety of related undecidability results. In addition to decidability questions, we briefly study the expressivities of universal MSO and weak universal MSO over expansions of (Rn,\beta). While the logics are incomparable in general, over expansions of (Rn,\beta), formulae of weak universal MSO translate into equivalent formulae of universal MSO.

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Authors (3)
  1. Antti Kuusisto (48 papers)
  2. Jeremy Meyers (2 papers)
  3. Jonni Virtema (42 papers)

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