Paper detail

Undecidable First-Order Theories of Affine Geometries

Tarski initiated a logic-based approach to formal geometry that studies first-order structures with a ternary betweenness relation (β) and a quaternary equidistance relation (\equiv). Tarski established, inter alia, that the first-order (FO) theory of (R^2,β,\equiv) is decidable. Aiello and van Benthem (2002) conjectured that the FO-theory of expansions of (R^2,β) with unary predicates is decidable. We refute this conjecture by showing that for all n>1, the FO-theory of monadic expansions of (R^2,β) is Π^1_1-hard and therefore not even arithmetical. We also define a natural and comprehensive class C of geometric structures (T,β), where T is a subset of R^2, and show that for each structure (T,β) in C, the FO-theory of the class of monadic expansions of (T,β) is undecidable. We then consider classes of expansions of structures (T,β) 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,β). While the logics are incomparable in general, over expansions of (R^n,β), formulae of weak universal MSO translate into equivalent formulae of universal MSO. This is an extended version of a publication in the proceedings of the 21st EACSL Annual Conferences on Computer Science Logic (CSL 2012).