Decidability of definability issues in the theory of real addition
Abstract
Given a subset of X⊂eq Rn we can associate with every point x∈ Rn a vector space V of maximal dimension with the property that for some ball centered at x, the subset X coincides inside the ball with a union of lines parallel with V. A point is singular if V has dimension 0. In an earlier paper we proved that a (R, +,< ,Z)-definable relation X is actually definable in (R, +,< ,1) if and only if the number of singular points is finite and every rational section of X is (R, +,< ,1)-definable, where a rational section is a set obtained from X by fixing some component to a rational value. Here we show that we can dispense with the hypothesis of X being (R, +,< ,Z)-definable by assuming that the components of the singular points are rational numbers. This provides a topological characterization of first-order definability in the structure (R, +,< ,1). It also allows us to deliver a self-definable criterion (in Muchnik's terminology) of (R, +,< ,1)- and (R, +,< ,Z)-definability for a wide class of relations, which turns into an effective criterion provided that the corresponding theory is decidable. In particular these results apply to the class of k-recognizable relations on reals, and allow us to prove that it is decidable whether a k-recognizable relation (of any arity) is l-recognizable for every base l ≥ 2.