Decidability of definability issues in the theory of real addition

Abstract

Given a subset of $X\subseteq {\mathbb{R}}^n$ we can associate with every point $x\in {\mathbb{R}}^n$ 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 $(\mathbb{R},+,<,\mathbb{Z})$-definable relation $X$ is actually definable in $(\mathbb{R},+,<,1)$ if and only if the number of singular points is finite and every rational section of $X$ is $(\mathbb{R},+,<,1)$-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 $(\mathbb{R},+,<,\mathbb{Z})$-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 $(\mathbb{R},+,<,1)$. It also allows us to deliver a self-definable criterion (in Muchnik's terminology) of $(\mathbb{R},+,<,1)$- and $(\mathbb{R},+,<,\mathbb{Z})$-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 $l\ge 2$.

References in corpus (2)

• Decomposition of Decidable First-Order Logics over Integers and Reals
• Generalizing a theorem of Bès and Choffrut