A strong version of Cobham's theorem

TL;DR

This paper strengthens Cobham's theorem by showing that if two sets are recognizable in multiplicatively independent bases but not definable in Presburger arithmetic, then the combined logical theory is undecidable.

Contribution

It proves a stronger version of Cobham's theorem, demonstrating undecidability of the combined structure when both sets are non-definable in Presburger arithmetic.

Findings

Undecidability of the first-order theory of (N,+,X,Y) under given conditions

Contrasts with B"uchi's theorem on decidability of (N,+,X)

Shows limitations of recognizability in multiple bases

Abstract

Let $k,\ell\geq 2$ be two multiplicatively independent integers. Cobham's famous theorem states that a set $X\subseteq \mathbb{N}$ is both $k$-recognizable and $\ell$-recognizable if and only if it is definable in Presburger arithmetic. Here we show the following strengthening: let $X\subseteq \mathbb{N}^m$ be $k$-recognizable, let $Y\subseteq \mathbb{N}^n$ be $\ell$-recognizable such that both $X$ and $Y$ are not definable in Presburger arithmetic. Then the first-order logical theory of $(\mathbb{N},+,X,Y)$ is undecidable. This is in contrast to a well-known theorem of B\"uchi that the first-order logical theory of $(\mathbb{N},+,X)$ is decidable.