On the Decidability of Presburger Arithmetic Expanded with Powers

TL;DR

This paper proves the decidability of the existential theory of integers with powers of two bases, but shows that the same for natural numbers would imply major breakthroughs in number theory.

Contribution

It establishes decidability results for Presburger arithmetic extended with powers of two bases and highlights the complexity of extending these results to natural numbers.

Findings

Existential fragment of integer structure with powers is decidable.

Decidability for natural numbers would imply breakthroughs in transcendental number theory.

Highlights the boundary between decidability and number-theoretic complexity.

Abstract

We prove that for any integers $\alpha, \beta > 1$, the existential fragment of the first-order theory of the structure $\langle \mathbb{Z}; 0,1,<, +, \alpha^{\mathbb{N}}, \beta^{\mathbb{N}}\rangle$ is decidable (where $\alpha^{\mathbb{N}}$ is the set of positive integer powers of $\alpha$, and likewise for $\beta^{\mathbb{N}}$). On the other hand, we show by way of hardness that decidability of the existential fragment of the theory of $\langle \mathbb{N}; 0,1, <, +, x\mapsto \alpha^x, x \mapsto \beta^x\rangle$ for any multiplicatively independent $\alpha,\beta > 1$ would lead to mathematical breakthroughs regarding base-$\alpha$ and base-$\beta$ expansions of certain transcendental numbers.