Year: 2026
Type: preprint
Abstract: For multiplicatively independent natural numbers $α$ and $β$, Villemaire showed in 1992 that the first-order theory of Presburger arithmetic expanded with both Büchi predicates $V_α$ and $V_β$ is undecidable, as it encodes multiplication. In recent years, Hieronymi and Schulz showed that Presburger arithmetic expanded with the weaker power predicates $α^\mathbb{N} = \{α^n: n \in \mathbb{N}\}$ and $β^\mathbb{N}$ is also undecidable, while Karimov et al. showed that the existential fragment of this theory is decidable. These results left open the natural problem of determining the decidability of the existential fragment of Villemaire's original expansion. We settle this question for coprime $α$ and $β$. Specifically, we give a quantifier-elimination argument that proves the decidability of the existential fragment of $\mathsf{FO}(\mathbb{Z};<,+, V_α, V_β)$.
Source: arXiv (Cornell University)
Author: Nieuwveld, Joris
Institution: University of Oxford
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: University of Oxford
OpenAlex ID: W7204267728
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