# An efficient quantifier elimination procedure for Presburger arithmetic

arXiv: https://arxiv.org/abs/2405.01183 · HTML: https://mlsanity.com/paper/2405.01183/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 May 2024.

## Built on by (most cited first, 1 of 1)

- [Presburger Functional Synthesis: Complexity and Tractable Normal Forms](https://mlsanity.com/paper/2508.07207/) (2025)

## Abstract (arXiv)

All known quantifier elimination procedures for Presburger arithmetic require doubly exponential time for eliminating a single block of existentially quantified variables. It has even been claimed in the literature that this upper bound is tight. We observe that this claim is incorrect and develop, as the main result of this paper, a quantifier elimination procedure eliminating a block of existentially quantified variables in singly exponential time. As corollaries, we can establish the precise complexity of numerous problems. Examples include deciding (i) monadic decomposability for existential formulas, (ii) whether an existential formula defines a well-quasi ordering or, more generally, (iii) certain formulas of Presburger arithmetic with Ramsey quantifiers. Moreover, despite the exponential blowup, our procedure shows that under mild assumptions, even NP upper bounds for decision problems about quantifier-free formulas can be transferred to existential formulas. The technical basis of our results is a kind of small model property for parametric integer programming that generalizes the seminal results by von zur Gathen and Sieveking on small integer points in convex polytopes.