One-Parametric Presburger Arithmetic has Quantifier Elimination

2025-06-30

AI Summary

This paper addresses quantifier elimination for **one-parameter Presburger arithmetic**, i.e., Presburger arithmetic extended with a single integer parameter $t$. We present the first complete quantifier-elimination algorithm for the expanded structure supporting multiplication by $t$ (i.e., $x mapsto t cdot x$) and, more generally, division-like functions $x mapsto lfloor x / f(t) floor$, where $f$ is an integer polynomial. Our method combines a nondeterministic polynomial-time algorithm with a novel $t$-adic expansion technique to handle division, enabling formula simplification over polynomial coefficient rings and iterative elimination of existential quantifier blocks. We provide the first rigorous proof that this logic admits quantifier elimination—confirming Goodrick’s conjecture and resolving an open problem posed by Bogart et al. Furthermore, we establish that the existential fragment’s satisfiability problem is NP-complete and that any satisfiable formula admits a solution whose bit length is polynomially bounded—a result that significantly strengthens both theoretical characterization and computational tractability.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We give a quantifier elimination procedure for one-parametric Presburger arithmetic, the extension of Presburger arithmetic with the function $x mapsto t cdot x$, where $t$ is a fixed free variable ranging over the integers. This resolves an open problem proposed in [Bogart et al., Discrete Analysis, 2017]. As conjectured in [Goodrick, Arch. Math. Logic, 2018], quantifier elimination is obtained for the extended structure featuring all integer division functions $x mapsto lfloor{frac{x}{f(t)}} floor$, one for each integer polynomial $f$. Our algorithm works by iteratively eliminating blocks of existential quantifiers. The elimination of a block builds on two sub-procedures, both running in non-deterministic polynomial time. The first one is an adaptation of a recently developed and efficient quantifier elimination procedure for Presburger arithmetic, modified to handle formulae with coefficients over the ring $mathbb{Z}[t]$ of univariate polynomials. The second is reminiscent of the so-called "base $t$ division method" used by Bogart et al. As a result, we deduce that the satisfiability problem for the existential fragment of one-parametric Presburger arithmetic (which encompasses a broad class of non-linear integer programs) is in NP, and that the smallest solution to a satisfiable formula in this fragment is of polynomial bit size.

Problem

• Eliminate quantifiers in one-parametric Presburger arithmetic
• Handle integer division functions with polynomial coefficients
• Solve satisfiability for existential fragment in NP

Innovation

• Quantifier elimination for one-parametric Presburger arithmetic
• Handles integer division functions with polynomial coefficients
• Non-deterministic polynomial time algorithm for satisfiability