Parametric Presburger Arithmetic: Complexity of Counting and Quantifier Elimination

TL;DR

This paper explores the complexity of counting solutions in parametric Presburger arithmetic with multiple parameters, showing that the counting function's behavior becomes computationally hard with two or more parameters, unlike the single-parameter case.

Contribution

It demonstrates that for two or more parameters, the counting function cannot be expressed as an eventual quasi-polynomial and can be computationally intractable, contrasting with the single-parameter scenario.

Findings

Counting functions are not polynomial-time computable for two or more parameters.

Single-parameter counting functions are eventually quasi-polynomial.

For multiple parameters, counting functions can be expressed using gcd and similar functions.

Abstract

We consider an expansion of Presburger arithmetic which allows multiplication by $k$ parameters $t_1,\ldots,t_k$. A formula in this language defines a parametric set $S_\mathbf{t} \subseteq \mathbb{Z}^{d}$ as $\mathbf{t}$ varies in $\mathbb{Z}^k$, and we examine the counting function $|S_\mathbf{t}|$ as a function of $\mathbf{t}$. For a single parameter, it is known that $|S_t|$ can be expressed as an eventual quasi-polynomial (there is a period $m$ such that, for sufficiently large $t$, the function is polynomial on each of the residue classes mod $m$). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming \textbf{P} $\neq$ \textbf{NP}) we construct a parametric set $S_{t_1,t_2}$ such that $|S_{t_1, t_2}|$ is not even polynomial-time computable on input $(t_1,t_2)$. In contrast, for parametric sets $S_\mathbf{t} \subseteq \mathbb{Z}^d$ with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that $|S_\mathbf{t}|$ is always polynomial-time computable in the size of $\mathbf{t}$, and in fact can be represented using the gcd and similar functions.