# Parametric Presburger arithmetic: complexity of counting and quantifier elimination

## Abstract

We consider an expansion of Presburger arithmetic which allows multiplication by k parameters t1, horizontal ellipsis ,tk. A formula in this language defines a parametric set St subset of Zd as t varies in Zk, and we examine the counting function |St| as a function of t. For a single parameter, it is known that |St| 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 P not equal NP) we construct a parametric set St1,t2 such that |St1,t2| is not even polynomial-time computable on input (t1,t2). In contrast, for parametric sets St subset of Zd with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that |St| is always polynomial-time computable in the size of t, and in fact can be represented using the gcd and similar functions.

## Repository Citation

Bogart, Tristram, John Goodrick, Danny Nguyen, and Kevin Woods. 2019. "Parametric Presburger arithmetic: complexity of counting and quantifier elimination." Mathematical Logic Quarterly 65(2): 237-250.

## Publisher

Wiley

## Publication Date

9-1-2019

## Publication Title

Mathematical Logic Quarterly

## Department

Mathematics

## Document Type

Article

## DOI

https://dx.doi.org/10.1002/malq.201800068

## Keywords

Rational generating-functions

## Language

English

## Format

text