# Presburger arithmetic, rational generating functions, and quasi-polynomials

## Abstract

A Presburger formula is a Boolean formula with variables in ℕ that can be written using addition, comparison (≤, =, etc.), Boolean operations (and, or, not), and quantifiers (∀ and ∃). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, if p = (p 1,…,p n ) are a subset of the free variables in a Presburger formula, we can define a counting function g(p) to be the number of solutions to the formula, for a given p. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. In the full version of this paper, we also translate known computational complexity results into this setting and discuss open directions.

## Repository Citation

Woods, Kevin. 2013. “Presburger arithmetic, rational generating functions, and quasi-polynomials.” In Automata, Languages, and Programming: 40th International Colloquium, ICALP 2013, Proceedings, Part II, edited by F.V. Fomin, et al. Lecture Notes in Computer Science 7966, 410-421. New York: Springer.

## Publisher

Springer

## Publication Date

1-1-2013

## Department

Mathematics

## Document Type

Book Chapter

## DOI

https://dx.doi.org/10.1007/978-3-642-39212-2_37

## ISBN

9783642392115

## Language

English

## Format

text