Bounding quantification in parametric expansions of Presburger arithmetic

TL;DR

This paper extends quantifier elimination techniques in Presburger arithmetic to parametric families, showing they can be defined with polynomially bounded quantifiers in an expanded language, applicable to broader algebraic structures.

Contribution

It provides a new proof that parametric Presburger families are definable with polynomially bounded quantifiers, generalizing Cooper's method and applying to expansions with scalar multiplication.

Findings

Parametric Presburger families are definable with polynomially bounded quantifiers.

The method generalizes to expansions with multiplication by scalars from any ring of functions into Z.

A new proof technique for quantifier elimination in Presburger arithmetic is introduced.

Abstract

We generalize Cooper's method of quantifier elimination for classical Presburger arithmetic to give a new proof that all parametric Presburger families (as defined by Kevin Woods) are definable by formulas with polynomially bounded quantifiers in an expanded language with predicates for divisibility by f(t) for every polynomial f over the integers. In fact, this quantifier bounding method works more generally in expansions of Presburger arithmetic with multiplication by scalars from any ring of functions into Z.