An efficient quantifier elimination procedure for Presburger arithmetic

TL;DR

This paper introduces a new quantifier elimination method for Presburger arithmetic that operates in singly exponential time, improving upon the previously believed doubly exponential bounds, and enables precise complexity analysis of various problems.

Contribution

It presents the first singly exponential time quantifier elimination procedure for Presburger arithmetic, challenging prior assumptions about complexity bounds.

Findings

Deciding monadic decomposability for existential formulas in polynomial time.

Determining if an existential formula defines a well-quasi ordering is now feasible in singly exponential time.

Certain Presburger formulas with Ramsey quantifiers can be decided efficiently using the new procedure.

Abstract

All known quantifier elimination procedures for Presburger arithmetic require doubly exponential time for eliminating a single block of existentially quantified variables. It has even been claimed in the literature that this upper bound is tight. We observe that this claim is incorrect and develop, as the main result of this paper, a quantifier elimination procedure eliminating a block of existentially quantified variables in singly exponential time. As corollaries, we can establish the precise complexity of numerous problems. Examples include deciding (i) monadic decomposability for existential formulas, (ii) whether an existential formula defines a well-quasi ordering or, more generally, (iii) certain formulas of Presburger arithmetic with Ramsey quantifiers. Moreover, despite the exponential blowup, our procedure shows that under mild assumptions, even NP upper bounds for decision problems about quantifier-free formulas can be transferred to existential formulas. The technical basis of our results is a kind of small model property for parametric integer programming that generalizes the seminal results by von zur Gathen and Sieveking on small integer points in convex polytopes.