An Efficient Quantifier Elimination Procedure for Presburger Arithmetic

Authors Christoph Haase , Shankara Narayanan Krishna , Khushraj Madnani , Om Swostik Mishra , Georg Zetzsche

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Christoph Haase
  • Department of Computer Science, University of Oxford, UK
Shankara Narayanan Krishna
  • Department of Computer Science & Engineering, IIT Bombay, India
Khushraj Madnani
  • Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany
Om Swostik Mishra
  • Department of Mathematics, IIT Bombay, India
Georg Zetzsche
  • Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany


We are grateful to (i) Pascal Bergsträßer, Moses Ganardi, and Anthony W. Lin for discussions about Weispfenning’s lower bound, (ii) Pascal Baumann, Eren Keskin, Roland Meyer for discussions on polyhedra, (iii) Anthony W. Lin and Matthew Hague for explaining some aspects of their results on monadic decomposability, and (iv) Gaëtan Regaud for proofreading.

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Christoph Haase, Shankara Narayanan Krishna, Khushraj Madnani, Om Swostik Mishra, and Georg Zetzsche. An Efficient Quantifier Elimination Procedure for Presburger Arithmetic. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 142:1-142:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • Presburger arithmetic
  • quantifier elimination
  • parametric integer programming
  • convex geometry


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