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DOI: 10.4230/LIPIcs.MFCS.2025.72

One-Parametric Presburger Arithmetic Has Quantifier Elimination

Authors: Alessio Mansutti and Mikhail R. Starchak

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract

We give a quantifier elimination procedure for one-parametric Presburger arithmetic, the extension of Presburger arithmetic with the function x ↦ t ⋅ x, where t is a fixed free variable ranging over the integers. This resolves an open problem proposed in [Bogart et al., Discrete Analysis, 2017]. As conjectured in [Goodrick, Arch. Math. Logic, 2018], quantifier elimination is obtained for the extended structure featuring all integer division functions x ↦ ⌊x/(f(t))⌋, one for each integer polynomial f. Our algorithm works by iteratively eliminating blocks of existential quantifiers. The elimination of a block builds on two sub-procedures, both running in non-deterministic polynomial time. The first one is an adaptation of a recently developed and efficient quantifier elimination procedure for Presburger arithmetic, modified to handle formulae with coefficients over the ring ℤ[t] of univariate polynomials. The second is reminiscent of the so-called "base t division method" used by Bogart et al. As a result, we deduce that the satisfiability problem for the existential fragment of one-parametric Presburger arithmetic (which encompasses a broad class of non-linear integer programs) is in NP, and that the smallest solution to a satisfiable formula in this fragment is of polynomial bit size.

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Alessio Mansutti and Mikhail R. Starchak. One-Parametric Presburger Arithmetic Has Quantifier Elimination. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 72:1-72:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mansutti_et_al:LIPIcs.MFCS.2025.72,
  author =	{Mansutti, Alessio and Starchak, Mikhail R.},
  title =	{{One-Parametric Presburger Arithmetic Has Quantifier Elimination}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{72:1--72:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.72},
  URN =		{urn:nbn:de:0030-drops-241794},
  doi =		{10.4230/LIPIcs.MFCS.2025.72},
  annote =	{Keywords: decision procedures, quantifier elimination, non-linear integer arithmetic}
}

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Track B: Automata, Logic, Semantics, and Theory of Programming

DOI: 10.4230/LIPIcs.ICALP.2024.132

Integer Linear-Exponential Programming in NP by Quantifier Elimination

Authors: Dmitry Chistikov, Alessio Mansutti, and Mikhail R. Starchak

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

This paper provides an NP procedure that decides whether a linear-exponential system of constraints has an integer solution. Linear-exponential systems extend standard integer linear programs with exponential terms 2^x and remainder terms (x mod 2^y). Our result implies that the existential theory of the structure (ℕ,0,1,+,2^(⋅),V_2(⋅,⋅), ≤) has an NP-complete satisfiability problem, thus improving upon a recent EXPSPACE upper bound. This theory extends the existential fragment of Presburger arithmetic with the exponentiation function x ↦ 2^x and the binary predicate V_2(x,y) that is true whenever y ≥ 1 is the largest power of 2 dividing x. Our procedure for solving linear-exponential systems uses the method of quantifier elimination. As a by-product, we modify the classical Gaussian variable elimination into a non-deterministic polynomial-time procedure for integer linear programming (or: existential Presburger arithmetic).

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Dmitry Chistikov, Alessio Mansutti, and Mikhail R. Starchak. Integer Linear-Exponential Programming in NP by Quantifier Elimination. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 132:1-132:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chistikov_et_al:LIPIcs.ICALP.2024.132,
  author =	{Chistikov, Dmitry and Mansutti, Alessio and Starchak, Mikhail R.},
  title =	{{Integer Linear-Exponential Programming in NP by Quantifier Elimination}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{132:1--132:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.132},
  URN =		{urn:nbn:de:0030-drops-202758},
  doi =		{10.4230/LIPIcs.ICALP.2024.132},
  annote =	{Keywords: decision procedures, integer programming, quantifier elimination}
}

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Documents authored by Starchak, Mikhail R.

One-Parametric Presburger Arithmetic Has Quantifier Elimination

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Integer Linear-Exponential Programming in NP by Quantifier Elimination

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