License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.ICALP.2023.112
URN: urn:nbn:de:0030-drops-181641
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Benedikt, Michael ; Chistikov, Dmitry ; Mansutti, Alessio

The Complexity of Presburger Arithmetic with Power or Powers

LIPIcs-ICALP-2023-112.pdf (0.8 MB)


We investigate expansions of Presburger arithmetic (Pa), i.e., the theory of the integers with addition and order, with additional structure related to exponentiation: either a function that takes a number to the power of 2, or a predicate 2^ℕ for the powers of 2. The latter theory, denoted Pa(2^ℕ(·)), was introduced by Büchi as a first attempt at characterizing the sets of tuples of numbers that can be expressed using finite automata; Büchi’s method does not give an elementary upper bound, and the complexity of this theory has been open. The former theory, denoted as Pa(λx.2^|x|), was shown decidable by Semenov; while the decision procedure for this theory differs radically from the automata-based method proposed by Büchi, Semenov’s method is also non-elementary. And in fact, the theory with the power function has a non-elementary lower bound. In this paper, we show that while Semenov’s and Büchi’s approaches yield non-elementary blow-ups for Pa(2^ℕ(·)), the theory is in fact decidable in triply exponential time, similarly to the best known quantifier-elimination algorithm for Pa. We also provide a NExpTime upper bound for the existential fragment of Pa(λx.2^|x|), a step towards a finer-grained analysis of its complexity. Both these results are established by analyzing a single parameterized satisfiability algorithm for Pa(λx.2^|x|), which can be specialized to either the setting of Pa(2^ℕ(·)) or the existential theory of Pa(λx.2^|x|). Besides the new upper bounds for the existential theory of Pa(λx.2^|x|) and Pa(2^ℕ(·)), we believe our algorithm provides new intuition for the decidability of these theories, and for the features that lead to non-elementary blow-ups.

BibTeX - Entry

  author =	{Benedikt, Michael and Chistikov, Dmitry and Mansutti, Alessio},
  title =	{{The Complexity of Presburger Arithmetic with Power or Powers}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{112:1--112:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-181641},
  doi =		{10.4230/LIPIcs.ICALP.2023.112},
  annote =	{Keywords: arithmetic theories, exponentiation, decision procedures}

Keywords: arithmetic theories, exponentiation, decision procedures
Collection: 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)
Issue Date: 2023
Date of publication: 05.07.2023

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