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Documents authored by Nieuwveld, Joris

Document
DOI: 10.4230/LIPIcs.MFCS.2025.80
On Expansions of Monadic Second-Order Logic with Dynamical Predicates

Authors: Joris Nieuwveld and Joël Ouaknine

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

Abstract
Expansions of the monadic second-order (MSO) theory of the structure ⟨ℕ;<⟩ have been a fertile and active area of research ever since the publication of the seminal papers of Büchi and Elgot & Rabin on the subject in the 1960s. In the present paper, we establish decidability of the MSO theory of ⟨ℕ;<,P⟩, where P ranges over a large class of unary "dynamical" predicates, i.e., sets of non-negative values assumed by certain integer linear recurrence sequences. One of our key technical tools is the novel concept of (effective) prodisjunctivity, which we expect may also find independent applications further afield.
Cite as

Joris Nieuwveld and Joël Ouaknine. On Expansions of Monadic Second-Order Logic with Dynamical Predicates. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 80:1-80:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{nieuwveld_et_al:LIPIcs.MFCS.2025.80,
  author =	{Nieuwveld, Joris and Ouaknine, Jo\"{e}l},
  title =	{{On Expansions of Monadic Second-Order Logic with Dynamical Predicates}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{80:1--80:17},
  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.2025.80},
  URN =		{urn:nbn:de:0030-drops-241879},
  doi =		{10.4230/LIPIcs.MFCS.2025.80},
  annote =	{Keywords: Monadic second-order logic, linear recurrence sequences, decidability, Baker’s theorem}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2023.130
Positivity Problems for Reversible Linear Recurrence Sequences

Authors: George Kenison, Joris Nieuwveld, Joël Ouaknine, and James Worrell

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
It is a longstanding open problem whether there is an algorithm to decide the Positivity Problem for linear recurrence sequences (LRS) over the integers, namely whether given such a sequence, all its terms are non-negative. Decidability is known for LRS of order 5 or less, i.e., for those sequences in which every new term depends linearly on the previous five (or fewer) terms. For simple LRS (i.e., those sequences whose characteristic polynomials have no repeated roots), decidability of Positivity is known up to order 9. In this paper, we focus on the important subclass of reversible LRS, i.e., those integer LRS ⟨u_n⟩_{n=0}^∞ whose bi-infinite completion ⟨u_n⟩_{n=-∞}^∞ also takes exclusively integer values; a typical example is the classical Fibonacci (bi-)sequence ⟨ … , 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, … ⟩. Our main results are that Positivity is decidable for reversible LRS of order 11 or less, and for simple reversible LRS of order 17 or less.
Cite as

George Kenison, Joris Nieuwveld, Joël Ouaknine, and James Worrell. Positivity Problems for Reversible Linear Recurrence Sequences. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 130:1-130:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kenison_et_al:LIPIcs.ICALP.2023.130,
  author =	{Kenison, George and Nieuwveld, Joris and Ouaknine, Jo\"{e}l and Worrell, James},
  title =	{{Positivity Problems for Reversible Linear Recurrence Sequences}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{130:1--130:17},
  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 =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.130},
  URN =		{urn:nbn:de:0030-drops-181821},
  doi =		{10.4230/LIPIcs.ICALP.2023.130},
  annote =	{Keywords: The Positivity Problem, Linear Recurrence Sequences, Verification}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2022.20
Skolem Meets Schanuel

Authors: Yuri Bilu, Florian Luca, Joris Nieuwveld, Joël Ouaknine, David Purser, and James Worrell

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract
The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost 90 years old, decidability of the Skolem Problem remains open. The main contribution of this paper is an algorithm to solve the Skolem Problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a stand-alone certificate that its output is correct - a set of zeros together with a collection of witnesses that no further zeros exist. We give a proof that the algorithm always terminates assuming two classical number-theoretic conjectures: the Skolem Conjecture (also known as the Exponential Local-Global Principle) and the p-adic Schanuel Conjecture. Preliminary experiments with an implementation of this algorithm within the tool Skolem point to the practical applicability of this method.
Cite as

Yuri Bilu, Florian Luca, Joris Nieuwveld, Joël Ouaknine, David Purser, and James Worrell. Skolem Meets Schanuel. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilu_et_al:LIPIcs.MFCS.2022.20,
  author =	{Bilu, Yuri and Luca, Florian and Nieuwveld, Joris and Ouaknine, Jo\"{e}l and Purser, David and Worrell, James},
  title =	{{Skolem Meets Schanuel}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{20:1--20:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.20},
  URN =		{urn:nbn:de:0030-drops-168180},
  doi =		{10.4230/LIPIcs.MFCS.2022.20},
  annote =	{Keywords: Skolem Problem, Skolem Conjecture, Exponential Local-Global Principle, p-adic Schanuel Conjecture}
}
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