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Positivity Problems for Reversible Linear Recurrence Sequences

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

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Author Details

George Kenison
  • Institute of Logic and Computation, TU Wien, Austria
Joris Nieuwveld
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Joël Ouaknine
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
James Worrell
  • Department of Computer Science, University of Oxford, UK

Funding

  • Kenison, George: Work supported by WWTF ICT19-018 grant ProbInG and ERC Consolidator Grant ARTIST 101002685.
  • Ouaknine, Joël: Also affiliated with Keble College, Oxford as https://emmy.network Fellow, and supported by DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science).
  • Worrell, James: Work supported by UKRI Frontier Research Grant EP/X033813/1.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ICALP.2023.130

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Computing methodologies → Algebraic algorithms
Keywords
  • The Positivity Problem
  • Linear Recurrence Sequences
  • Verification

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