License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2021.99
URN: urn:nbn:de:0030-drops-141682
URL: https://drops.dagstuhl.de/opus/volltexte/2021/14168/
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Neumann, Eike ; Ouaknine, Joël ; Worrell, James

Decision Problems for Second-Order Holonomic Recurrences

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LIPIcs-ICALP-2021-99.pdf (0.7 MB)


Abstract

We study decision problems for sequences which obey a second-order holonomic recurrence of the form f(n + 2) = P(n) f(n + 1) + Q(n) f(n) with rational polynomial coefficients, where P is non-constant, Q is non-zero, and the degree of Q is smaller than or equal to that of P. We show that existence of infinitely many zeroes is decidable. We give partial algorithms for deciding the existence of a zero, positivity of all sequence terms, and positivity of all but finitely many sequence terms. If Q does not have a positive integer zero then our algorithms halt on almost all initial values (f(1), f(2)) for the recurrence. We identify a class of recurrences for which our algorithms halt for all initial values. We further identify a class of recurrences for which our algorithms can be extended to total ones.

BibTeX - Entry

@InProceedings{neumann_et_al:LIPIcs.ICALP.2021.99,
  author =	{Neumann, Eike and Ouaknine, Jo\"{e}l and Worrell, James},
  title =	{{Decision Problems for Second-Order Holonomic Recurrences}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{99:1--99:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14168},
  URN =		{urn:nbn:de:0030-drops-141682},
  doi =		{10.4230/LIPIcs.ICALP.2021.99},
  annote =	{Keywords: holonomic sequences, Positivity Problem, Skolem Problem}
}

Keywords: holonomic sequences, Positivity Problem, Skolem Problem
Collection: 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)
Issue Date: 2021
Date of publication: 02.07.2021


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