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Decision Problems for Second-Order Holonomic Recurrences

Eike Neumann Orcid Logo, Joël Ouaknine, James Worrell

48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), Volume: 198, Pages: 99:1 - 99:20

Swansea University Author: Eike Neumann Orcid Logo

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DOI (Published version): 10.4230/LIPIcs.ICALP.2021.99

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...

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Published in: 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)
ISBN: 978-3-95977-195-5
ISSN: 1868-8969
Published: Dagstuhl, Germany Schloss Dagstuhl -- Leibniz-Zentrum für Informatik 2021
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URI: https://cronfa.swan.ac.uk/Record/cronfa60144
first_indexed 2022-06-07T12:59:34Z
last_indexed 2023-01-11T14:41:55Z
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spelling 2022-10-31T14:36:04.6380468 v2 60144 2022-06-07 Decision Problems for Second-Order Holonomic Recurrences 1bf535eaa8d6fcdfbd464a511c1c0c78 0009-0003-2907-1566 Eike Neumann Eike Neumann true false 2022-06-07 MACS 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. Conference Paper/Proceeding/Abstract 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) 198 99:1 99:20 Schloss Dagstuhl -- Leibniz-Zentrum für Informatik Dagstuhl, Germany 978-3-95977-195-5 1868-8969 Holonomic sequences, Positivity Problem, Skolem Problem 2 7 2021 2021-07-02 10.4230/LIPIcs.ICALP.2021.99 https://drops.dagstuhl.de/opus/volltexte/2021/14168 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University Joël Ouaknine: ERC grant AVS-ISS (648701) and DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science). James Worrell: EPSRC Fellowship EP/N008197/1. 2022-10-31T14:36:04.6380468 2022-06-07T13:55:57.3081475 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Eike Neumann 0009-0003-2907-1566 1 Joël Ouaknine 2 James Worrell 3 60144__24403__b06968a1f54147b9a5637b811d2bfa48.pdf 60144.pdf 2022-06-28T14:22:39.2947868 Output 724232 application/pdf Version of Record true © Eike Neumann, Joël Ouaknine, and James Worrell; licensed under Creative Commons License CC-BY 4.0 true eng https://creativecommons.org/licenses/by/4.0/
title Decision Problems for Second-Order Holonomic Recurrences
spellingShingle Decision Problems for Second-Order Holonomic Recurrences
Eike Neumann
title_short Decision Problems for Second-Order Holonomic Recurrences
title_full Decision Problems for Second-Order Holonomic Recurrences
title_fullStr Decision Problems for Second-Order Holonomic Recurrences
title_full_unstemmed Decision Problems for Second-Order Holonomic Recurrences
title_sort Decision Problems for Second-Order Holonomic Recurrences
author_id_str_mv 1bf535eaa8d6fcdfbd464a511c1c0c78
author_id_fullname_str_mv 1bf535eaa8d6fcdfbd464a511c1c0c78_***_Eike Neumann
author Eike Neumann
author2 Eike Neumann
Joël Ouaknine
James Worrell
format Conference Paper/Proceeding/Abstract
container_title 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)
container_volume 198
container_start_page 99:1
publishDate 2021
institution Swansea University
isbn 978-3-95977-195-5
issn 1868-8969
doi_str_mv 10.4230/LIPIcs.ICALP.2021.99
publisher Schloss Dagstuhl -- Leibniz-Zentrum für Informatik
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Computer Science{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Computer Science
url https://drops.dagstuhl.de/opus/volltexte/2021/14168
document_store_str 1
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description 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.
published_date 2021-07-02T07:56:43Z
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score 11.056659

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