License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2020.37
URN: urn:nbn:de:0030-drops-118982
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/11898/
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Akshay, S. ; Balaji, Nikhil ; Murhekar, Aniket ; Varma, Rohith ; Vyas, Nikhil

Near-Optimal Complexity Bounds for Fragments of the Skolem Problem

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Abstract

Given a linear recurrence sequence (LRS), specified using the initial conditions and the recurrence relation, the Skolem problem asks if zero ever occurs in the infinite sequence generated by the LRS. Despite active research over last few decades, its decidability is known only for a few restricted subclasses, by either restricting the order of the LRS (upto 4) or by restricting the structure of the LRS (e.g., roots of its characteristic polynomial).
In this paper, we identify a subclass of LRS of arbitrary order for which the Skolem problem is easy, namely LRS all of whose characteristic roots are (possibly complex) roots of real algebraic numbers, i.e., roots satisfying x^d = r for r real algebraic. We show that for this subclass, the Skolem problem can be solved in NP^RP. As a byproduct, we implicitly obtain effective bounds on the zero set of the LRS for this subclass. While prior works in this area often exploit deep results from algebraic and transcendental number theory to get such effective results, our techniques are primarily algorithmic and use linear algebra and Galois theory. We also complement our upper bounds with a NP lower bound for the Skolem problem via a new direct reduction from 3-CNF-SAT, matching the best known lower bounds.

BibTeX - Entry

@InProceedings{akshay_et_al:LIPIcs:2020:11898,
  author =	{S. Akshay and Nikhil Balaji and Aniket Murhekar and Rohith Varma and Nikhil Vyas},
  title =	{{Near-Optimal Complexity Bounds for Fragments of the Skolem Problem}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{37:1--37:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Christophe Paul and Markus Bl{\"a}ser},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/11898},
  URN =		{urn:nbn:de:0030-drops-118982},
  doi =		{10.4230/LIPIcs.STACS.2020.37},
  annote =	{Keywords: Linear Recurrences, Skolem problem, NP-completeness, Weighted automata}
}

Keywords: Linear Recurrences, Skolem problem, NP-completeness, Weighted automata
Collection: 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)
Issue Date: 2020
Date of publication: 04.03.2020

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