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Near-Optimal Complexity Bounds for Fragments of the Skolem Problem

Authors S. Akshay , Nikhil Balaji , Aniket Murhekar, Rohith Varma, Nikhil Vyas

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Author Details

S. Akshay
  • IIT Bombay, India
Nikhil Balaji
  • University of Oxford, UK
Aniket Murhekar
  • University of Illinois, Urbana Champaign, Urbana, IL, USA
Rohith Varma
  • Indian Institute of Technology Palakkad, India
Nikhil Vyas
  • MIT, Cambridge, MA, USA

Funding

This work was partly supported by DST/CEFIPRA/INRIA Associated team EQuaVE.
  • Akshay, S.: Partly supported by DST-INSPIRE Faculty Award [IFA12-MA-17] and SERB Matrices grant MTR/2018/000744.
  • Vyas, Nikhil: Supported by NSF CCF-1909429.

Acknowledgements

This research was supported in part by the International Centre for Theoretical Sciences (ICTS) during a visit for participating in the program - Workshop on Algebraic Complexity Theory (Code: ICTS/wact2019/03).

Cite As Get BibTex

S. Akshay, Nikhil Balaji, Aniket Murhekar, Rohith Varma, and Nikhil Vyas. Near-Optimal Complexity Bounds for Fragments of the Skolem Problem. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.STACS.2020.37

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Linear Recurrences
  • Skolem problem
  • NP-completeness
  • Weighted automata

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