Near-Optimal Complexity Bounds for Fragments of the Skolem Problem
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.
Linear Recurrences
Skolem problem
NP-completeness
Weighted automata
Theory of computation~Problems, reductions and completeness
37:1-37:18
Regular Paper
This work was partly supported by DST/CEFIPRA/INRIA Associated team EQuaVE.
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).
S.
Akshay
S. Akshay
IIT Bombay, India
https://orcid.org/0000-0002-2471-5997
Partly supported by DST-INSPIRE Faculty Award [IFA12-MA-17] and SERB Matrices grant MTR/2018/000744.
Nikhil
Balaji
Nikhil Balaji
University of Oxford, UK
https://orcid.org/0000-0001-9234-0683
Aniket
Murhekar
Aniket Murhekar
University of Illinois, Urbana Champaign, Urbana, IL, USA
Rohith
Varma
Rohith Varma
Indian Institute of Technology Palakkad, India
Nikhil
Vyas
Nikhil Vyas
MIT, Cambridge, MA, USA
https://orcid.org/0000-0002-4055-7693
Supported by NSF CCF-1909429.
10.4230/LIPIcs.STACS.2020.37
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S. Akshay, Nikhil Balaji, Aniket Murhekar, Rohith Varma, and Nikhil Vyas
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