On Positivity and Minimality for Second-Order Holonomic Sequences
An infinite sequence ⟨u_n⟩_n of real numbers is holonomic (also known as P-recursive or P-finite) if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is said to be positive if each u_n ≥ 0, and minimal if, given any other linearly independent sequence ⟨v_n⟩_n satisfying the same recurrence relation, the ratio u_n/v_n → 0 as n → ∞.
In this paper we give a Turing reduction of the problem of deciding positivity of second-order holonomic sequences to that of deciding minimality of such sequences. More specifically, we give a procedure for determining positivity of second-order holonomic sequences that terminates in all but an exceptional number of cases, and we show that in these exceptional cases positivity can be determined using an oracle for deciding minimality.
Holonomic sequences
Minimal solutions
Positivity Problem
Theory of computation~Logic and verification
67:1-67:15
Regular Paper
This work was partly carried out during a visit of Florian Luca at the Max Planck Institute for Software Systems in Saarbrücken, Germany from September 2020 to March 2021. He thanks the institution for its hospitality and excellent working conditions. The authors would like to thank the anonymous referees for their detailed comments, which have led to significant improvements and clarifications in the final version of this paper.
George
Kenison
George Kenison
Institute for Logic and Computation, The Technical University of Vienna, Austria
WWTF Grant ProbInG ICT19-018 and the ERC Consolidator Grant ARTIST 101002685.
Oleksiy
Klurman
Oleksiy Klurman
School of Mathematics, University of Bristol, UK
Max Planck Institute for Mathematics, Bonn, Germany
Engel
Lefaucheux
Engel Lefaucheux
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Florian
Luca
Florian Luca
School of Mathematics, University of the, Witwatersrand, Johannesburg, South Africa
Research Group in Algebraic Structures & , Applications, King Abdulaziz University, Riyadh, Saudi Arabia
Centro de Ciencias Matemáticas UNAM, Morelia, Mexico
Pieter
Moree
Pieter Moree
Max Planck Institute for Mathematics, Bonn, Germany
https://orcid.org/0000-0002-5318-2587
Joël
Ouaknine
Joël Ouaknine
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
https://orcid.org/0000-0003-0031-9356
ERC grant AVS-ISS (648701), and DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science). Joël Ouaknine is also affiliated with Keble College, Oxford as http://emmy.network/ Fellow.
Markus A.
Whiteland
Markus A. Whiteland
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
https://orcid.org/0000-0002-6006-9902
James
Worrell
James Worrell
Department of Computer Science, University of Oxford, UK
EPSRC Fellowship EP/N008197/1.
10.4230/LIPIcs.MFCS.2021.67
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George Kenison, Oleksiy Klurman, Engel Lefaucheux, Florian Luca, Pieter Moree, Joël Ouaknine, Markus A. Whiteland, and James Worrell
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