[Paper Review] Positivity problems for low-order linear recurrence sequences

Joël Ouaknine, James Worrell|arXiv (Cornell University)|Jan 5, 2014

Polynomial and algebraic computation46 references57 citations

TL;DR

This paper establishes decidability of the Positivity and Ultimate Positivity problems for linear recurrence sequences (LRS) of order 5 or less, showing the former lies in the Counting Hierarchy and the latter is solvable in polynomial time. It further demonstrates that extending decidability to order 6 would require breakthroughs in Diophantine approximation of transcendental numbers.

ABSTRACT

We consider two decision problems for linear recurrence sequences (LRS) over the integers, namely the Positivity Problem (are all terms of a given LRS positive?) and the Ultimate Positivity Problem (are all but finitely many terms of a given LRS positive?). We show decidability of both problems for LRS of order 5 or less, with complexity in the Counting Hierarchy for Positivity, and in polynomial time for Ultimate Positivity. Moreover, we show by way of hardness that extending the decidability of either problem to LRS of order 6 would entail major breakthroughs in analytic number theory, more precisely in the field of Diophantine approximation of transcendental numbers.

Motivation & Objective

• To determine the decidability of the Positivity Problem—whether all terms of a given integer linear recurrence sequence (LRS) are positive—for low-order sequences.
• To investigate the decidability of the Ultimate Positivity Problem—whether all but finitely many terms of a given LRS are positive—within the same order constraints.
• To establish complexity bounds for both problems in the case of LRS of order 5 or less.
• To explore the theoretical limits of decidability by showing that extending the results to order 6 would require major advances in analytic number theory.

Proposed method

• Reduction of the Positivity and Ultimate Positivity problems to decision procedures over real closed fields using algebraic techniques.
• Employment of the Counting Hierarchy as a complexity measure for the Positivity problem, leveraging results from computational model theory.
• Application of polynomial-time algorithms for the Ultimate Positivity problem via sign analysis of recurrence roots and asymptotic behavior.
• Use of structural properties of linear recurrence sequences, including characteristic roots and their multiplicities, to analyze term sign patterns.
• Establishing hardness results by reducing known hard problems in Diophantine approximation to the LRS decision problems.
• Demonstrating that decidability for order 6 would imply breakthroughs in the approximation of transcendental numbers by rationals.

Experimental results

Research questions

• RQ1 Is the Positivity Problem decidable for linear recurrence sequences of order 5 or less?
• RQ2 Can the Ultimate Positivity Problem be solved in polynomial time for LRS of order 5 or less?
• RQ3 What is the precise complexity class of the Positivity Problem for low-order LRS?
• RQ4 To what extent is decidability of these problems limited by number-theoretic barriers?
• RQ5 Would extending decidability to order 6 sequences require new results in Diophantine approximation?

Key findings

• The Positivity Problem for integer linear recurrence sequences of order 5 or less is decidable and lies within the Counting Hierarchy.
• The Ultimate Positivity Problem for such sequences is decidable in polynomial time.
• Decidability for order 6 or higher would imply major advances in Diophantine approximation of transcendental numbers.
• The results establish tight theoretical boundaries for decidability in low-order LRS.
• The hardness results show that extending the decidability results to order 6 is unlikely without breakthroughs in analytic number theory.