Ultimate Positivity is Decidable for Simple Linear Recurrence Sequences

Joel Ouaknine; James Worrell

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arXiv · 1309.1914

Ultimate Positivity is Decidable for Simple Linear Recurrence Sequences

Joel Ouaknine·James Worrell

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Abstract

We consider the decidability and complexity of the Ultimate Positivity Problem, which asks whether all but finitely many terms of a given rational linear recurrence sequence (LRS) are positive. Using lower bounds in Diophantine approximation concerning sums of S-units, we show that for simple LRS (those whose characteristic polynomial has no repeated roots) the Ultimate Positivity Problem is decidable in polynomial space. If we restrict to simple LRS of a fixed order then we obtain a polynomial-time decision procedure. As a complexity lower bound we show that Ultimate Positivity for simple LRS is hard for co$\exists\mathbb{R}$, i.e., the class of problems solvable in the universal theory of the reals (which lies between coNP and PSPACE).

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Joel Ouaknine, James Worrell. 2013-09-07. Ultimate Positivity is Decidable for Simple Linear Recurrence Sequences. https://arxiv.org/abs/1309.1914

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