## On robustness for the skolem, positivity and ultimate positivity problems

*In Logical Methods in Computer Science*

## Abstract

The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial config- uration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: the best known decidability results are for LRS with special properties (e.g., low order recurrences). On the other hand, these problems are much easier for “uninitialised”uninitialised variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided by polynomial time algorithms (Tiwari in 2004, Braverman in 2006).In this paper, we consider problems that lie between the initialised and uninitialised variants. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighbourhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighbourhood is given as part of the input, then robust Skolem and robust positivity are Diophantine hard, i.e., solving either would entail major breakthroughs in Diophantine approximations, as happens for (non-robust) positivity. Interestingly, this is the first Diophantine hardness result on a variant of the Skolem problem. On the other hand, if one asks whether such a neighbourhood exists, then the problems turn out to be decidable in their full generality, with PSPACE complexity. Our analysis is based on the set of initial configurations such that positivity holds, which leads to new insights into these difficult problems, and interesting geometrical interpretations.Our techniques also allow us to tackle robustness for ultimate positivity, which asks whether there is a bound on the number of steps after which the LRS remains positive. There are two natural robust variants depending on whether we ask for a “uniform”uniform bound on this number of steps, independent of the starting configuration in the neighbourhood. We show that for the uniform variant, results are similar to positivity. On the other hand, for the non-uniform variant, robust ultimate positivity has different properties when the neighbourhood is open and when it is closed. When it is open, the problem turns out to be tractable, even when the neighbourhood is given as part of the input.