Positivity of Nearly Linearly Recurrent Sequences

Nearly linear recurrences are a generalisation of linear recurrences and are instances of linear time-invariant systems in control theory and linear constraint loops in program analysis. In this paper we formulate the Positivity Problem for such recurrences. This asks whether all sequences satisfying a given recurrence with given initial conditions are positive. This problem is a generalisation of the Positivity Problem for linear recurrence sequences, and is a special case of the non-reachability problem for linear time-invariant systems. Our main contribution is a decision procedure for the Positivity Problem for recurrences of order two. The termination proof of our procedure relies on a new transcendence result for infinite series that is of independent interest.