On the Complexity of Robust Eventual Inequality Testing for C-Finite Functions

We study the computational complexity of a robust version of the problem of testing two univariate C-finite functions for eventual inequality at large times. Specifically, working in the bit-model of real computation, we consider the eventual inequality testing problem for real functions that are specified by homogeneous linear Cauchy problems with arbitrary real coefficients and initial values. In order to assign to this problem a well-defined computational complexity, we develop a natural notion of polynomial-time decidability of subsets of computable metric spaces which extends our recently introduced notion of maximal partial decidability. We show that eventual inequality of C-finite functions is polynomial-time decidable in this sense.