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.

翻译：我们研究了单变量C-有限函数在大时间尺度下最终不等式测试问题的鲁棒版本的计算复杂性。具体而言，在实数计算的位模型框架下，我们考虑了由具有任意实系数和初始值的齐次线性柯西问题所定义的实函数的最终不等式测试问题。为了给该问题赋予明确定义的计算复杂性，我们提出了一种关于可计算度量空间子集的多项式时间可判定性的自然概念，该概念扩展了我们近期引入的最大部分可判定性概念。我们证明，在此意义下，C-有限函数的最终不等式问题是多项式时间可判定的。