We prove that functions over the reals computable in polynomial time can be characterised using discrete ordinary differential equations (ODE), also known as finite differences. We also provide a characterisation of functions computable in polynomial space over the reals. In particular, this covers space complexity, while existing characterisations were only able to cover time complexity, and were restricted to functions over the integers. We prove furthermore that no artificial sign or test function is needed even for time complexity. At a technical level, this is obtained by proving that Turing machines can be simulated with analytic discrete ordinary differential equations. We believe this result opens the way to many applications, as it opens the possibility of programming with ODEs, with an underlying well-understood time and space complexity.

翻译：我们证明，可以通过离散常微分方程（即有限差分）刻画在多项式时间内可计算的实数函数。我们还给出了在实数域上多项式空间可计算函数的刻画。值得注意的是，这涵盖了空间复杂度，而现有刻画仅能覆盖时间复杂度，且限于整数域上的函数。我们进一步证明，即使对于时间复杂度也无需使用人工符号或测试函数。从技术层面看，这一结论通过证明图灵机可由解析离散常微分方程模拟来实现。我们相信这一结果为许多应用开辟了道路，因为它开启了基于常微分方程进行编程的可能性，且时间复杂度与空间复杂度具有良好的理论基础。