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：我们证明了在多项式时间内可计算的实数函数可以通过离散常微分方程（也称为有限差分）来刻画。我们还给出了实数域上多项式空间可计算函数的刻画。特别地，这涵盖了空间复杂度，而现有的刻画仅能覆盖时间复杂度，并且仅限于整数函数。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函数或测试函数。在技术层面上，这是通过证明可以用解析离散常微分方程模拟图灵机来实现的。我们相信这一结果为众多应用开辟了道路，因为它开启了用常微分方程进行编程的可能性，同时其时间和空间复杂度具有清晰的理论基础。