Polynomial time computable functions over the reals characterized using discrete ordinary differential equations

The class of functions from the integers to the integers computable in polynomial time has been characterized recently using discrete ordinary differential equations (ODE), also known as finite differences. In the framework of ordinary differential equations, this is very natural to try to extend the approach to classes of functions over the reals, and not only over the integers. Recently, an extension of previous characterization was obtained for functions from the integers to the reals, but the method used in the proof, based on the existence of a continuous function from the integers to a suitable discrete set of reals, cannot extend to functions from the reals to the reals, as such a function cannot exist for clear topological reasons. In this article, we prove that this is indeed possible to provide an elegant and simple algebraic characterization of functions from the reals to the reals: we provide a characterization of such functions as the smallest class of functions that contains some basic functions, and that is closed by composition, linear length ODEs, and a natural effective limit schema. This is obtained using an alternative proof technique based on the construction of specific suitable functions defined recursively, and a barycentric method. Furthermore, we also extend previous characterizations in several directions: First, we prove that there is no need of multiplication. We prove a normal form theorem, with a nice side effect related to formal neural networks. Indeed, given some fixed error and some polynomial time t(n), our settings produce effectively some neural network that computes the function over its domain with the given precision, for any t(n)-polynomial time computable function f .