A characterization of polynomial time computable functions from the integers to the reals using discrete ordinary differential equations

TL;DR

This paper extends the characterization of polynomial-time computable functions from integers to reals using discrete ordinary differential equations, incorporating limits and classical ODE tools for a broader class of functions.

Contribution

It introduces a new characterization of polynomial-time computable real functions via discrete ODEs, including limits and classical ODE techniques.

Findings

Characterization of polynomial-time computable real functions using discrete ODEs.

Inclusion of limits and classical ODE tools in the characterization.

Extension from integer to real functions in polynomial time.

Abstract

In a recent article, the class of functions from the integers to the integers computable in polynomial time has been characterized using discrete ordinary differential equations (ODE), also known as finite differences. Doing so, we pointed out the fundamental role of linear (discrete) ODEs and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming. In this article, we extend the approach to a characterization of functions from the integers to the reals computable in polynomial time in the sense of computable analysis. In particular, we provide a characterization of such functions in terms of 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.