Simulation of Turing machines with analytic discrete ODEs: FPTIME and FPSPACE over the reals characterised with discrete ordinary differential equations

TL;DR

This paper characterizes polynomial time and space computability over the reals using discrete ordinary differential equations, extending previous work limited to time and integers, and shows Turing machines can be simulated with these ODEs.

Contribution

It introduces a novel characterization of real-valued functions computable in polynomial time and space via discrete ODEs, including simulation of Turing machines without artificial functions.

Findings

Functions in P over reals characterized by discrete ODEs

Functions in PSPACE over reals characterized by discrete ODEs

Turing machines simulated with analytic discrete ODEs

Abstract

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.