Recursion schemes, discrete differential equations and characterization of polynomial time computation

TL;DR

This paper introduces a novel framework based on discrete ODEs to characterize polynomial time computation, unifying various complexity classes and providing new insights into the relationship between analog and classical computation models.

Contribution

It presents an original framework using discrete ODEs for characterizing complexity classes, unifying existing approaches and deepening understanding of computation through differential equations.

Findings

Discrete ODEs effectively characterize polynomial time complexity.

The framework unifies multiple existing complexity characterizations.

Linear discrete ODEs and classical tools are fundamental in understanding computability.

Abstract

This papers studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs). It presents a new framework using discrete ODEs as a central tool for computation and provides several implicit characterizations of complexity and computability classes. The proposed framework presents an original point of view on complexity and computability classes. It also unifies in an elegant settings various constructions that have been proposed for characterizing these classes. This includes Cobham's and, Bellantoni and Cook's definition of polynomial time and later extensions on the approach, as well as recent characterizations of computability and complexity by classes of ordinary differential equations. It also helps understanding the relationships between analog computations and classical discrete models of computation theory. At a more technical point of view, this paper points out the fundamental role of linear (discrete) ordinary differential equations and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming various algorithms.