A characterization of functions over the integers computable in polynomial time using discrete differential equations

TL;DR

This paper introduces a novel framework based on discrete ordinary differential equations to characterize functions over integers that are computable in polynomial time, unifying various complexity class characterizations.

Contribution

It develops a comprehensive theory of discrete ODEs for computation, linking classical and modern complexity characterizations and providing new insights into algorithm design.

Findings

Discrete ODEs can characterize polynomial-time computable functions.

The framework unifies implicit complexity and continuous ODE approaches.

Linear discrete ODEs play a fundamental role in capturing complexity measures.

Abstract

This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations as a central tool for computation and algorithm design. We present the general theory of discrete ODEs for computation theory, we illustrate this with various examples of algorithms, and we provide several implicit characterizations of complexity and computability classes. The proposed framework presents an original point of view on complexity and computation classes. It unifies several constructions that have been proposed for characterizing these classes including classical approaches in implicit complexity using restricted recursion schemes, as well as recent characterizations of computability and complexity by classes of continuous 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) ODEs and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming many algorithms.