Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length

TL;DR

This paper characterizes the class PTIME using polynomial ordinary differential equations, providing a continuous and elegant perspective on classical computational complexity without relying on discrete models.

Contribution

It offers the first implicit characterization of PTIME via polynomial ODEs, extending to functions over reals and providing a new continuous framework for complexity classes.

Findings

PTIME characterized by polynomial ODEs

Extension to polynomial-time computable functions over reals

Provides a continuous perspective on classical complexity

Abstract

We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class $\operatorname{PTIME}$ of languages computable in polynomial time in terms of differential equations with polynomial right-hand side. This result gives a purely continuous (time and space) elegant and simple characterization of $\operatorname{PTIME}$. This is the first time such classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the reals in the sense of computable analysis. This extends to deterministic complexity classes above polynomial time. This may provide a new perspective on classical complexity, by giving a way to define complexity classes, like $\operatorname{PTIME}$, in a very simple way, without any reference to a notion of (discrete) machine. This may also provide ways to state classical questions about computational complexity via ordinary differential equations, i.e.~by using the framework of analysis.