Lipschitz Continuous Ordinary Differential Equations are Polynomial-Space Complete

TL;DR

This paper demonstrates that initial value problems with Lipschitz continuous functions can have solutions that are complete for polynomial space, answering longstanding questions in computational complexity.

Contribution

It establishes the polynomial-space completeness of solutions to Lipschitz continuous differential equations, extending to Volterra integral equations, and introduces a new class of polynomial-space computation tableaux.

Findings

Lipschitz condition implies weak feedback in differential equations

Polynomial-space completeness of solutions to certain differential equations

Resolution of Ko's questions on Volterra integral equations

Abstract

In answer to Ko's question raised in 1983, we show that an initial value problem given by a polynomial-time computable, Lipschitz continuous function can have a polynomial-space complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomial-space computation tableaux with equally weak feedback, and show that they are still polynomial-space complete. The same technique also settles Ko's two later questions on Volterra integral equations.