Continuous Ordinary Differential Equations and Transfinite Computations

TL;DR

This paper explores the connection between continuous ordinary differential equations with greedy solutions and ordinal-based computation models, revealing their equivalence and implications for computability and mathematical analysis.

Contribution

It establishes a strong correspondence between greedy solutions of certain ODEs and models of ordinal computation, extending the understanding of ODEs in computability theory.

Findings

Greedy solutions of CODEs are equivalent to ordinal-based computation models.

Reachability problems in these models have the same computational hardness.

Provides a bridge between differential equations and generalized computability theory.

Abstract

We consider Continuous Ordinary Differential Equations (CODE) y'=f(y), where f is a continuous function. They are known to always have solutions for a given initial condition y(0)=y0, these solutions being possibly non unique. We restrict to our attention to a class of continuous functions, that we call greedy: they always admit unique greedy solutions, i.e. going in greedy way in some fixed direction. We prove that they can be seen as models of computation over the ordinals and conversely in a very strong sense. In particular, for such ODEs, to a greedy trajectory can be associated some ordinal corresponding to some time of computation, and conversely models of computation over the ordinals can be associated to some CODE. In particular, analyzing reachability for one or the other concept with respect to greedy trajectories has the same hardness. This also brings new perspectives on analysis in Mathematics, by providing ways to translate results for ITTMs to CODEs. This also extends some recent results about the relations between ordinary differential equations and Turing machines, and more widely with (generalized) computability theory.