Computability of Initial Value Problems

TL;DR

This paper uses Weihrauch complexity to analyze the computability of initial value problems, establishing their equivalence to weak König's lemma and deriving implications for the computability of solutions.

Contribution

It demonstrates that solving continuous initial value problems is Weihrauch equivalent to weak König's lemma, providing a uniform perspective and new computability results.

Findings

Solutions with maximal domains are non-deterministically computable.

For computable instances, solutions are low in the function space.

Finite solution sets are computable and can be found with finite mind-change algorithms.

Abstract

We demonstrate that techniques of Weihrauch complexity can be used to get easy and elegant proofs of known and new results on initial value problems. Our main result is that solving continuous initial value problems is Weihrauch equivalent to weak K\H{o}nig's lemma, even if only solutions with maximal domains of existence are considered. This result simultaneously generalizes negative and positive results by Aberth and by Collins and Gra\c{c}a, respectively. It can also be seen as a uniform version of a Theorem of Simpson. Beyond known techniques we exploit for the proof that weak K\H{o}nig's lemma is closed under infinite loops. One corollary of our main result is that solutions with maximal domain of existence of continuous initial value problems can be computed non-deterministically, and for computable instances there are always solutions that are low as points in the function space. Another corollary is that in the case that there is a fixed finite number of solutions, these solutions are all computable for computable instances and they can be found uniformly in a finite mind-change computation.