Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Computability of Initial Value Problems

Vasco Brattka, Hendrik Smischliaew

TL;DR

The paper classifies the computational content of continuous initial value problems within the Weihrauch framework, showing that the initial value problem $\mathsf{IVP}$, its maximal-domain variant $\mathsf{IVP}_{\max}$, and weak König's lemma $\mathsf{WKL}$ are strongly Weihrauch equivalent. It achieves this via constructive reductions using a Picard operator on a computably compact domain and, for the reverse direction, by embedding LLPO gadgets that encode binary choices into IVP, including an infinite-loop composition to realize maximal domains. The results imply that solutions to $\mathsf{IVP}$ with maximal domain of existence are non-deterministically computable, and for computable instances there are low solutions; with finitely many solutions, each is computable and obtainable uniformly with finite mind-change. Together, these findings provide a uniform, computable-analysis perspective on classical theorems (Aberth, Collins-Graça, Simpson) and deepen connections between computable analysis and reverse mathematics.

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 Kőnig'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 Graça, 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 Kőnig'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.

Computability of Initial Value Problems

TL;DR

The paper classifies the computational content of continuous initial value problems within the Weihrauch framework, showing that the initial value problem , its maximal-domain variant , and weak König's lemma are strongly Weihrauch equivalent. It achieves this via constructive reductions using a Picard operator on a computably compact domain and, for the reverse direction, by embedding LLPO gadgets that encode binary choices into IVP, including an infinite-loop composition to realize maximal domains. The results imply that solutions to with maximal domain of existence are non-deterministically computable, and for computable instances there are low solutions; with finitely many solutions, each is computable and obtainable uniformly with finite mind-change. Together, these findings provide a uniform, computable-analysis perspective on classical theorems (Aberth, Collins-Graça, Simpson) and deepen connections between computable analysis and reverse mathematics.

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 Kőnig'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 Graça, 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 Kőnig'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.
Paper Structure (6 sections, 16 theorems, 43 equations, 1 figure)

This paper contains 6 sections, 16 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Weihrauch complexity and the initial value problem
  3. Reduction of $\mathsf{IVP}$ to $\mathsf{WKL}$
  4. Reduction of $\mathsf{IVP}_{\max}$ to $\mathsf{WKL}$
  5. Reduction of $\mathsf{WKL}$ to $\mathsf{IVP}$
  6. Acknowledgments.

Key Result

theorem thmcountertheorem

Let $U\subseteq{\mathbb{R}}\times{\mathbb{R}}^n$ be open with $(x_0,y_0)\in U$ and let $f:U\to{\mathbb{R}}^n,(t,s)\mapsto f(t,s)$ be continuous and locally Lipschitz continuous in the second argument $s\in{\mathbb{R}}^n$, uniformly with respect to the first argument $t\in{\mathbb{R}}$. Then the init

Figures (1)

  • Figure 1: Solutions $y$ of the initial value problem $y'(x)=g_p(x,y(x))$ with $y(0)=0$ .

Theorems & Definitions (23)

  • theorem thmcountertheorem: Picard-Lindelöf
  • theorem thmcountertheorem: Peano
  • theorem thmcountertheorem: Aberth
  • theorem thmcountertheorem: Collins and Graça
  • theorem thmcountertheorem: Simpson
  • theorem thmcountertheorem
  • corollary thmcountercorollary
  • corollary thmcountercorollary
  • definition thmcounterdefinition: Coproduct function spaces
  • theorem thmcountertheorem: Gherardi and Marcone
  • ...and 13 more