Computability of Initial Value Problems

2024-12-31

AI Summary

This study investigates the computability of initial value problems (IVPs) for ordinary differential equations, aiming to precisely characterize their computational difficulty within the Weihrauch complexity framework. Employing tools from computable analysis, recursion theory, and Weihrauch reducibility—combined with closure arguments under the Weak König Lemma (WKL) and techniques from nondeterministic and low-point computation—we establish, for the first time, that the continuous IVP solution operator is Weihrauch-equivalent to WKL. This unifies and generalizes the classical noncomputability results of Aberth and of Collins–Graça, while also yielding a uniform version. Key contributions include: (i) the maximal domain solution is nondeterministically computable; (ii) all solutions corresponding to computable initial values are low (i.e., Turing below the halting problem); and (iii) in the case of finite solution sets, the solution set is uniformly computable and fully enumerable within a bounded number of mind-changes.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 KH{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 Grac{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 KH{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.

Problem

• Computability
• Initial Value Problems
• Conditions for Solvability

Innovation

• Weihrauch complexity
• equivalence proof
• computable solutions