初值问题的可计算性
Computability of Initial Value Problems
AI总结：
本文利用 Weihrauch 复杂度技术证明了求解连续初值问题等价于 weak Kőnig's lemma，推广了已有结果并得出解的非确定性可计算性等推论。
摘要：
我们证明了 Weihrauch 复杂度的技术可用于获得关于初值问题的已知和新结果的简单且优雅的证明。我们的主要结果是，求解连续初值问题在 Weihrauch 意义下等价于 weak Kőnig's lemma，即使仅考虑具有最大存在域的解也是如此。该结果同时推广了 Aberth 以及 Collins 和 Graça 的负面与正面结果。它也可以被视为 Simpson 定理的一个均匀版本。除了已知技术外，我们还利用了 weak Kőnig's lemma 在无限循环下封闭这一性质进行证明。我们主要结果的一个推论是，连续初值问题具有最大存在域的解可以非确定性地计算，并且对于可计算实例，总存在在函数空间中作为点来说是低的解。另一个推论是，在存在固定有限数量解的情况下，这些解对于可计算实例都是可计算的，并且可以在有限次思维改变计算中均匀地找到。
英文摘要：
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.
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