It is known that there exist mathematical problems of practical relevance which cannot be computed on a Turing machine. An important example is the calculation of the first derivative of continuously differentiable functions. This paper precisely classifies the non-computability of the first derivative, and of the maximum-norm of the first derivative in the Zheng-Weihrauch hierarchy. Based on this classification, the paper investigates whether it is possible that a Turing machine detects this non-computability of the first derivative by observing the data of the problem, and whether it is possible to detect upper bounds for the peak value of the first derivative of continuously differentiable functions. So from a practical point of view, the question is whether it is possible to implement an exit-flag functionality for observing non-computability of the first derivative. This paper even studies two different types of exit-flag functionality. A strong one, where the Turing machine always has to stop, and a weak one, where the Turing machine stops if and only if the input lies within the corresponding set of interest. It will be shown that non-computability of the first derivative is not detectable by a Turing machine for two concrete examples, namely for the problem of computing the input–output behavior of simple analog circuits and for solutions of the three-dimensional wave equation. In addition, it is shown that it is even impossible to detect an upper bound for the maximum norm of the first derivative. In particular, it is shown that all three problems are not even semidecidable. Finally, we briefly discuss implications of these results for analog and quantum computing.

中文翻译：

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论数字计算机上电路和波动方程解不可计算性的不可测性和不可计算性的程度

众所周知，存在无法在图灵机上计算的实际相关数学问题。一个重要的例子是计算连续可微函数的一阶导数。本文对一阶导数的不可计算性和Zheng-Weihrauch层次中一阶导数的最大范数进行了精确分类。基于这种分类，本文研究了图灵机是否有可能通过观察问题的数据来检测一阶导数的这种不可计算性，以及是否有可能检测到一阶导数峰值的上界连续可微函数。所以从实际的角度来看，问题是是否有可能实现退出标志功能来观察一阶导数的不可计算性。本文甚至研究了两种不同类型的退出标志功能。一个强的，图灵机总是必须停止，一个弱的，当且仅当输入位于相应的感兴趣集合内时，图灵机才停止。对于两个具体示例，即计算简单模拟电路的输入-输出行为的问题和三维波动方程的解，图灵机无法检测到一阶导数的不可计算性。此外，它表明甚至不可能检测一阶导数的最大范数的上限。尤其是，结果表明，所有三个问题甚至都不是半可判定的。最后，我们简要讨论了这些结果对模拟和量子计算的影响。