On non-detectability of non-computability and the degree of non-computability of solutions of circuit and wave equations on digital computers

TL;DR

This paper classifies the non-computability of the first derivative and its maximum norm within the Zheng-Weihrauch hierarchy, showing that such non-computability cannot be detected by Turing machines in practical problems like circuit behavior and wave equations.

Contribution

It provides a precise classification of the non-computability of derivatives and demonstrates the impossibility of detecting this non-computability or bounds via Turing machines in specific physical problems.

Findings

Non-computability of the first derivative is not detectable by Turing machines.

It is impossible to detect upper bounds for the maximum norm of the first derivative.

Problems like analog circuit behavior and wave equations are not even semi-decidable regarding derivative non-computability.

Abstract

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.