On non-detectability of non-computability and the degree of non-computability of solutions of circuit and wave equations on digital computers (2205.12626v1)

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.

• The paper demonstrates that Turing machines cannot autonomously detect non-computable derivatives in digital simulations.
• It employs the Zheng-Weihrauch hierarchy to rigorously classify non-computability in both circuit and wave equation models.
• The study highlights the need for alternative computing paradigms to address limitations in traditional digital simulation approaches.

Non-Detectability of Non-Computability in Circuit and Wave Equations

The paper, "On non-detectability of non-computability and the degree of non-computability of solutions of circuit and wave equations on digital computers," explores a critical analysis of the limitations of Turing machines in detecting non-computability within certain mathematical frameworks. The authors, Holger Boche and Volker Pohl, present an exploration of the computability issues associated with analog circuit simulations and wave equations, offering a comprehensive examination of these phenomena through the lens of computational theory.

Key Concepts and Findings

The paper introduces and utilizes the Zheng-Weihrauch hierarchy to classify the degree of non-computability for several critical mathematical problems, notably focusing on the non-computability of derivatives and solutions related to circuit and wave equations. This hierarchy aids in understanding the nuanced levels of non-computability, providing a clearer framework for analysis.

1. Non-Computability of Derivatives:
   • The authors precisely classify the non-computability of the first derivative of continuously differentiable functions. It is established that for certain functions within a defined class $U$, $u'(0)$ and $\|u'\|_\infty$ are non-computable. Despite being computable in other contexts, these derivatives defy computation when constrained by the limitations of digital machines.
2. Non-Detectability via Turing Machines:
   • A crucial aspect of the paper is the demonstration that no Turing machine can autonomously detect non-computability of these derivatives by examining the problem's data. This result emphasizes the inherent limitations of digital computers, as opposed to analog systems, in handling these types of problems.
3. Implications for Circuit Theory:
   • The inability to detect non-computability in ordinary differential operators presents significant challenges for computer-aided design and simulation in circuit theory. The paper highlights that algorithms requiring derivative calculations may frequently encounter non-computable solutions, questioning the reliability of digital simulations for certain classes of circuits.
4. Wave Equation Solutions:
   • Extending the investigation to the three-dimensional wave equation, the paper concludes that non-computability of wave solutions cannot be detected using Turing machines. This reinforces the theoretical boundaries faced when simulating continuous systems using discrete computation models.

Implications and Future Directions

The implications of this research are profound, particularly in fields dependent on numerical simulation, such as communications, signal processing, and quantum computing. The inability to detect non-computability has direct impacts on the development of algorithms that rely on computability assumptions. The research suggests a shift towards exploring alternative computational paradigms, such as analog computing, to overcome these fundamental limitations.

Future developments in AI and computing may focus on leveraging these insights to design systems that strategically circumvent non-computability rather than confronting it directly. The findings call for an interdisciplinary approach that incorporates both theoretical advancements and practical implementations, fostering innovations in computational science.

As computational models evolve, understanding and addressing non-computable problems will be critical in shaping the capabilities of future technologies. This research contributes significantly to this dialogue, providing a foundational understanding of the constraints of digital computation in modeling the physical world.