标题： 数字孪生和信号处理中算法信息的连续有序层次结构 显示英文标题

标题： Continuously Ordered Hierarchies of Algorithmic Information in Digital Twinning and Signal Processing

摘要： 我们考虑了分数微积分中的一个连续算法信息层次结构的例子，并探讨了其在数字孪生潜在应用中的情境。数字孪生指的是控制工程中不同新兴的方法论，涉及创建某个物理实体的数字副本。从可计算性理论的角度来看，确保数字孪生完整性的问题——即保持其状态与物理对应物一致——涉及到一种算法信息概念，该概念决定了我们可以可靠地通过算法分析其数字孪生推断出物理系统的哪些属性。本研究调查了周期函数的分数微积分——特别是，我们考虑了维纳代数——作为算法信息概念的一个典型应用。我们建立了周期函数空间之间的一个连续有序的算法信息层次结构——根据它们的分数光滑度程度而定——其中排序关系决定了某个函数的特定表示是否比另一个包含“更多”或“更少”的信息。此外，我们在lp空间之间建立了类似的层次结构，这些空间构成了（传统）数字信号处理的核心部分。值得注意的是，这两个层次结构在数学上是“对偶”的。从实用角度来看，我们的方法最终归类于形式验证和（广义）形式方法。 显示英文摘要

摘要： We consider a fractional-calculus example of a continuous hierarchy of algorithmic information in the context of its potential applications in digital twinning. Digital twinning refers to different emerging methodologies in control engineering that involve the creation of a digital replica of some physical entity. From the perspective of computability theory, the problem of ensuring the digital twin's integrity -- i.e., keeping it in a state where it matches its physical counterpart -- entails a notion of algorithmic information that determines which of the physical system's properties we can reliably deduce by algorithmically analyzing its digital twin. The present work investigates the fractional calculus of periodic functions -- particularly, we consider the Wiener algebra -- as an exemplary application of the algorithmic-information concept. We establish a continuously ordered hierarchy of algorithmic information among spaces of periodic functions -- depending on their fractional degree of smoothness -- in which the ordering relation determines whether a certain representation of some function contains ``more'' or ``less'' information than another. Additionally, we establish an analogous hierarchy among lp-spaces, which form a cornerstone of (traditional) digital signal processing. Notably, both hierarchies are (mathematically) ``dual'' to each other. From a practical perspective, our approach ultimately falls into the category of formal verification and (general) formal methods.