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

TL;DR

This paper explores a hierarchy of algorithmic information in fractional calculus and signal processing, with applications to digital twinning, formal verification, and the analysis of physical systems' digital replicas.

Contribution

It introduces a novel hierarchy of algorithmic information based on fractional smoothness and establishes dual relationships between function spaces and lp-spaces.

Findings

Hierarchy of algorithmic information depending on fractional smoothness

Duality between function space hierarchy and lp-space hierarchy

Application to digital twinning and formal verification

Abstract

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.