Remote State Estimation and Blum–Shub–Smale Machines—A Computability Analysis With Applications to Virtual Twinning

This article analyzes aspects of the problem of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">remote state estimation</i> (RSE) <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">via noisy communication channels</i> for their <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Blum–Shub–Smale</i> (BSS) computability, motivated by an exemplary application to a formal model of virtual twinning subject to stringent integrity requirements. Computability theory provides a unique framework for the formal and mathematically rigorous analysis of algorithms and computing machines. Therefore, computability theory is essential in the domain of safety- and life-critical technology, where the formal verification of automated systems is necessary. Based on the RSE problem, we establish a simple mathematical model of virtual-twin systems that entails a formal notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">integrity</i> (i.e., a state where the virtual entity accurately mirrors its physical counterpart). The model's notion of integrity is related to the question of whether the system under consideration is capable of computing the communication channel's zero-error capacity and corresponding zero-error codes. While this task is known to exceed the theoretical capabilities of Turing computers, we prove its formal feasibility within the model BSS machines. As different authors have proposed BSS machines as potential model of some forms of analog computing, this article serves as a proof-of-concept for a theoretical <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">analog supremacy</i> of unconventional information-processing hardware. Considering recent advances in the development of such hardware, forms of analog supremacy will likely become relevant in the future of cyber-physical systems and information technology.