On the Semidecidability of the Remote State Estimation Problem

In this article, we consider the decision problem associated with the task of remotely estimating the state of a dynamic plant via a noisy communication channel. Given a machine-readable description of the plant's and channel's characteristics, does there exist an algorithm that decides whether remote state estimation is possible? From an analytic point of view, this problem has been shown to involve the zero-error capacity of the communication channel. By applying results from Turing machine theory and zero-error coding, we analyze several related variants of the decision problem mentioned above. Our analysis also incorporates a weakened form of the state estimation objective, which has been shown to depend on the classical Shannon Capacity instead. In the broadest sense, our results yield a fundamental limit to the capabilities of computer-aided design tools and adaptive autonomous systems, assuming they are based on digital hardware.