Computability of the Channel Reliability Function and Related Bounds
Holger Boche, Christian Deppe
TL;DR
This work proves that the channel reliability function $E(W,R)$ and several closely related bounds are not Turing computable when viewed as functions of the channel $W$ and rate $R$. It develops computability-theoretic tools for computable channels, analyzes $R_ ∞(W)$ and the zero-error feedback capacity $C_0^{FB}(W)$, and shows these quantities are not Banach–Mazur computable; in addition, sequences such as the $k$-letter expurgation bounds fail to yield a computable approximation of $E(W,R)$. The results further reveal intricate tensor-product behavior, including superadditivity phenomena for expurgation-rate bounds and for $C_0^{FB}$ under certain conditions. Collectively, the findings indicate that no simple, algorithmically computable closed-form expression can, in general, capture the channel reliability function or its key bounds, with significant implications for automatic verification and certification of reliability in next-generation networks. The work thus raises fundamental questions about the computability of performance functions that underpin trustworthiness and QoS guarantees in communication systems.
Abstract
The channel reliability function is an important tool that characterizes the reliable transmission of messages over communication channels. For many channels, only upper and lower bounds of the function are known. We analyze the computability of the reliability function and its related functions. We show that the reliability function is not a Turing computable performance function. The same also applies to the functions related to the sphere packing bound and the expurgation bound. Furthermore, we consider the $R_\infty$ function and the zero-error feedback capacity, since they play an important role in the context of the reliability function. Both the $R_\infty$ function and the zero-error feedback capacity are not Banach Mazur computable.