Computability of the Channel Reliability Function and Related Bounds

TL;DR

This paper investigates the computability of the channel reliability function and related bounds, demonstrating that key functions like the reliability function, sphere packing bound, and zero-error feedback capacity are not Turing or Banach Mazur computable.

Contribution

It establishes the non-computability of fundamental information-theoretic functions, providing new insights into their theoretical limitations.

Findings

Reliability function is not Turing computable.

Sphere packing and expurgation bounds are not Turing computable.

Zero-error feedback capacity is not Banach Mazur computable.

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.