Computability of the Zero-Error Capacity of Noisy Channels

TL;DR

This paper proves that the zero-error capacity of noisy channels cannot be computed algorithmically and explores its relationship with other capacities, highlighting fundamental limits in their computability and semi-decidability.

Contribution

It establishes the non-computability of zero-error capacity and links its semi-decidability to that of Shannon capacity of graphs and other related capacities.

Findings

Zero-error capacity is not Banach-Mazur computable.

Important questions about semi-decidability are equivalent across capacities.

The Borel-Turing computability of Shannon capacity remains an open problem.

Abstract

Zero-error capacity plays an important role in a whole range of operational tasks, in addition to the fact that it is necessary for practical applications. Due to the importance of zero-error capacity, it is necessary to investigate its algorithmic computability, as there has been no known closed formula for the zero-error capacity until now. We show that the zero-error capacity of noisy channels is not Banach-Mazur computable and therefore not Borel-Turing computable. We also investigate the relationship between the zero-error capacity of discrete memoryless channels, the Shannon capacity of graphs, and Ahlswede's characterization of the zero-error-capacity of noisy channels with respect to the maximum error capacity of 0-1-arbitrarily varying channels. We will show that important questions regarding semi-decidability are equivalent for all three capacities. So far, the Borel-Turing computability of the Shannon capacity of graphs is completely open. This is why the coupling with semi-decidability is interesting.