Computability of the Zero-Error Capacity of Noisy Channels
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.
Keywords
Cite
@article{arxiv.2010.06873,
title = {Computability of the Zero-Error Capacity of Noisy Channels},
author = {Holger Boche and Christian Deppe},
journal= {arXiv preprint arXiv:2010.06873},
year = {2024}
}