Computability of the Zero-Error Capacity of Noisy Channels
Holger Boche, Christian Deppe
TL;DR
The paper investigates whether the zero-error capacity $C_0(W)$ of discrete memoryless channels is algorithmically computable. Using Banach-Mazur and Borel-Turing computability frameworks and leveraging Ahlswede’s correspondence to 0-1 AVCs and Shannon’s graph capacity $\Theta(G)$ (with $2^{C_0(W)}=\Theta(G_W)$), it shows that $C_0$ is not Banach-Mazur computable and that related semi-decidability questions transfer across $C_0$, $\Theta$, and $C_{max}$. It further analyzes the computability landscape for 0-1 AVCs, proving that the average-error capacity $C_{av}$ is computable and Borel-Turing computable, while the status of $C_{max}$ and $\Theta$ remains intricately linked and open in general. The paper also introduces Kolmogorov oracle-based results, showing decidability of certain $\Theta$-level sets under an oracle, but leaving unconditional computability of $C_0$ via such oracles unresolved, thereby highlighting fundamental limits on computing zero-error capacities from channel descriptions.
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.