# The Computability of the Channel Reliability Function and Related Bounds
## 7 Conclusions
In fact, if a function is not Banach-Mazur computable, then it cannot be computable under any other standard notion of computability. This underscores the foundational and maximal character of Banach-Mazur computability within the hierarchy of computability concepts. Moreover, as shown in [63], there exist even total functions--functions defined on all computable real numbers--that are Banach-Mazur computable but not Turing computable. For readers interested in a deeper understanding of computability theory--how to determine whether a function is computable, along with illustrative examples and detailed explanations--we recommend the comprehensive work by [PERSON] [64] and [PERSON]'s _New Computational Paradigms_[65]. Practical implications of these theoretical analyses, especially their relevance to real-world applications, are further explored in [66], which may be of particular interest to those seeking connections between theory and practice. The contributions of both authors are equal across all categories. All authors have read and agreed to the published version of the manuscript. The authors gratefully acknowledge the financial support provided by the Federal Ministry of Education and Research (BMBF) of Germany under the \"Souveran. Digital. Vernetzt.\" program, specifically through the joint project 6G-life, project identification numbers 16 KISK002 and 16 KISK263. [PERSON] and [PERSON] also acknowledge the financial support from the BMBF's quantum program QuaPhySI under grants 16 KIS1598K and 16 KIS2234, as well as from the QUET project under grants 16 KISQ093 and 16 KISQ0170. Additionally, they were supported by the QC-CamNetz project under grants 16 KISQ077 and 16 KISQ169. Furthermore, they received funding from the DFG through the project \"Post Shannon Theorie und Implementierung,\" under grants BO 1734/38-1 and DE 1915/2-1. Special thanks also go to the BMBF within the national initiative for their support of [PERSON] under grant 16 KIS1003K and of [PERSON] under grant 16 KIS1005. The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author(s). \begin{table}
\begin{tabular}{l l} \hline \hline
**Problem** & **Result** \\ \hline \multirow{3}{*}{Computability of capacity} & \(R_{\infty}\): Yes \\  & \(R_{0}\): Unknown \\  & \(R_{0}\) for alphabet size \(<5\): Yes \\ \hline \multirow{3}{*}{Semi-decidability of capacity \(>\lambda\)} & \(R_{\infty}\): No \\  & \(C_{0}^{FB}\): No \\ \hline \multirow{3}{*}{Semi-decidability of capacity \(<\lambda\)} & \(R_{\infty}\): Yes \\  & \(C_{0}^{FB}\): Yes \\ \hline \multirow{3}{*}{Computability of capacity function} & \(R_{\infty}\): No \\  & \(C_{0}^{FB}\): No \\ \hline \multirow{3}{*}{Computability of performance function} & \(E(W,R)\): No \\  & \(\{E_{ex}(\cdot,\cdot,\cdot)\}_{k\in\mathbb{N}}\): No \\ \hline \multirow{3}{*}{Additivity} & \(R_{\infty}\): Yes \\  & \(C_{0}^{FB}\): Unknown \\ \cline{1-1}  & \(C_{0}^{FB}\) for alphabet size 2: Yes \\ \cline{1-1}  & \(R_{ex}^{k}\): No \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Overview of results. **Acknowledgments:** [PERSON] would like to thank [PERSON] for insightful discussions and questions regarding the theory of the channel reliability function and the trustworthiness of numerical simulations of this function on digital computers. He also expresses his gratitude to [PERSON] and [PERSON] for their discussions at ISIT 2019 in Paris, which sparked the research leading to the results presented in this paper. Finally, we express our appreciation to [PERSON] for his helpful and insightful comments. **Conflicts of Interest:** The authors declare no conflicts of interest.