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The channel reliability function is an important tool that characterizes thereliable transmission of messages over communication channels. For manychannels, only the upper and lower bounds of the function are known. In thispaper we analyze the computability of the reliability function and its relatedfunctions. We show that the reliability function is not a Turing computableperformance function. The same also applies to the functions of the spherepacking bound and the expurgation bound. Furthermore, we consider the$R_\infty$ function and the zero-error feedback capacity, since they play animportant role in the context of the reliability function. Both the $R_\infty$function and the zero-error feedback capacity are not Banach Mazur computable.We show that the $R_\infty$ function is additive. The zero-error feedbackcapacity is super-additive and we characterize its behavior.

Computability of the Channel Reliability Function and Related Bounds