  The channel reliability function is an important tool that characterizes the
reliable transmission of messages over communications channels. For many
channels only the upper and lower bounds of the function are known. In this
paper we analyze the computability of the reliability function and its related
functions. We show that the reliability function is not a Turing computable
performance function. The same also applies to the functions of the sphere
packing bound and the expurgation bound. Furthermore, we consider the
$R_\infty$ function and the zero-error feedback capacity, both of them play an
important role for 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 feedback capacity is
super-additive and we characterize it's behavior.
