The capacity of injective semi-deterministic two-way channels

The capacity region of the class of injective semi-deterministic two-way channels (TWCs) is investigated in this paper. To characterize this capacity, two conditions under which Shannon's bounds on the capacity region of TWCs are tight are first given. Using those conditions, it is shown that the capacity of this class of TWCs is characterized by the rectangle formed by the one-way capacities. This proves that adaptation is not needed for this class. This class encompasses, among others, all memoryless additive channels with input-independent noise, and hence, adaptation is useless for all such channels. This also shows that there exist continuous additive TWCs not of the exponential family type for which adaptation is not necessary. An example of a Cauchy TWC is given, and its capacity is characterized in closed form under a logarithmic constraint. Finally, the impact of the dependence of the noise on the inputs is discussed, and it is shown that adaptation may still be useless in such cases.