To Feed or Not to Feedback

We study communication over finite state channels (FSCs), where the encoder and the decoder can control the availability or the quality of noise-free feedback, which is fed back from the decoder to the encoder. Specifically, the instantaneous feedback is a function of an action taken by the encoder, an action taken by the decoder, and the channel output. Encoder and decoder actions take values from finite alphabet sets and may be subject to average cost constraints. We prove capacity results for such a setting by constructing a sequence of codes, using a simple scheme based on code tree, which generates channel input symbols along with encoder and decoder actions. We prove that the limit of this sequence exists, and provide an upper bound on the maximum achievable rate. Our upper and lower bounds coincide and hence yield the capacity for the case where the probability of initial state is positive for all states. Next, the capacity is given for indecomposable channels without intersymbol interference as the limit of normalized directed information between the input and output sequences, maximized over an appropriate set of causally conditioned distributions. As a special case of our framework, we characterize the capacity of coding on the backward link in FSCs, i.e., when the decoder sends limited-rate instantaneous coded noise-free feedback on the backward link. Finally, we propose an extension of the Blahut-Arimoto algorithm for evaluating the capacity when actions can be cost constrained and demonstrate its application in a few examples. Among these examples are those of to feed or not to feedback where the encoder takes binary actions that determine whether the current channel output will be fed back to the encoder, with a constraint on the fraction of channel outputs that are fed back.