  We study the communication over Finite State Channels (FSCs), where the
encoder and the decoder can control the availability or the quality of the
noise-free feedback. 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 in finite alphabets, and may be
subject to average cost constraints. We prove capacity results for such a
setting by constructing a sequence of achievable rates, using a simple scheme
based on 'code tree' generation, that generates channel input symbols along
with encoder and decoder actions. We prove that the limit of this sequence
exists. For a given block length and probability of error, we give 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. Further, for stationary indecomposable channels
without intersymbol interference (ISI), the capacity is given as the limit of
normalized directed information between the input and output sequence,
maximized over an appropriate set of causally conditioned distributions. As an
important special case, we consider the framework of 'to feed or not to feed
back' where either the encoder or the decoder takes binary actions, which
determine whether current channel output will be fed back to the encoder, with
a constraint on the fraction of channel outputs that are fed back. As another
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 on a few
examples.
