The problem of state communication over a discrete memoryless channel with discrete memoryless state is studied when the state information is available strictly causally at the encoder. It is shown that block Markov encoding, in which the encoder communicates a description of the state sequence in the previous block by incorporating side information about the state sequence at the decoder, yields the minimum state estimation error. When the same channel is used to send additional independent information at the expense of a higher channel state estimation error, the optimal tradeoff between the rate of the independent information and the state estimation error is characterized via the capacitydistortion function. It is shown that any optimal tradeoff pair can be achieved via rate-splitting. These coding theorems are then extended optimally to the case of causal channel state information at the encoder using the Shannon strategy.

Channels with action-dependent states, as defined in [1], are considered. While [1] investigated the scenario of message dependent non-adaptive action sequences, this work focuses on adaptive action sequences, where the action is a strictly causal function of the message and the state sequences. The capacity of such a channel is characterized for the case when the state information is available non-causally at the channel encoder and it is shown that the adaptive action is not useful in increasing the capacity. The capacity of the action-dependent additive Gaussian channel, which was left open in [1], is then characterized by showing the equivalence of the current setting to the problem of the cooperative multiple access channel (MAC) with asymmetric state information at the encoders [2]. The problem setting is then extended to characterize the rate-distortion region of the source coding dual of the adaptive action-dependent channel coding setup, where a source is to be communicated with some fidelity to a decoder with adaptive action-dependent side information. In this setting adaptivity offers no help in enlarging the ratedistortion region.

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A problem of state information transmission over a state-dependent discrete memoryless channel (DMC) with independent and identically distributed (i.i.d.) states, known strictly causally at the transmitter is investigated. It is shown that block-Markov encoding coupled with channel state estimation conditioned on treating the decoded message and received channel output as side information at the decoder yields the minimum state estimation error. This same channel can also be used to send additional independent information at the expense of a higher channel state estimation error. The optimal tradeoff between the rate of the independent information that can be reliably transmitted and the state estimation error is characterized via the capacity-distortion function. It is shown that any optimal tradeoff pair can be achieved via a simple rate-splitting technique, whereby the transmitter appropriately allocates its rate between pure information transmission and state estimation.

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