We consider finite-state channels (FSCs) with feedback and state information known causally at the encoder. This setting is quite general and includes: a memoryless channel with i.i.d. state (the Shannon strategy), Markovian states that include look-ahead (LA) access to the state and energy harvesting. We characterize the feedback capacity of the general setting as the directed information between auxiliary random variables with memory to the channel outputs. We also propose two methods for computing the feedback capacity: (i) formulating an infinite-horizon average-reward dynamic program; and (ii) a single-letter lower bound based on auxiliary directed graphs called$Q$-graphs. We demonstrate our computation methods on three examples. In the first example, we introduce a channel with LA and establish a closed-form, analytic lower bound on its feedback capacity. Furthermore, we extend the channel with general parameters, and derive numerical lower bounds for each parameter. In the second example, we show that the mentioned methods achieve the feedback capacity of known unifilar FSCs such as the Ising channel. Finally, in the last example, we generalize the Ising channel such that the state is stochastically dependent on the input, and investigate its feedback capacity.

We consider finite-state channels (FSCs) where the channel state is stochastically dependent on the previous channel output. We refer to these as Noisy Output is the STate (NOST) channels. We derive the feedback capacity of NOST channels in two scenarios: with and without causal state information (CSI) available at the encoder. If CSI is unavailable, the feedback capacity is$C_{\text {FB}}= \max _{P(x|y')} I(X;Y|Y')$, while if it is available at the encoder, the feedback capacity is$C_{\text {FB-CSI}}= \max _{P(u|y'),x(u,s')} I(U;Y|Y')$, where$U$is an auxiliary RV with finite cardinality. In both formulas, the output process is a Markov process with stationary distribution. The derived formulas generalize special known instances from the literature, such as where the state is i.i.d. and where it is a deterministic function of the output.$C_{\text {FB}}$and$C_{\text {FB-CSI}}$are also shown to be computable via convex optimization problem formulations. Finally, we present an example of an interesting NOST channel for which CSI available at the encoder does not increase the feedback capacity.

We consider finite state channels (FSCs) with feedback and state known causally at the encoder. This setting is general and includes both a channel with a Markovian state in which the state is input-independent, but also many other cases where the state is input-dependent such as the energy harvesting model. We characterize the capacity as a multi-letter expression that includes auxiliary random variables with memory. We derive a single-letter computable lower bound based on auxiliary directed graphs that are used to provide an auxiliary structure for the channel outputs and are called Q-graphs. This method is implemented for binary energy-harvesting model with a unitsized battery and the noiseless channel, whose exact capacity has remained an open problem. We identify a structure of Q-graphs, with achievable rates that outperform the best achievable rates known in the literature.