  In this paper, we investigate the capacity of finite-state channels (FSCs) in
presence of delayed feedback. We show that the capacity of a FSC with delayed
feedback can be computed as that of a new FSC with instantaneous feedback and
an extended state. Consequently, graph-based methods to obtain computable upper
and lower bounds on the delayed feedback capacity of unifilar FSCs are
proposed. Based on these methods, we establish that the capacity of the
trapdoor channel with delayed feedback of two time instances is given by
$\log_2(3/2)$. In addition, we derive an analytical upper bound on the delayed
feedback capacity of the binary symmetric channel with a no consecutive ones
input constraint. This bound also serves as a novel upper bound on its
non-feedback capacity, which outperforms all previously known bounds. Lastly,
we demonstrate that feedback does improve the capacity of the dicode erasure
channel.
