  In this paper, a general binary-input binary-output (BIBO) channel is
investigated in the presence of feedback and input constraints. The feedback
capacity and the optimal input distribution of this setting are calculated for
the case of an $(1,\infty)$-RLL input constraint, that is, the input sequence
contains no consecutive ones. These results are obtained via explicit solution
of a corresponding dynamic programming optimization problem. A simple coding
scheme is designed based on the principle of posterior matching, which was
introduced by Shayevitz and Feder for memoryless channels. The posterior
matching scheme for our input-constrained setting is shown to achieve capacity
using two new ideas: \textit{message history}, which captures the memory
embedded in the setting, and \textit{message splitting}, which eases the
analysis of the scheme. Additionally, in the special case of an S-channel, we
give a very simple zero-error coding scheme that is shown to achieve capacity.
For the input-constrained BSC, we show using our capacity formula that feedback
increases capacity when the cross-over probability is small.
