- In this paper, a general binary-input binary-output 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 an equivalent 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: history bits , which captures the memory embedded in our setting, and message-interval 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 binary symmetric channel, we show using our capacity formula that feedback increases capacity when the cross-over probability is small.