$2.$ Recall the following example from the lecture, encoding adding $1$ in binary:

$\backslash [$

$1\backslash$mid $1$

$\backslash ]$

Write down a transducer performing the inverse action, i$.$e$.$ a transducer that takes a binary number $($parsed from right to left$)$ and outputs one less $($in binary form, read from right to left$),$ with the exception that $\backslash (00\backslash$ldots $0\backslash )$ is rewritten to $\backslash (11\backslash$ldots $1\backslash ).$