Alien Operation

Consider the following function $f$, defined on pairs of boolean values: $$ f(x, y) = \begin{cases} 1, \text{ if } x = y = 0 \\ 0, \text{ otherwise} \end{cases} $$ So $f(0, 0) = 1$, and for any other input $f(x, y) = 0$. You're given a binary string $S$ of length $N$. You can perform the following operation on it: - Choose an index $i$ ($1 \lt i \lt N$) and do **both**: 1. Set $S_{i-1} = f