Let's say we have $n$ binary unknowns (i.e. either 0 or 1), $x_1$ to $x_n$. And we have some equations similar to the following:
$x_1 + x_2 + \dots + x_n = a_1$,
$x_1.x_2 + x_3.x_4 + \dots + x_{n-1}.x_n = a_2$,
$\dots$
$(1-x_1).x_2.x_3.(1-x_4) + (1-x_5).x_6.x_7.(1-x_8) + \dots = a_i$,
$\dots$
In linear equations systems, we can determine whether there is a solution or not based on number of unknowns vs. equations.
What I wonder is, is there a rule, similar to the linear equations for above kind of equations? If not, how should I approach solving these kinds of equations?
Thank you