3-CNF to "independent form"
Is it possible to convert all logical formulae into a form such that each variable ends up in exactly 1 "factor" of the and operation? ($\land$)


In general, it is impossible to separate variables in that way. Consider the formula $a \leftrightarrow b$, and assume it can be written in the form $f(a) \land g(b)$.

Up to logical equivalence, $f(a)$ must be either $a$ or $\lnot a$. Ditto for $g(b)$. We therefore have that $f(a) \land g(b)$ must be one of these formulas:

$$
a\land b \qquad
a\land \lnot b \qquad
\lnot a\land b \qquad
\lnot a\land \lnot b \qquad
$$

However, none of the above ones is equivalent to $a \leftrightarrow b$.

(Well, to be pedantic, $f(a)$ and $g(b)$ could also be the constantly true or constantly false function. Those choices won't work either, obviously.)