Given
$\mathbb{B} = \{true, false\}$,
and function $f: \mathbb{B} \times \mathbb{B} \times \mathbb{B} \to \mathbb{B}, f(a,b,c) = a \land b \lor c,~ \forall a,b,c \in \mathbb{B}$.
I want to find a function $g: \mathbb{B} \times \mathbb{B} \times \mathbb{B} \times \mathbb{B} \times \mathbb{B} \times \mathbb{B} \to \mathbb{B}$ such that $g(a,b,c,\neg a, \neg b, \neg c) = \neg f(a,b,c) ~\forall a,b,c \in \mathbb{B}$.
A significant constraint is that $g$ may only use function $f$ (with parameters $a,b,c,\neg a, \neg b, \neg c$), constants $true$ and $false$, logical conjunction ($\land$) and disjunction ($\lor$). In particular, negation ($\neg$) and input parameters $a,b,c,\neg a, \neg b, \neg c$ must not be used (directly).
Is there any systematic way to find such a function $g$? How can I know whether such a function exists at all?
Sorry for the bad title of this question. I didn't know better.