I read in these two papers http://www.ccs.neu.edu/home/lieber/courses/csg260/f06/materials/papers/max-sat/p216-schaefer.pdf and http://people.csail.mit.edu/madhu/papers/noneed/fullbook.ps that if we have a boolean formula that is $0-valid$ then (of course) SAT problem is in $\mathcal{P}$ but finding a solution with maximum true literals is $\mathcal{NP}-$hard.
N.B. As defined in the previous papers, a $0-valid$ boolean formula $f$ is a boolean formula $f: \{0, 1\}^n\rightarrow\{0, 1\}$ that satisfies $f(0, \dotsc, 0)=1$.
My question is:
Can I represent a general $0-valid$ boolean formula on the variables $x=\left(x_1, \dotsc, x_n\right)$ by the following one:
$f(x)=\bigwedge\limits_{i=1}^{L}\bigvee\limits_{i\in\mathcal{S}_l}\neg\;x_i$?
Where $L$ is the number of clauses and $\mathcal{S}_l\;\forall\;l\in\{1,\dotsc, L\}$ is a subset of $\{1, \dotsc, n\}$.