We know that $3SAT \in NP$, and the definition of $NP$ is as follows:
$NP$ is the class of languages that have polynomial time verifiers.
But I have a question:
what does input mean for the $3SAT$ question? Is it the number of variables $n$ or the number of clauses $m$,
Let's take an example, given a CNF expression, whose clauses has exactly 3 literals. The CNF expression consists of $m$ clauses involving $n$ variables $x_1, ..., x_n$.
If the input refers to the number of variables $n$, and the Boolean formula has $2^n$ clauses, it is obvious that an assignment of the validation formula requires $3*2^n$ steps, not polynomial time.This contradicts that $3SAT \in NP$
If the input refers to the number of clauses $m$, an assignment of the validation formula requires 3*m steps, which is polynomial time, which makes sense
So I think for the 3sat problem, the input refers to the number of clauses $m$, Is there a problem with my idea?