what does **input** mean for the $3SAT$ question? Is it the number of variables $n$ or the number of clauses $m$

Question

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？

Answer 1

Neither. The input is the Boolean formula $\phi$. You are probably asking what is the length of the input, as computational complexity tries to express the running time of the algorithm as a function of the length of the input. The length of $\phi$ depends on how it is encoded, but typically the length will depend on both $n$ and $m$. A reasonable estimate, for a typical encoding, is $\Theta(m \log n)$. If you want to think of the length of the input as $m$ that is a plausible approximation that is likely to be good enough for many purposes. As long as you use a reasonable encoding, the exact details are unlikely to change whether an algorithm is polynomial-time or not.

Answer 2

The input is just a Boolean formula $\phi$.

$|\phi|$ depends on the way in which $\phi$ has been encoded.