PCP theorem states that $$PCP(O(\log n),O(1))=NP.$$ Could we not run through $O(\log n)$ bits deterministically?
Does PCP theorem statement mean any set of $O(\log n)$ random bits out of $2^{O(\log n)}$ choices of such bits suffice? If so, why cannot I choose all $0$ bits? If not why not?
The statement of the PCP theorem is essentially equivalent to the following statement:
For every problem in NP there is a reduction $f$ to $O(1)$-FUNCTION-SAT such that if $x$ is a YES instance then $f(x)$ is satisfiable, whereas if $x$ is a NO instance then at most half of the clauses of $f(x)$ can be satisfied.
Here an instance of $k$-FUNCTION-SAT consists of a collection of "clauses", each an arbitrary function on $k$ inputs.
Where did the parameters $O(\log n),O(1)$ in the PCP theorem disappear? The first parameter is the logarithm of the number of clauses, so there are $2^{O(\log n)}$ clauses; the second parameter is the arity of the clauses. Together, these parameters guarantee that the FUNCTION-SAT instance has polynomial size, and so the reduction runs in polynomial time.
PCPs are not always described this way, but the descriptions are equivalent. Proof of the equivalence left to the reader.
You are enumerating over all choices of $O(\log n)$ when you are constructing the FUNCTION-SAT instance. But it only allows you to construct the instance, not to tell whether it is satisfiable or not; that is an NP task.
The first parameter is the number of random bits, and the second the number of queries to the oracle string $\Pi$ the verifier may use.
The length of $\Pi$ is not bounded; therefore, having only access to $O(1)$ symbols of $\Pi$ (and being allowed only relatively few random bits for choosing which) is a non-trivial restriction.
See here for an example illustrating this.
there are multiple comments stating the question is "uninformed" etc and makes no logical sense and suggesting "read a survey on PCP" but not actually (helpfully) volunteering/ citing any. here is a nice one run across by Ryan O'Donnell an expert on boolean functions. its more of an overview but states the basics, possibly enough to remove some beginner misconceptions, and points to many other papers with more details. consider this something like a "hint" answer.
