# Can one use the PCP theorem to prove correctness of deternimistic algorithms?

I am thinking of the equality "PCP(O(log(n)),0) = P"

Say I have a deterministic polynomial time algorithm $A$ whose correctness I can't prove immediately. But say I create a probabilistic version of this algorithm say $A_P$ such that it uses log(n) random bits. Now if I prove somekind of (exactly what ?) expectation correctness of $A_P$ then does it imply that $A$ was correct?

If the above is not right then I would like if someone can point me to what is the closest correct thing to this!

The statement you mention is much easier to prove. It says in effect that a polytime algorithm that uses $O(\log n)$ random bits can be made deterministic while keeping it polytime (given reasonable completeness and soundness properties). The proof is very simple – we can run the algorithm over the $2^{O(\log n)} = n^{O(1)}$ many choices of random bits, check how many times the algorithm accepts, and so determine whether the input belongs to the language.