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I'm studying the PCP theorem.

While it is easy to prove that $\mathsf{P}=\text{PCP}(O(\log n),0)$ , proving that $\text{PCP}(O(\log n),1)\subseteq \mathsf{P}$ i.e. PCP that uses $O(\log n)$ random bits and read 1 bit of the proof is less obvious, what I tried to do is to take some proof $\pi$ of length $n^{O(1)}$ (because effectively the message sent by the prover is bounded by $2^{r(n)}q(n)=2^{O(\log n)}=n^{O(1)}$) then try all the coin tosses each time the verifier read some bit of the message so if the proof is not correct we flip the bit in the proof!

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By the basic equivalence of PCPs and MaxCSP problems, there exists a conjunction $C$ of polynomially many boolean literals, s.t.

$\exists \text{ proof } \pi$ which makes the verifier $V$ accept with probability $\geq p$ $\Leftrightarrow$ $\exists$ an assignment that satisfies at least a $p$ fraction of the literals in $C$.

So you just need to show that the following problem is polytime solvable: given a conjunction $C$ of boolean literals, what is the largest number of literals that can be simultaneously satisfied. That's a very easy problem.

The idea here is that a PCP with query complexity $q$ corresponds to some MaxCSP problem with arity $q$. If the PCP has completeness $c$ and soundness $s$, then the language accepted by the PCP is reducible to a promise version of an arrity $q$ MaxCSP problem where in a 'no' instance at most an $s$ fraction of the constraints can be satisfied, and in a 'yes' instance at least a $c$ fraction of the constraints can be satisfied. These reductions are standard, look up any reference on PCPs.

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  • $\begingroup$ thank you for the great help ... I am trying to understand these things and you help me a lot ... although I knew the reduction but I didn't know it's usefulness thank to you know I understand it very good ... $\endgroup$ – Fayez Abdlrazaq Deab May 26 '13 at 9:22
  • $\begingroup$ I am glad you found this helpful! $\endgroup$ – Sasho Nikolov May 26 '13 at 22:15
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The PCP theorem states that $\mathrm{NP}=\mathrm{PCP}[O(\log n),O(1)]$. So showing that $\mathrm{PCP}[O(\log n), O(1)] \subseteq \mathrm{P}$ should be pretty difficult...

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  • $\begingroup$ I didn't mean O(1) but just 1 query $\endgroup$ – Fayez Abdlrazaq Deab May 25 '13 at 21:48

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