User:Devanden/PCP Theorem

Overview
The PCP Theorem is a theorem in Computational complexity theory

Theorem
$$NP = PCP(O(\log n), O(1)) \ $$.

Proof.
The proof is in two parts: First, prove that $$NP \subset PCP(O(\log n),O(1)) \ $$ and then that $$PCP(O(\log n),O(1)) \subset NP \ $$.

If $$L \in \mbox{PCP}(O(\log n),O(1))$$, then $$L \le ( \mbox{SAT}, \epsilon-\mbox{UNSAT})$$.

Proof. Construct a 3CNF from the PCP. Variables: each bit in the proof $$y_1, y_2, \cdots, y_m$$. Encode the clauses when the verifier accepts. For each random string r, confuct a sub-formula $$\phi_r$$. Then $$\phi=\bigwedge_r \phi_r$$. Fixed r, can determine all possible variables queried $$(\le 2^q)$$ from the depth q dicision tree.