Editing Probabilistically checkable proof (section)

{{More footnotes|date=November 2012}}

In [[computational complexity theory]], a '''probabilistically checkable proof''' ('''PCP''') is a type of [[mathematical proof|proof]] that can be checked by a [[randomized algorithm]] using a bounded amount of randomness and reading a bounded number of bits of the proof. The algorithm is then required to accept correct proofs and reject incorrect proofs with very high probability. A standard proof (or [[Certificate (complexity)|certificate]]), as used in the [[Formal verification|verifier]]-based definition of the [[complexity class]] [[NP (complexity)|NP]], also satisfies these requirements, since the checking procedure deterministically reads the whole proof, always accepts correct proofs and rejects incorrect proofs. However, what makes them interesting is the existence of probabilistically checkable proofs that can be checked by reading only a few bits of the proof using randomness in an essential way.

Probabilistically checkable proofs give rise to many complexity classes depending on the number of queries required and the amount of randomness used. The class {{math|1={{sans-serif|PCP}}[''r''(''n''),''q''(''n'')]}} refers to the set of [[decision problem]]s that have probabilistically checkable proofs that can be verified in polynomial time using at most ''r''(''n'') random bits and by reading at most ''q''(''n'') bits of the proof.{{r|ab07}} Unless specified otherwise, correct proofs should always be accepted, and incorrect proofs should be rejected with probability greater than 1/2. The [[PCP theorem]], a major result in computational complexity theory, states that {{math|1={{sans-serif|PCP}}[''O''(log ''n''),''O''(1)] = {{sans-serif|NP}}}}.