Probabilistically checkable proof

In computational complexity theory, a probabilistically checkable proof (PCP) is a type of 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), as used in the verifier-based definition of the complexity class 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 PCP[r(n),q(n)] refers to the set of decision problems 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. 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 PCP[O(log n),O(1)] = NP.

The complexity class PCP is the class of decision problems that have probabilistically checkable proofs with completeness 1, soundness α < 1/2, randomness complexity O(log n) and query complexity O(1).

A probabilistically checkable proof system with completeness c(n) and soundness s(n) over alphabet Σ for a decision problem L, where 0 ≤ s(n) ≤ c(n) ≤ 1, is a randomized oracle Turing Machine V (the verifier) that, on input x and oracle access to a string π ∈ Σ^* (the proof), satisfies the following properties:

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