PCP theorem

The PCP theorem states that every language in NP has proofs verifiable using logarithmically many random bits and a constant number of queries to a proof.

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Probabilistically checkable proof: A proof encoded so a probabilistic verifier can check its validity by reading only a small number of locations. The theorem guarantees such proof systems for every language in NP.
Hardness of approximation: The study of limits on how closely computationally difficult optimization problems can be approximated efficiently. PCP-based reductions establish sharp limits on approximation algorithms.
NP: The class of decision problems whose yes-instances have polynomial-size certificates verifiable in polynomial time. The theorem says every language in this class admits a PCP verifier.
Sanjeev Arora: An Indian-American theoretical computer scientist known for work on probabilistically checkable proofs and approximation algorithms. He co-authored the 1992 proof of the PCP theorem.
Arthur–Merlin protocol: An interactive proof system in which a probabilistic verifier and an all-powerful prover exchange messages. Its communication model differs from a verifier’s static, locally queried proof.
PCP verifier: A randomized algorithm that checks a proof by querying selected positions and accepting or rejecting. Its randomness and query limits state the theorem’s central resource bounds.
Maximum satisfiability: The optimization problem of finding an assignment that satisfies as many Boolean clauses as possible. PCP reductions prove that approximating its optimum is difficult.
P: The class of decision problems solvable by a deterministic algorithm in polynomial time. PCP verification uses randomness and proof access rather than requiring an ordinary polynomial-time decision procedure.
Shmuel Safra: An Israeli theoretical computer scientist whose work includes probabilistically checkable proofs and computational complexity. His 1992 work with Arora established the theorem’s central characterization.
Probabilistically checkable argument: A proof system whose prover is computationally bounded and whose verifier checks a proof probabilistically. It relaxes the PCP model by restricting the prover’s computational power.