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In computational complexity theory, the PCP theorem states that every decision problem in the NP complexity class has probabilistically checkable proofs of constant query complexity and logarithmic randomness complexity.
The PCP theorem says that for some universal constant K, for every n, any mathematical proof for a statement of length n can be rewritten as a different proof of length poly that is formally verifiable with 99% accuracy by a randomized algorithm that inspects only K letters of that proof.
The PCP theorem is the cornerstone of the theory of computational hardness of approximation, which investigates the inherent difficulty in designing efficient approximation algorithms for various optimization problems. It has been described by Ingo Wegener as "the most important result in complexity theory since Cook's theorem" and by Oded Goldreich as "a culmination of a sequence of impressive works rich in innovative ideas".