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PCP Theorem


The PCP theorem states that every NP-problem has a probabilistically checkable proof system whose randomized verifier uses O(logn) random bits and reads only O(1) bits of the proof, with a constant upper bound on the probability of accepting an incorrect proof. In complexity notation, the theorem is

 NP=PCP(O(logn),O(1)).

It is a fundamental source of results showing that approximation problems are NP-hard.


See also

Closest Vector Problem, NP-Hard Problem, NP-Problem, Proof, Projection Games Conjecture, Satisfiability Problem

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References

Arora, S.; Lund, C.; Motwani, R.; Sudan, M.; and Szegedy, M. "Proof Verification and the Hardness of Approximation Problems." J. ACM 45, 501-555, 1998. https://doi.org/10.1145/278298.278306.Arora, S. and Safra, S. "Probabilistic Checking of Proofs: A New Characterization of NP." J. ACM 45, 70-122, 1998. https://doi.org/10.1145/273865.273901.

Cite this as:

Weisstein, Eric W. "PCP Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PCPTheorem.html

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