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Unformatted text preview: 6.841 Advanced Complexity Theory Apr 13, 2009 Lecture 18 Lecturer: Madhu Sudan Scribe: Jinwoo Shin In this lecture, we will discuss about PCP ( probabilistically checkable proof ). We first overview the history followed by the formal definition of PCP and then study its connection to inapproximability results. 1 History of PCP Interactive proofs were first independently investigated by GoldwasserMicaliRackoffand Babai. Goldwasser MicaliWigderson were focused on the cryptographic implications of this technique; they are interested in zeroknowledge proofs, which prove that one has a solution of a hard problem without revealing the solution itself. Their approach is based on the existence of oneway functions. Another development on this line was made by BenOrGoldwasserKilianWigderson who introduced several provers. 2 IP has two provers who may have agreed on their strategies before the interactive proof starts. However, during the interactive proof, no informationexchange is allowed between two provers. The verifier can them questions in any order. The authors showed that with 2 provers, zeroknowledge proofs are possible without assuming the existence of oneway functions. Clearly, 2 IP is at least as powerful as IP since the verifier could just ignore the second prover. Further more, one can define 3 IP , 4 IP , ...,MIP and naturally ask whether those classes get progressively more powerful. It turns out that the answer is no by FortnowRompelSipser. They proved that 2 IP = 3 IP = ... = MIP = OIP , where OIP is called oracle interactive proof . OIP is an interactive proof system with a memoryless oracle; all answers are committed to the oracle before the interactive proof starts and the oracle just answer them (without considering the previous queryhistory) when the verifier query them. Now we consider the difference between OIP and NEXPTIME . They both have an input x of n bits and a proof(certificate) of length 2 n c for some constant c . The difference relies on the verifier V . If L NEXPTIME , there exists a deterministicexptime V such that If x L , then there exists a such that V ( x, ) accepts, If x / L , then , V ( x, ) rejects. Similarly, if L OIP , there exists a randomizedpolytime V such that If x L , then there exists a such that V ( x, ) accepts with high probability, If x / L , then , V ( x, ) accepts with low probability. Hence the relation OIP NEXPTIME is obvious since one can build the verifier for NEXPTIME by simulating all random coins of V in OIP . More interestingly, BabaiFortnowLund showed the other direction OIP NEXPTIME . Using this relation MIP = OIP = NEXPTIME , FeigeGoldwasser LovaszSafraSzegedy proved that the maximum cliquesize of a graph is hard to approximate and later...
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This note was uploaded on 04/02/2010 for the course CS 6.841 taught by Professor Madhusudan during the Spring '09 term at MIT.
 Spring '09
 MadhuSudan

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