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lect18 - 6.841 Advanced Complexity Theory Lecture 18...

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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 Goldwasser-Micali-Rackoff and Babai. Goldwasser- Micali-Wigderson were focused on the cryptographic implications of this technique; they are interested in zero-knowledge 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 one-way functions. Another development on this line was made by BenOr-Goldwasser-Kilian-Wigderson 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 information-exchange is allowed between two provers. The verifier can them questions in any order. The authors showed that with 2 provers, zero-knowledge proofs are possible without assuming the existence of one-way 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 Fortnow-Rompel-Sipser. 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 query-history) when the verifier query them. Now we consider the difference between OIP and NEXPT IME . They both have an input x of n bits and a proof(certificate) π of length 2 n c for some constant c . The difference relies on the verifier V . If L NEXPT IME , there exists a deterministic-exptime V such that If x L , then there exists a π such that V ( x, π ) accepts, If x / L , then π , V ( x, π ) rejects. Similarly, if L OIP , there exists a randomized-polytime V such that If x L , then there exists a π such that V ( x, π ) accepts with high probability, If x / L , then π , V ( x, π ) accepts with low probability. Hence the relation OIP NEXPT IME is obvious since one can build the verifier for NEXPT IME by simulating all random coins of V in OIP . More interestingly, Babai-Fortnow-Lund showed the other direction OIP NEXPT IME . Using this relation MIP = OIP = NEXPT IME , Feige-Goldwasser-
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