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lect15 - 6.841 Advanced Complexity Theory April 1, 2009...

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Unformatted text preview: 6.841 Advanced Complexity Theory April 1, 2009 Lecture 15 Lecturer: Madhu Sudan Scribe: Rotem Oshman 1 Todays topics Private coins public coins (that is, IP[ k ] AM) Goldwasser-Sipser approximate counting protocol Towards protocols for the Permanent, in the goal of showing that #P IP. 2 Review of last lecture 2.1 Graph Non-Isomorphism The graph non-isomorphism problem is to decide the language GNI = { ( G ,G 1 ) | G negationslash G 1 } . Last lecture we saw a private-coin 2-round protocol for GNI: The Verifier chooses a permutation R S n , where n is the number of nodes in the graph, and a bit b R { , 1 } . It sends H = ( G b ) to the Prover. The Prover returns with a bit b prime . The Verifier accepts if b = b prime . If G G 1 , then the bit b is independent of H , which means that the Prover is essentially guessing b prime and has a probability of 1 / 2 of being correct. If G negationslash G 1 , then H identifies b uniquely and the Prover will always be correct. 2.2 Kilians protocol: IP[ k ] AM[poly] Last time we saw a public-coin protocol through which the Prover could convince the Verifier that there are many coin tosses that would have made the Verifier accept in the original private-coin protocol. Kilians protocol is public coin and has completeness of 1, but the number of rounds in the protocol depends on the number of random coins used in the original private-coin protocol. For example, the public-coin protocol we would get for GNI would have O ( n log n ) rounds instead of 2. 15-1 3 The Goldwasser-Sipser Protocol Let S { , 1 } n be a set, such that membership in S is verifiable in AM. We are interested in solving the promise problem given by YES = { S : | S | f ( n ) } NO = braceleftbigg S : | S | < f ( n ) 10 n 2 bracerightbigg 3.1 Initial attempt Suppose that f ( n ) is very large: f ( n ) 2 n (actually a little less than that). In this case, for YES instances we have | S | |{ , 1 } n | 1, and for NO instances we have | S | |{ , 1 } n | < 1 n 2 . In other words, when we choose a random string, for YES instances our chance of hitting a member of...
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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.

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lect15 - 6.841 Advanced Complexity Theory April 1, 2009...

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