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lec15 - Arthur-Merlin Games IP permits verifier to keep...

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1 CS151 Complexity Theory Lecture 15 May 17, 2011 May 17, 2011 2 Arthur-Merlin Games IP permits verifier to keep coin-flips private necessary feature? GNI protocol breaks without it Arthur-Merlin game : interactive protocol in which coin-flips are public Arthur (verifier) may as well just send results of coin-flips and ask Merlin (prover) to perform any computation Arthur would have done May 17, 2011 3 Arthur-Merlin Games Clearly Arthur-Merlin IP – “private coins are at least as powerful as public coins” Proof that IP = PSPACE actually shows PSPACE Arthur-Merlin IP PSPACE – “public coins are at least as powerful as private coins” ! May 17, 2011 4 Arthur-Merlin Games Delimiting # of rounds: AM[k] = Arthur-Merlin game with k rounds, Arthur (verifier) goes first MA[k] = Arthur-Merlin game with k rounds, Merlin (prover) goes first Theorem : AM[k] ( MA[k] ) equals AM[k] ( MA[k] ) with perfect completeness. Theorem : for all constant k 2 AM[k] = AM[2] . May 17, 2011 5 MA and AM Two important classes: MA = MA[2] AM = AM[2] definitions without reference to interaction: L MA iff 9 poly-time language R x L 9 m Pr r [(x, m, r) R] = 1 x L 8 m Pr r [(x, m, r) R] ½ L AM iff 9 poly-time language R x L Pr r [ 9 m (x, m, r) R] = 1 x L Pr r [ 9 m (x, m, r) R] ½ May 17, 2011 6 MA and AM P NP coNP Σ 2 Π 2 AM coAM MA coMA
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