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lecture18 - INTERACTIVE PROOFS CSci 5403 COMPLEXITY THEORY...

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1 COMPLEXITY THEORY CSci 5403 LECTURE XVIII: INTERACTIVE PROOFS II INTERACTIVE PROOFS Definition. An interactive proof system has two players: Arbitrary Prover P : Σ * Σ * α 1 = V(x;r) α 2 = P(x, α 1 ) α k = P(x, α 1 , α 2 ,…, α k-1 ) α k+1 = accept/reject The conversation (V P)(x;r) is α 1 , α 2 ,…, α k+1 . α k-1 = V(x, α 1 , α 2 ,…, α k-2 ;r) We say that (V P)(x;r) has k rounds. (V P)(x;r) accepts if α k+1 = accept PPT Verifier V : accept, reject Σ * Σ * ˫ { } INTERACTIVE PROOFS Definition. (P,V) is an interactive proof system for L 1. x ˥ L ˰ Pr[V accepts x] 2/3 2. x L ˰ Pr[V accepts x] 1/3 Definition. L ˥ IP[k] if ˳ k(n)-round interactive proof system (P,V) for L and constant c such that ˲ x,r |(P V)(x;r)| |x| c Definition. IP = ˫ c IP[n c ] Definition. Pr[(P V)(x) accepts] = Pr r [(P V)(x;r) accepts] Pr[V accepts x] = max P Pr[(P V)(x) accepts] (Completeness) (Soundness) ARTHUR-MERLIN PROOFS Merlin Arthur x ˥ L α 1 = random r 1 α 2 = P(x,r 1 ) α 3 = random r 2 α k = P(x, α 1 ,…, α k-1 ) V(x, α 1 ,…, α k ) = accept/reject Definition. L ˥ AM[k] if there exists k(n)-round Arthur-Merlin interactive proof system for L. Definition. AM = AM[2]. MA = MA[2] An Arthur-Merlin verifier has no private randomness:

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2 Theorem. For every ƒ, IP[2ƒ(n)] ˧ AM[2ƒ(n)+3].
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This note was uploaded on 10/21/2011 for the course CSCI 5403 taught by Professor Sturtivant,c during the Spring '08 term at Minnesota.

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lecture18 - INTERACTIVE PROOFS CSci 5403 COMPLEXITY THEORY...

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