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hwsol5.1 - leave it to Arthur to verify in polynomial time...

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Izaksonas-Smith, Lofgren, Musbach CSci 5403— Homework 5 Problem 1 Suppose that L BP · NP , meaning there is a polynomial-time computable function f such that for any x L , Pr r [ f x ; r ) 3SAT ] 2 / 3 and for any x / L , Pr r [ f x ; r ) 3SAT ] 1 / 3. Then Arthur can send a random string r to Merlin of necessity) and accept iF Merlin’s reply certi±es that f x ; r ) 3SAT . We have completeness because Merlin is capable of generating a certi±cate for f x ; r ) if it is in 3SAT , which happens with probability at least 2 / 3 if x L . We have soundness because Merlin is incapable of generating a certi±cate for f x ; r ) if it is not in 3SAT , which happens with probability at least 2 / 3 if x / L . Conversely, suppose that L AM . Let L T be the set of pairs x r ) such that after Arthur says r , Merlin can convince Arthur to accept. Then L T NP because Merlin’s argument serves as a certi±cate just
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Unformatted text preview: leave it to Arthur to verify in polynomial time). Since Arthur runs in polynomial time, by the Cook-Levin reduction there is a 3SAT formula φ x r with free variables a such that φ x r a ) iF Arthur accepts Merlin’s argument a , and L T = x r ) : ∃ aφ x r a ) } . Let f x ; r ) = φ x r . If x ∈ L , then with probability at least 2 / 3, Merlin can convince Arthur to accept, meaning x r ) ∈ L T and φ x r ∈ 3SAT . In other words, if x ∈ L then Pr r [ f x ; r ) ∈ 3SAT ] ≥ 2 / 3. If x / ∈ L , then with probability at most 1 / 3, Merlin can convince Arthur to accept, meaning x r ) ∈ L T . In other words, if x / ∈ L then Pr r [ f x ; r ) ∈ 3SAT ] ≤ 1 / 3....
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