Lect12

Lect12 - 6.841 Advanced Complexity Theory March 19, 2007...

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Unformatted text preview: 6.841 Advanced Complexity Theory March 19, 2007 Lecture 12 Lecturer: Madhu Sudan Scribe: Suho Oh, Jose Soto 1 Overview Randomized Reductions. Valiant-Vazirani: SAT RP Unique- SAT . Todas Theorem: PH P # P . 2 The Theorem of Valiant-Vazirani. To state this theorem we will need some definitions first: Definition 1 ( Unique- SAT promise problem) . Unique- SAT = ( U Y ES ,U NO ) . U Y ES = { | has 1 satisfying assignment } . U NO = { | has 0 satisfying assignment } . Definition 2 (Randomized Reductions) Given two promise problems = ( Y ES , NO ) and = ( Y ES , NO ) . We say that reduces to under a BP randomized reduction BP if there exists a probabilistic polynomial time algorithm A , a polynomial p ( n ) and a polynomial time computable function s ( n ) such that: x Y ES = A ( x ) Y ES w.p. s ( n ) + 1 p ( n ) . x NO = A ( x ) negationslash NO w.p. s ( n ) . [ A ( x ) NO w.p. 1 s ( n )] . When s ( n ) = 0 we say that it is a RP randomized reduction and we denote it by RP . Using the previous definition we can state the theorem as follows: Theorem 1 (Valiant-Vazirani) SAT RP Unique- SAT. To find an RP reduction a natural idea is to map an instance ( x ) of SAT into a new formula ( x ) = ( x ) f ( x ), where f ( x ) is a sufficiently nice formula. In that way if ( x ) SAT NO then we would know that ( x ) has no satisfying assignment, and so ( x ) U NO . The problem is to determine a nice f ( x ) such that if SAT Y ES , then ( x ) has exactly one satisfying assignment with enough probability....
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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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Lect12 - 6.841 Advanced Complexity Theory March 19, 2007...

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