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Unformatted text preview: 6.841 Advanced Complexity Theory April 8, 2009 Lecture 17 Lecturer: Madhu Sudan Scribe: Jean Yang 1 Overview Last lecture we showed PERM ∈ IP ⇒ # P ∈ IP by constructing an IP protocol involving verifying a polynomial using a curve. In this lecture we review the method for showing PERM ∈ IP and apply the method first for showing # SAT ∈ IP and then for showing PSPACE ⊆ IP . In the second half of lecture we discussed “knowledge,” its definition, and its relationship to crytography and complexity. 1.1 Administrative notes • No lecture on Monday. • Madhu will be away–Swastik will lecture on PCP. 2 Review from last time: # P ∈ IP We showed that PERM ∈ IP problem using a polynomial construction sequence. A cool part is that this method would work for any problem that we can deconstruct into a sequence of polynomials, so we can also apply this method to showing # SAT ∈ IP and, eventually, PSPACE ⊆ IP . 2.1 Polynomial construction sequence Recall from last lecture that we can compute the permanent in field F using the sequence of n polynomials P 1 ,P 2 ,...,P ‘- 1 ,P ‘ where P i is the permanent of an i × i matrix. We have the following properties: Each polynomial of degree ≤ d Each polynomial has number of variables ≤ m P is computable in time ≤ t P i is computable in time t with oracle for P i- 1 with # calls ≤ w = n We get the last property from the downward self-reducibility of the permanent: P n ( M ) = ∑ n i =1 m 1 i P n- 1 ( M i ). Given a = ( a 1 ,...,a m ) ∈ F m and b ∈ F , we can prove interactively that P ‘ ( a ) = b in time polynomial in ‘,d, F ,m,t,w ), provided that | F | is sufficiently large. We showed in the previous lecture a method for verifying PERM by running a curve through the space and then asking about a random point on the curve. We can work backwards from P n- 1 . 2.2 Typical phase of interaction We show how to verify the question “ P i ( a ( i ) ) = b ( i ) ?” for polynomial P i in a sequence of polynomials. To verify P n ( M ) = a , we can compute the w inputs M 1 ,M 2 ,...,M w to use the oracle for P n- 1 so that we can easily compute P n ( M ) from P n- 1 ( M ) ...P n- 1 ( M w ). For verification purposes we use a curve C : Z p → Z v p (where Z v p is our input space) such that C (1) = M 1 ,...,C ( w ) = M w . Note deg( C ) ≤ w (actually deg( C ) = w- 1). The interaction goes as follows: • We ask the prover “ P n- 1 ◦ C =?” 17-1 • The prover comes back with an answer h with degree ≤ d · w . • The verifier verifies that “ P n ( M ) = a ” is consistent with P n- 1 ( M i ) = h ( i ) by picking a t at random and verifies P n- 1 ( C ( t )) = h ( t )....
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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.
- Spring '09