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Unformatted text preview: Notes on Complexity Theory Last updated: November, 2011 Lecture 22 Jonathan Katz 1 NP PCP ( poly , 1) We show here a probabilistically checkable proof for NP in which the verifier reads only a constant number of bits from the proof (and uses only polynomially many random bits). In addition to being of independent interest, this result is used as a key step in the proof of the PCP theorem itself. To show the desired result, we will work with the NPcomplete language of satisfiable quadratic equations . Instances of this problem consist of a system of m quadratic equations n X i,j =1 c ( k ) i,j x i x j = c ( k ) m k =1 (1) (over the field F 2 ) in the n variables x 1 ,...,x n . (Note that we can assume no linear terms since x i = x i x i in F 2 and the summations above include the case i = j .) A system of the above form is said to be satisfiable if there is an assignment to the { x i } for which every equation is satisfied. It is obvious that this problem is in NP . To show that it is NPcomplete we reduce an instance of 3 SAT to an instance of the above. Given a 3 SAT formula on n variables, using arithmetization we can express each of its clauses as a cubic equation. (One way to do this is as follows: arithmetize the literal x j by the term 1 x j and the literal x j by the term x j ; a clause 1 2 3 is arithmetized by the product of the arithmetization of its literals. Then ask whether there is an assignment under which the arithmetization of all the clauses of equal 0.) To reduce the degree to quadratic, we introduce the dummy variables { x i,j } n i,j =1 and then: (1) replace monomials of the form x i x j x k with a monomial of the form x i,j x k , and (2) introduce n 2 new equations of the form x i,j x i x j = 0. We remark that there is no hope of reducing the degree further (unless P = NP ) since a system of linear equations can be solved using standard linear algebra. 1.1 A PCP for Satisfiable Quadratic Equations: An Overview For the remainder of these notes, we will assume a system of m equations in the n variables { x i } , as in Eq. (1). The proof string will be a boolean string { , 1 } n 2 that we index by a binary vector ~v of length n 2 . Equivalently, we will view as a function : { , 1 } n 2 { , 1 } . For a given system of satisfiable quadratic equations, should be such that ( ~v ) def = n X i,j =1 a i a j v i,j for some satisfying assignment ( a 1 ,...,a n ), where ~v = ( v 1 , 1 ,...,v 1 ,n ,...,v n, 1 ,...,v n,n ) Note that with ( a 1 ,...,a n ) fixed, is a linear function of ~v ; i.e., is just the dot product of the input with the fixed string ( a 1 , 1 ,...,a n,n )....
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This note was uploaded on 01/13/2012 for the course CMSC 652 taught by Professor Staff during the Fall '08 term at Maryland.
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