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# lecture22 - Notes on Complexity Theory Last updated...

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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 NP-complete 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 NP-complete we reduce an instance of 3 SAT to an instance of the above. Given a 3 SAT formula φ 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 φ 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 π will be a boolean string π ∈ { , 1 } n 2 that we index by a binary vector ~v of length n 2 . Equivalently, we will view π as a function π : { , 1 } n 2 → { , 1 } . For a given system of satisfiable quadratic equations, π should be such that π ( ~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, π is a linear function of ~v ; i.e., π is just the dot product of the input with the fixed string ( a 1 , 1 ,...,a n,n )....
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