lec17 - CS151 Complexity Theory Lecture 17 May 24, 2011 The...

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CS151 Complexity Theory Lecture 17 May 24, 2011
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May 24, 2011 2 The outer verifier Theorem : NP PCP[log n, polylog n] Proof (first steps): define: Polynomial Constraint Satisfaction (PCS) problem prove: PCS gap problem is NP -hard
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May 24, 2011 3 NP PCP[log n, polylog n] algebraic version: MAX- k -PCS given: • variables x 1 , x 2 , …, x n taking values from field F q n = q m for some integer m • k -ary constraints C 1 , C 2 , …, C t – assignment viewed as f:(F q ) m F q output: max. # of constraints simultaneously satisfiable by an assignment that has deg. ≤ d
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May 24, 2011 4 NP PCP[log n, polylog n] MAX- k -PCS gap problem : given: • variables x 1 , x 2 , …, x n taking values from field F q n = q m for some integer m • k -ary constraints C 1 , C 2 , …, C t – assignment viewed as f:(F q ) m F q YES: some degree d assignment satisfies all constraints NO: no degree d assignment satisfies more than (1- ε ) fraction of constraints
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May 24, 2011 5 NP PCP[log n, polylog n] Lemma : for every constant 1 > ε > 0, the MAX- k -PCS gap problem with t = poly(n) k-ary constraints with k = polylog(n) field size q = polylog(n) n = q m variables with m = O(log n / loglog n) degree of assignments d = polylog(n) gap ε is NP -hard.
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May 24, 2011 6 NP PCP[log n, polylog n] t = poly(n) k-ary constraints with k = polylog(n) field size q = polylog(n) n = q m variables with m = O(log n / loglog n) degree of assignments d = polylog(n) check: headed in right direction O(log n) random bits to pick a constraint query assignment in O(polylog(n)) locations to determine if it is satisfied completeness 1; soundness 1- ε (if prover keeps promise to supply degree d polynomial)
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May 24, 2011 7 NP PCP[log n, polylog n] Low-degree testing: want: randomized procedure that is given d , oracle access to f:(F q ) m F q runs in poly(m, d) time always accepts if deg(f) ≤ d rejects with high probability if deg(f) > d too much to ask. Why?
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May 24, 2011 8 NP PCP[log n, polylog n] Definition : functions f, g are δ-close if Pr x [f(x) ≠ g(x)] δ Lemma : 5 δ > 0 and a randomized procedure that is given d , oracle access to f:(F q ) m F q runs in poly(m, d) time – uses O(m log |F q |) random bits always accepts if deg(f) ≤ d rejects with high probability if f is not δ-close to any g with deg(g) ≤ d
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May 24, 2011 9 NP PCP[log n, polylog n] idea of proof: restrict to random line L check if it is low degree always accepts if deg(f) ≤ d other direction much more complex ( F q ) m
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10 NP PCP[log n, polylog n] can only force prover to supply function f that is close to a low-degree polynomial how to bridge the gap? recall low-degree polynomials form an
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lec17 - CS151 Complexity Theory Lecture 17 May 24, 2011 The...

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