lec17 - The outer verifier Theorem: NP CS151 Complexity...

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1 CS151 Complexity Theory Lecture 17 May 24, 2011 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 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 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- ) fraction of constraints 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 is NP -hard. 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- (if prover keeps promise to supply degree d polynomial)
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2 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? 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 : 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 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 May 24, 2011 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 - The outer verifier Theorem: NP CS151 Complexity...

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