lec9 - Derandomization Goal try to simulate BPP in...

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1 CS151 Complexity Theory Lecture 9 April 26, 2011 April 26, 2011 2 Derandomization Goal : try to simulate BPP in subexponential time (or better) use Pseudo-Random Generator (PRG): often: PRG “good” if it passes (ad -hoc) statistical tests seed output string G t bits m bits April 26, 2011 3 Derandomization ad-hoc tests not good enough to prove BPP has non-trivial simulations Our requirements: G is efficiently computable – “ stretches t bits into m bits – “ fools ” small circuits: for all circuits C of size at most s : |Pr y [C(y) = 1] Pr z [C(G(z)) = 1]| ≤ ε April 26, 2011 4 Simulating BPP using PRGs Recall: L BPP implies exists p.p.t.TM M x L Pr y [M(x,y) accepts] ≥ 2/3 x L Pr y [M(x,y) rejects] ≥ 2/3 given an input x: convert M into circuit C(x, y) simplification: pad y so that |C| = |y| = m hardwire input x to get circuit C x Pr y [C x (y) = 1] ≥ 2/3 (“yes”) Pr y [C x (y) = 1] ≤ 1/3 (“no”) April 26, 2011 5 Simulating BPP using PRGs Use a PRG G with output length m seed length t « m error ε < 1/6 fooling size s = m Compute Pr z [C x (G(z)) = 1] exactly evaluate C x (G(z)) on every seed z {0,1} t running time (O(m)+(time for G)) 2 t April 26, 2011 6 Simulating BPP using PRGs knowing Pr z [C x (G(z)) = 1] , can distinguish between two cases: 0 1/3 1/2 2/3 1 “yes”: ε 0 1/3 1/2 2/3 1 “no”: ε
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