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# lec13 - CS151 Complexity Theory Lecture 13 Alternating...

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CS151 Complexity Theory Lecture 13 May 10, 2011

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May 10, 2011 2 Alternating quantifiers Pleasing viewpoint: P NP coNP Δ 3 Σ 2 Π 2 Δ 2 9 8 98 89 Σ 3 Π 3 PSPACE PH 989 898 9898989 …” Σ i Π i …” …” const. # of alternations poly(n) alternations
May 10, 2011 3 PH collapse Theorem : if Σ i = Π i then for all j > i Σ j = Π j = Δ j = Σ i “the polynomial hierarchy collapses to the i-th level” Proof: – sufficient to show Σ i = Σ i+1 – then Σ i+1 = Σ i = Π i = Π i+1 ; apply theorem again P NP coNP Σ 3 Π 3 Δ 3 PH Σ 2 Π 2 Δ 2

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May 10, 2011 4 PH collapse – recall: L Σ i+1 iff expressible as L = { x | 9 y (x, y) R } where R Π i – since Π i = Σ i , R expressible as R = { (x,y) | 9 z ( (x, y), z) R’ } where R’ Π i-1 together: L = { x | (y, z) (x, (y, z)) R’} – conclude L Σ i
May 10, 2011 5 Karp-Lipton we know that P = NP implies SAT has polynomial-size circuits. suppose SAT has poly-size circuits any consequences? might hope: SAT P/poly PH collapses to P , same as if SAT P Theorem (KL): if SAT has poly-size circuits then PH collapses to the second level.

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May 10, 2011 6 BPP PH Recall: don’t know BPP different from EXP Theorem (S,L,GZ): BPP ( Π 2 Σ 2 ) • don’t know Π 2 Σ 2 different from EXP but believe much weaker
May 10, 2011 7 BPP PH Proof: BPP language L: 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 strong error reduction : p.p.t. TM M’ use n random bits (|y’| = n) # strings y’ for which M’(x, y’) incorrect is at most 2 n/3 (can’t achieve with naïve amplification)

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May 10, 2011 8 BPP PH view y’ = ( w , z ) , each of length n/2 consider output of M’(x, ( w , z )) : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 w =       000…00          000…01          000…10     …       111… 11 x L x L so few ones, not enough for whole disk
May 10, 2011 9 BPP PH proof (continued): strong error reduction: # bad y’ < 2 n/3 y’ = (w, z) with |w| = |z| = n/2 Claim: L = {x : 9 w 8 z M’(x, (w, z)) = 1 } x L: suppose 8 w 9 z M’(x, (w, z)) = 0 implies 2 n/2 0’s; contradiction x L: suppose 9 w 8 z M’(x, (w, z)) = 1 implies 2 n/2 1’s; contradiction

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May 10, 2011 10 BPP PH given BPP language L: 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 showed L = {x : 9 w 8 z M’(x, (w, z)) = 1} – thus BPP Σ 2 BPP closed under complement BPP Π 2 – conclude: BPP ( Π 2 Σ 2 )
May 10, 2011 11 New Topic The complexity of counting

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May 10, 2011 12 Counting problems So far, we have ignored function problems
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lec13 - CS151 Complexity Theory Lecture 13 Alternating...

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