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lec13 - Alternating quantifiers Pleasing viewpoint CS151...

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1 CS151 Complexity Theory Lecture 13 May 10, 2011 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 989 …” 898 …” 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 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 | 9 (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. 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
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