Shafi Goldwasser, Michael Sipser Shafi Goldwasser, Michael Sipser l - PowerPoint PPT Presentation

Shafi Goldwasser, Michael Sipser s i ( x 1 can always contain w )

Shafi Goldwasser, Michael Sipser

Shafi Goldwasser, Michael Sipser ∑ l (| w |) (uniformly)

Shafi Goldwasser, Michael Sipser 2 –poly( n ) ≤ e ≤ 1/2–1/poly( n )

Shafi Goldwasser, Michael Sipser

Shafi Goldwasser, Michael Sipser

Shafi Goldwasser, Michael Sipser

Shafi Goldwasser, Michael Sipser Z Z Z

Shafi Goldwasser, Michael Sipser if 2 b /4 ≥ | C | ≥ 2 b /8 then Pr[ H ( C ) ∩ Z =Ø] ≤ 2 -l /8 if | C | ≤ 2 b /d, d>0, then Pr[ H ( C ) ∩ Z ≠Ø] ≤ l 3 / d

Shafi Goldwasser, Michael Sipser

Shafi Goldwasser, Michael Sipser

Shafi Goldwasser, Michael Sipser

Shafi Goldwasser, Michael Sipser if 2 b /4 ≥ | C | ≥ 2 b /8 then Pr[ H ( C ) ∩ Z =Ø] ≤ 2 -l /8 if | C | ≤ 2 b /d, d>0, then Pr[ H ( C ) ∩ Z ≠Ø] ≤ l 3 / d

Shafi Goldwasser, Michael Sipser

Shafi Goldwasser, Michael Sipser V ( w,r,# ) V ( w,r,#x#y ) NOTA: the α x are disjoint.

Shafi Goldwasser, Michael Sipser

Shafi Goldwasser, Michael Sipser e 2 l /2 b e 2 l /2 b

Shafi Goldwasser, Michael Sipser

Shafi Goldwasser, Michael Sipser ( Full proof )

Shafi Goldwasser, Michael Sipser ( Full proof )

Shafi Goldwasser, Michael Sipser

Shafi Goldwasser, Michael Sipser ( Full proof )

Shafi Goldwasser, Michael Sipser ( Full proof )

Shafi Goldwasser, Michael Sipser Arthur-Merlin Games Poly-Time ‹ † w ∈ L b 1 H , Z x i , y i , b i +1 x i ∈ H -1 ( Z ) ? (via s i -1 # x i many r would lead V to accept) 1≤ i ≤ g H , Z r (many r would lead V to accept) r ∈ H -1 ( Z ) ? 1≤ i ≤ g , V ( w,r,s i -1 ) = x i ? V ( w,r,s g ) = “accept” ? ∑ b i ≥ l - g log l ?

Shafi Goldwasser, Michael Sipser Arthur-Merlin Games Poly-Time ‹ † w ∈ L NOTA: [ x ]=ceiling( x ) b 1 =2+[log| γ max |] H ∈ R (∑ m → ∑ b 1 ) l , Z ∈ R ( ∑ m ) l 2 x i ∈ H -1 ( Z ) , α x i ∈ γ max , y i =P( s i -1 # x i ), b i +1 =2+[log| γ max |] x i ∈ H -1 ( Z ) ? 1≤ i ≤ g H ∈ R (∑ m → ∑ b i+1 ) l , Z ∈ R ( ∑ m ) l 2 r r ∈ H -1 ( Z ) ? 1≤ i ≤ g , V ( w,r,s i -1 ) = x i ? V ( w,r,s g ) = “accept” ? ∑ b i ≥ l - g log l ?

Shafi Goldwasser, Michael Sipser Arthur’s protocol ∑ b i ∑ b i (and sends to M) ( Full proof )

Shafi Goldwasser, Michael Sipser ( Full proof w ∈ L )

Shafi Goldwasser, Michael Sipser 1≤ i ≤ g Merlin’s protocol ( Full proof w ∈ L )

Shafi Goldwasser, Michael Sipser Merlin’s protocol ( Full proof w ∈ L )

Shafi Goldwasser, Michael Sipser if 2 b /4 ≥ | C | ≥ 2 b /8 then Pr[ H ( C ) ∩ Z =Ø] ≤ 2 -l /8 ( Full proof w ∈ L )

Shafi Goldwasser, Michael Sipser ( Claim 1/6 ) ( Full proof w ∈ L )

Shafi Goldwasser, Michael Sipser α x ={ r : ( VP )( w,r ) accepts via s i -1 # x } ( ) there are l possibilities for γ i , thus at least one is of size total/ l ( Claim 1/6 ) ( Full proof w ∈ L )

Shafi Goldwasser, Michael Sipser ( Claim 2/6 ) ( Full proof w ∈ L )

Shafi Goldwasser, Michael Sipser Arthur-Merlin Games Poly-Time † w ∉ L b 1 H ∈ R (∑ m → ∑ b 1 ) l , Z ∈ R ( ∑ m ) l 2 x i , y i , b i +1 x i ∈ H -1 ( Z ) ? 1≤ i ≤ g H ∈ R (∑ m → ∑ b i+1 ) l , Z ∈ R ( ∑ m ) l 2 r r ∈ H -1 ( Z ) ? 1≤ i ≤ g , V ( w,r,s i -1 ) = x i ? V ( w,r,s g ) = “accept” ? ∑ b i ≥ l - g log l ?

Shafi Goldwasser, Michael Sipser ( Full proof w ∉ L )

Shafi Goldwasser, Michael Sipser ( Claim 3/6 ) ( Full proof w ∉ L )

Shafi Goldwasser, Michael Sipser ( Claim 4/6 ) ( Full proof w ∉ L )

Shafi Goldwasser, Michael Sipser ( Claim 5/6 ) ( Full proof w ∉ L )

Shafi Goldwasser, Michael Sipser ( Claim 5/6 ) if | C | ≤ 2 b /d, d>0, then Pr[ H ( C ) ∩ Z ≠Ø] ≤ l 3 / d ( Full proof w ∉ L )

Shafi Goldwasser, Michael Sipser if | C | ≤ 2 b /d, d>0, then Pr[ H ( C ) ∩ Z ≠Ø] ≤ l 3 / d ( Claim 6/6 ) ( Full proof w ∉ L )

Shafi Goldwasser, Michael Sipser ( Full proof w ∉ L )

Shafi Goldwasser, Michael Sipser ( Full proof w ∉ L )

Shafi Goldwasser, Michael Sipser ( Full proof w ∉ L )

Shafi Goldwasser, Michael Sipser ( Full proof w ∉ L )

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