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

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

Nov 22, 2022 203 likes •685 views

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 1/21/poly( n ) Shafi

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Shafi Goldwasser, Michael Sipser ( x1 can always contain w )

si

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Shafi Goldwasser, Michael Sipser

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Shafi Goldwasser, Michael Sipser (uniformly)

∑l(|w|)

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Shafi Goldwasser, Michael Sipser

2–poly(n) ≤ e ≤ 1/2–1/poly(n)

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Shafi Goldwasser, Michael Sipser

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Shafi Goldwasser, Michael Sipser

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Shafi Goldwasser, Michael Sipser

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Shafi Goldwasser, Michael Sipser

Z Z Z

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Shafi Goldwasser, Michael Sipser

if 2b/4 ≥ |C| ≥ 2b/8 then Pr[H(C)∩Z=Ø] ≤ 2-l/8 if |C| ≤ 2b/d, d>0, then Pr[H(C)∩Z≠Ø] ≤ l3/d

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Shafi Goldwasser, Michael Sipser

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Shafi Goldwasser, Michael Sipser

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Shafi Goldwasser, Michael Sipser

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Shafi Goldwasser, Michael Sipser

if 2b/4 ≥ |C| ≥ 2b/8 then Pr[H(C)∩Z=Ø] ≤ 2-l/8 if |C| ≤ 2b/d, d>0, then Pr[H(C)∩Z≠Ø] ≤ l3/d

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Shafi Goldwasser, Michael Sipser

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Shafi Goldwasser, Michael Sipser NOTA: the αx are disjoint.

V(w,r,#) V(w,r,#x#y)

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Shafi Goldwasser, Michael Sipser

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Shafi Goldwasser, Michael Sipser

e2l/2b e2l/2b

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Shafi Goldwasser, Michael Sipser

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Shafi Goldwasser, Michael Sipser

( Full proof )

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Shafi Goldwasser, Michael Sipser

( Full proof )

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Shafi Goldwasser, Michael Sipser

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Shafi Goldwasser, Michael Sipser

( Full proof )

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Shafi Goldwasser, Michael Sipser

( Full proof )

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Shafi Goldwasser, Michael Sipser

Arthur-Merlin Games w∈L

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Poly-Time

b1 xi,yi,bi+1 r H,Z H,Z 1≤i≤g xi ∈H-1(Z) ? r ∈H-1(Z) ? 1≤i≤g, V(w,r,si-1) = xi ? V(w,r,sg) = “accept” ? ∑ bi ≥ l - g log l ? (via si-1#xi many r would lead V to accept) (many r would lead V to accept)

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Shafi Goldwasser, Michael Sipser

Arthur-Merlin Games w∈L

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Poly-Time

b1=2+[log|γmax|] xi∈H-1(Z), αxi∈γmax, yi=P(si-1#xi), bi+1=2+[log|γmax|] r H∈R(∑m→∑b1)l, Z∈R(∑m)l2 H∈R(∑m→∑bi+1)l, Z∈R(∑m)l2 1≤i≤g xi ∈H-1(Z) ? r ∈H-1(Z) ? 1≤i≤g, V(w,r,si-1) = xi ? V(w,r,sg) = “accept” ? ∑ bi ≥ l - g log l ? NOTA: [x]=ceiling(x)

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Shafi Goldwasser, Michael Sipser

( Full proof ) ∑bi ∑bi

(and sends to M)

Arthur’s protocol

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Shafi Goldwasser, Michael Sipser

( Full proof w∈L )

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Shafi Goldwasser, Michael Sipser

( Full proof w∈L )

Merlin’s protocol

1≤i≤g

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Shafi Goldwasser, Michael Sipser

( Full proof w∈L )

Merlin’s protocol

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Shafi Goldwasser, Michael Sipser

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

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Shafi Goldwasser, Michael Sipser

( Full proof w∈L ) ( Claim 1/6 )

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Shafi Goldwasser, Michael Sipser

( Full proof w∈L ) ( Claim 1/6 )

αx={r : (VP)(w,r) accepts via si-1#x} there are l possibilities for γi, thus at least one is of size total/l

( )

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Shafi Goldwasser, Michael Sipser

( Full proof w∈L ) ( Claim 2/6 )

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Shafi Goldwasser, Michael Sipser

Arthur-Merlin Games w∉L

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Poly-Time

b1 xi, yi, bi+1 r H∈R(∑m→∑b1)l, Z∈R(∑m)l2 H∈R(∑m→∑bi+1)l, Z∈R(∑m)l2 1≤i≤g xi ∈H-1(Z) ? r ∈H-1(Z) ? 1≤i≤g, V(w,r,si-1) = xi ? V(w,r,sg) = “accept” ? ∑ bi ≥ l - g log l ?

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Shafi Goldwasser, Michael Sipser

( Full proof w∉L )

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Shafi Goldwasser, Michael Sipser

( Full proof w∉L ) ( Claim 3/6 )

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Shafi Goldwasser, Michael Sipser

( Full proof w∉L ) ( Claim 4/6 )

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Shafi Goldwasser, Michael Sipser

( Full proof w∉L ) ( Claim 5/6 )

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Shafi Goldwasser, Michael Sipser

( Full proof w∉L ) ( Claim 5/6 ) if |C| ≤ 2b/d, d>0, then Pr[H(C)∩Z≠Ø] ≤ l3/d

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Shafi Goldwasser, Michael Sipser

( Full proof w∉L ) ( Claim 6/6 ) if |C| ≤ 2b/d, d>0, then Pr[H(C)∩Z≠Ø] ≤ l3/d

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Shafi Goldwasser, Michael Sipser

( Full proof w∉L )

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Shafi Goldwasser, Michael Sipser

( Full proof w∉L )

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Shafi Goldwasser, Michael Sipser

( Full proof w∉L )

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Shafi Goldwasser, Michael Sipser

( Full proof w∉L )

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