CSci 5403 Lecture 15

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CSci 5403
COMPLEXITY THEORY
LECTURE XVII INTERACTIVE PROOFS I
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WHAT IS A PROOF?
G ? HAM v1,v2,,vn
Prover
Verifier
A traditional proof is a static sequence of
symbols that establishes that x?L.
NP languages with short proofs
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INTERACTIVE PROOFS
x?L
?

?
Unbounded Prover
Efficient Verifier

almost certainly so.
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EXAMPLE
Recall that graphs H and G are isomorphic, written
HG, if there is a 1-1, onto p G ? H such that
(u,v) ? G iff (p(u),p(v)) ? H.
Then ISO (H,G) HG ? NP the proof is p.
Thus NONISO (H,G) H ? G ? coNP.
ISO is neither known to be in P nor NP-complete.
Is NONISO ? NP? Seems hard
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INTERACTIVE PROOFS
G0 ? G1
The proof is too long for you to read.
I am crushed.
Intriguing.
Unbounded Prover
Traditional Prover
Efficient Verifier
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INTERACTIVE PROOFS
Definition. An interactive proof system has two
players
?
The conversation (V?P)(xr) is a1,a2,,ak1.
(V?P)(xr) accepts if ak1 accept
We say that (V?P)(xr) has k rounds.
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INTERACTIVE PROOFS
Definition. Pr(P?V)(x) accepts
Prr(P?V)(xr) accepts PrV accepts x maxP
Pr(P?V)(x) accepts

• Definition. (P,V) is an interactive proof system
  for L
• x?L ? PrV accepts x 2/3
• x ? L ? PrV accepts x 1/3

(Completeness)
(Soundness)
Definition. L?IPk if ?k(n)-round interactive
proof system (P,V) for L and constant c such
that (P?V)(xr) xc
Definition. IP ?c IPnc
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DEFINITION ISSUES
Theorem. If L has an interactive proof system
with deterministic verifier, then L ? NP.
Theorem. IP ? PSPACE.
Proof. Given V, x we can find the set of
responses that maximize PrV accepts x in
polynomial space.
Theorem. Let pin(n) minx?Ln0,1n PrV accepts
x, pout(n) max x??n0,1n PrV accepts
x. If pin(n) pout(n) 1/poly(n), then L?IP.
Proof. Iterate O(1/(pin(n) pout(n))2) times.
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ARTHUR-MERLIN PROOFS
An Arthur-Merlin verifier has no private
randomness
x ? L
a1 random r1
a2 P(x,r1)
a3 random r2
?
ak P(x,a1,,ak-1)
Arthur
Merlin
V(x,a1,,ak) accept/reject
Definition. L ? AMk if there exists
k(n)-round Arthur-Merlin interactive proof system
for L.
Definition. AM AM2. MA MA2
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NONISO ?AMk?
b
G0 ? G1
Unbounded Prover
Efficient Verifier
Theorem. For every N ? N, IP(n) ?
AM(n)2.
We will show a simpler result NONISO ? AM.
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Consider the set S H H G0 or H G1 .
Assume neither Gb has nontrivial automorphisms.
Then G0G1 ? S n!, but G0?G1 ? S2n!
Idea use hashing to prove that S2n!
Lemma. Let H X ? T be a pairwise
independent hash family, and let W ? X . Then
W/T - ½ (W/T)2 Prh,t?w ? W. h(w)t
W/T.
Proof. Apply Inclusion-Exclusion, using ?w,h .
Prth(w)t 1/T and ?w1,w2,t.
Prhh(w1)h(w2)t 1/T2.
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NONISO ? AM2
Let H 0,1n² ? 1,2,,4n! be a family of
pairwise-independent hash functions.
h ?R H, t ?R 4n!
G, b, p
accept if h(G)t and G p(Gb)
Arthur
Merlin
PrV accepts Pr?G ? S. h(G)t.
G0G1 ? Sn! ? PrV accepts n!/4n! ¼.
G0?G1 ? PrV accepts n!/2n! ½(n!/2n!)2
3/8.
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Theorem. If L has a k-round AM proof system
with soundness error 1/3, it has a k-round AM
proof system with soundness error 2-n.
Theorem. If L has a k-round AM proof system with
completeness 2/3, then L has a k1 round AM
proof system with completeness 1.
Proof. See homework.
Theorem. AMO(1) AM.
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Theorem. If ISO is NP complete, then PHS2.
Proof. If ISO is NP complete, then NONISO
is coNP complete.
Then there exists such that (?) ? NONISO iff
?y.?(y).
Because NONISO ? AM, there exists V such that
maxa PrV(x,r,a) 1 1 when x ? NONISO and
maxa PrV(x,r,a) 1 lt 1 otherwise.
Thus ?r?x,a V((?(x,)),r,a) iff ?x?y ?(x,y).
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