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CS151
Complexity Theory
Lecture 17
May 24, 2011
May 24, 2011
2
The outer verifier
Theorem
:
NP
PCP[log n, polylog n]
Proof (first steps):
– define:
Polynomial Constraint Satisfaction
(PCS) problem
– prove: PCS gap problem is
NP
hard
May 24, 2011
3
NP
PCP[log n, polylog n]
• algebraic version: MAX
k
PCS
– given:
• variables x
1
, x
2
, …, x
n
taking values from
field
F
q
• n = q
m
for some integer
m
• k
ary constraints C
1
, C
2
, …, C
t
– assignment viewed as f:(F
q
)
m
F
q
– output:
max. # of constraints simultaneously
satisfiable by an assignment
that has deg. ≤
d
May 24, 2011
4
NP
PCP[log n, polylog n]
• MAX
k
PCS
gap problem
:
– given:
• variables x
1
, x
2
, …, x
n
taking values from
field
F
q
• n = q
m
for some integer
m
• k
ary constraints C
1
, C
2
, …, C
t
– assignment viewed as f:(F
q
)
m
F
q
– YES:
some degree
d
assignment satisfies
all
constraints
– NO:
no degree
d
assignment satisfies more
than (1
) fraction of constraints
May 24, 2011
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NP
PCP[log n, polylog n]
Lemma
: for every constant 1 > ε > 0, the
MAX
k
PCS gap problem with
t
= poly(n) kary constraints with
k
= polylog(n)
field size
q
= polylog(n)
n = q
m
variables with
m
= O(log n / loglog n)
degree of assignments
d
= polylog(n)
gap
is
NP
hard.
May 24, 2011
6
NP
PCP[log n, polylog n]
t
= poly(n) kary constraints with
k
= polylog(n)
field size
q
= polylog(n)
n = q
m
variables with
m
= O(log n / loglog n)
degree of assignments
d
= polylog(n)
• check: headed in right direction
– O(log n)
random bits to pick a constraint
– query assignment in
O(polylog(n))
locations to
determine if it is satisfied
– completeness 1; soundness 1
(if prover keeps promise to supply degree d polynomial)
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May 24, 2011
7
NP
PCP[log n, polylog n]
• Lowdegree testing:
– want: randomized procedure that is given
d
,
oracle access to
f:(F
q
)
m
F
q
• runs in
poly(m, d)
time
• always accepts
if deg(f) ≤ d
• rejects with high probability if
deg(f) > d
– too much to ask. Why?
May 24, 2011
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NP
PCP[log n, polylog n]
Definition
: functions
f, g
are
δclose
if
Pr
x
[f(x) ≠ g(x)]
δ
Lemma
:
δ > 0 and a randomized procedure that
is given
d
, oracle access to
f:(F
q
)
m
F
q
– runs in
poly(m, d)
time
– uses
O(m log F
q
)
random bits
– always accepts if
deg(f) ≤ d
– rejects with high probability if
f is not δclose
to any g with deg(g) ≤ d
May 24, 2011
9
NP
PCP[log n, polylog n]
• idea of proof:
– restrict to random line L
– check if it is low degree
– always accepts if
deg(f) ≤ d
– other direction much more complex
(
F
q
)
m
May 24, 2011
10
NP
PCP[log n, polylog n]
– can only force prover to supply function f that
is
close
to a lowdegree polynomial
– how to bridge the gap?
– recall lowdegree polynomials form an
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 Fall '09

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