1
CS151
Complexity Theory
Lecture 9
April 26, 2011
April 26, 2011
2
Derandomization
•
Goal
: try to simulate BPP in
subexponential time (or better)
•
use
Pseudo-Random Generator
(PRG):
•
often: PRG “good” if it passes (ad
-hoc)
statistical tests
seed
output string
G
t
bits
m
bits
April 26, 2011
3
Derandomization
•
ad-hoc tests not good enough to
prove
BPP has non-trivial simulations
•
Our requirements:
–
G is
efficiently computable
– “
stretches
”
t
bits into
m
bits
– “
fools
” small circuits: for all circuits C of size at
most
s
:
|Pr
y
[C(y) = 1]
–
Pr
z
[C(G(z)) = 1]| ≤
ε
April 26, 2011
4
Simulating
BPP
using PRGs
•
Recall: L
BPP
implies exists p.p.t.TM M
x
L
Pr
y
[M(x,y) accepts] ≥ 2/3
x
L
Pr
y
[M(x,y) rejects] ≥ 2/3
•
given an input x:
–
convert M into circuit C(x, y)
–
simplification: pad y so that |C| = |y| =
m
•
hardwire input x to get circuit C
x
Pr
y
[C
x
(y) = 1]
≥ 2/3
(“yes”)
Pr
y
[C
x
(y) = 1] ≤ 1/3
(“no”)
April 26, 2011
5
Simulating
BPP
using PRGs
•
Use a PRG G with
–
output length
m
–
seed length
t
« m
–
error
ε
< 1/6
–
fooling size
s
= m
•
Compute
Pr
z
[C
x
(G(z)) = 1]
exactly
–
evaluate
C
x
(G(z))
on every seed
z
{0,1}
t
•
running time
(O(m)+(time for G))
2
t
April 26, 2011
6
Simulating
BPP
using PRGs
•
knowing
Pr
z
[C
x
(G(z)) = 1]
, can distinguish
between two cases:
0
1/3
1/2
2/3
1
“yes”:
ε
0
1/3
1/2
2/3
1
“no”:
ε

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