1
CS151
Complexity Theory
Lecture 10
April 28, 2011
April 28, 2011
2
Worstcase vs. Averagecase
Theorem
(NW): if
E
contains 2
Ω(n)
unapp
roximable functions then
BPP
=
P
.
• How reasonable is unapproximability
assumption?
• Hope: obtain
BPP
=
P
from worstcase
complexity assumption
– try to fit into existing framework without new
notion of “unapproximability”
April 28, 2011
3
Worstcase vs. Averagecase
Theorem
(ImpagliazzoWigderson, SudanTrevisanVadhan)
If
E
contains functions that require size
2
Ω(n)
circuits, then
E
contains 2
Ω(n)
–unapp
roximable functions.
• Proof:
– main tool:
error correcting code
April 28, 2011
4
Codes and Hardness
0
1
0
0
1
0
1
0
m
:
0
1
0
0
1
0
1
0
Enc(m):
0
0
0
1
0
0
1
1
0
0
0
1
0
R:
0
1
0
0
0
D
C
f:{0,1}
log k
{0,1}
f ’:{0,1}
log n
{0,1}
small circuit C
approximating f’
decoding
procedure
i
2
{0,1}
log k
small circuit
that computes
f exactly
f(i)
April 28, 2011
5
Encoding
• use a (variant of)
ReedMuller
code
concatenated
with the
Hadamard
code
– q (field size), t (dimension), h (degree)
•
encoding procedure
:
– message m
2
{0,1}
k
– subset S
μ
F
q
of size h
– efficient 11 function Emb: [k]
!
S
t
– find coeffs of degree h polynomial
p
m
:
F
q
t
!
F
q
for which
p
m
(Emb(i)) = m
i
for all i
(linear algebra)
so, need h
t
≥ k
April 28, 2011
6
Encoding
•
encoding procedure
(continued):
– Hadamard code
Had:{0,1}
log q
!
{0,1}
q
• = ReedMuller with field size 2, dim. log q, deg. 1
• distance ½ by SchwartzZippel
– final codeword:
(Had(p
m
(
x
)))
x
2
F
q
t
• evaluate p
m
at all points, and encode each
evaluation with the Hadamard code
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
April 28, 2011
7
Encoding
0
1
0
0
1
0
1
0
m
:
Emb: [k]
!
S
t
S
t
F
q
t
p
m
degree h
polynomial with
p
m
(Emb(i)) = m
i
5
7
2
9
2
1
0
3
8
3
6
0
1
0
0
1
0
1
0
. . .
. . .
evaluate at
all
x
2
F
q
t
encode each symbol
with
Had:{0,1}
log q
!
{0,1}
q
April 28, 2011
8
Decoding
• small circuit C computing R, agreement
½ +
•
Decoding step 1
– produce circuit C’ from C
• given
x
2
F
q
t
outputs “guess” for p
m
(
x
)
• C’ computes
{z : Had(z) has agreement ½ +
with xth block},
outputs random z in this set
0
1
0
0
1
0
1
0
Enc(m):
0
0
0
1
0
1
1
0
0
0
1
0
R:
0
1
0
0
April 28, 2011
9
Decoding
•
Decoding step 1
(continued):
– for at least
/2
of blocks, agreement in block is
at least
½ +
/2
– Johnson Bound: when this happens, list size
is
S = O(1/
2
),
so probability C’ correct is
1/S
– altogether:
• Pr
x
[C’(x) = p
m
(x)] ≥
(
3
)
• C’ makes q queries to C
• C’ runs in time poly(q)
April 28, 2011
10
Decoding
• small circuit C’ computing R’,
agreement
’ =
(
3
)
•
Decoding step 2
– produce circuit C’’ from C’
• given
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Randomness, size, Hadamard code, small circuit

Click to edit the document details