Tokyo Institute of Technology Akinori Kawachi Layout Randomized vs Determinsitic Algorithms Primality Test General Framework for Derandomization Circuit Complexity Derandomization ID: 783632
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Slide1
Circuit Complexity and Derandomization
Tokyo Institute of Technology
Akinori
Kawachi
Slide2Layout
Randomized vs
Determinsitic
Algorithms
Primality
Test
General Framework for
Derandomization
Circuit Complexity
Derandomization
Circuits
Circuit Complexity and NP vs. P
Necessity of Circuit Complexity for
Derandomization
Summary
Slide3Deterministic
v.s
. Randomized Algorithms
for (Decision) Problems
Randomness is useful for real-world computation!
Decision problem: PRIME
Input: n-bit number x (0
x < 2
n
)
Output: “Yes” if x PRIME (x is prime)
“No” otherwise
Elementary Det. algorithm: O(2n/2) time [Eratosthenes, B.C. 2c]
Rand. algorithm: O(n3) time w/ succ. prob. 99% [Miller 1976, Rabin 1980]
Exponential-time speed-up!
n = input length
Slide4Deterministic v.s. Randomized Algorithms
for (Decision) Problems
How much randomness make computation strong?
Det. algorithm:
O(n
12
) time
[Agrawal, Kayal & Saxena 2004
Gödel Prize]
Rand. algorithm: O(n3) time w/ succ. prob. 99% [Miller 1976, Rabin 1980]
Input: n-bit number x (0
N < 2n)
Output
: “Yes” if x PRIME (x is prime)
“No” otherwise
Polynomial-time slow-down
Decision problem: PRIME
Slide5Derandomization Conjecture
BPP = P
Randomization yields
NO
exponential speed-up!
Always poly-time
derandomization
possible?
Conjecture
P = {problem: poly-time det. TM computes}
BPP = {problem: poly-time prob. TM computes
w/ bounded errors}
Slide6Class BPP
Class BPP (
B
ounded-error
P
rob.
P
oly-time)
L∈BPP
x∊L
x∉L
Def
Prr[A(
x,r) = Yes] > 2/3
r is uniform over {0,1}mm = |r| = poly(|x|)A(・,・): poly-time det. TM
Prr[A(x,r) = No] > 2/3
Slide7Nondeterministic Version
AM = NP
Conjecture
Class
AM
(
A
rthur-
M
erlin Games)L∈AM
x∊L
x∉L
Def
Pr
r[
w: A(x,w,r) = Yes] > 2/3
|r|,|w| = poly(|x|)A(・
,・,・): poly-time det. TM
Pr
r
[
w
: A(
x,w,r
) = No] > 2/3
Slide8Hardness vs. Randomness Trade-offs[Yao ’82, Blum &
Micali
’84
]
Hard problem exists
Good Pseudo-Random Generator (PRG)
exists.Simulate randomized algorithms det.ly with PRG!
Theorem
[Impagliazzo & Wigderson 1998]
2O(
)-time computable decision problem Hs.t. no 20.1
-size circuit can compute for every
BPP = P
(L is computed in prob. poly-time w/ bounded errors L is computed in det. poly-time)
Similar theorem holds in nondet
. version (AM=NP)[Klivans & van Melkebeek 2001]
Slide9Circuit
x
3
∧
x
1
x
2
0
￢
∨
∧
∧
∨
Gate set = {
∧
,
∨
,
￢
, 0, 1}
Slide10Gate set = {∧
,
∨
,
￢
, 0, 1}
Circuit
0
∧
1
1
￢
∨
∧
∧
∨
1
∧
1 = 1
1
1
1
1
0
0
0
1
1
0
￢
0
= 1
1
∧
0
=
0
0
∨
1
= 1
0
∧
1
=
0
0
1
∨
0
= 1
1
Input
= (1,1,0)
0
Size = 7
Depth = 5
Slide11Circuit Complexity
Size of circuits is measure for computational resource!
Circuit complexity of L
:= min { size of circuit family computing L }
s(n)-size
circuit family {C
n
:{0,1}
n
→{0,1}}n computes L
DefinitionDef
x L
C|x|(x) = 1
x
L C|x|(x) = 0
&
size of Cn
s(n)
Slide12Computational Power of Circuits
Circuit complexity of any problem = O(2
n
/n)
Theorem
[
Lupanov
1970]
any (even non-recursive) problem can be computed by some O(2n/n)-size circuit family.
P
SIZE(poly)
Theorem
[Fisher & Pippenger 1979]
Poly-time TM can be simulated by poly-size circuit family.
SIZE(poly) = {problem: poly-size circuit family can compute}
Slide13NP vs. P and Circuits
NP
≠
P
Conjecture
Some NP problem cannot be computed by any
poly-time TM.
NP
⊄ SIZE(poly)
ConjectureSome NP problem has superpoly
circuit complexity. Note: NP ⊄ SIZE(poly)
NP ≠ PProving super-poly circuit complexity in NP
solves NP vs. P!
Slide14NEXP
⊄
SIZE(poly
)
MA-EXP
⊄
SIZE(poly)
Current Status
Theorem (
Buhrman, Fortnow, & Thierauf 1998)
NEXP ⊄ ACC0(poly)
Theorem (Williams 2011)
Randomized version of NEXP
Const
-depth poly-size w/ Modulo gatesGrand Challenge
Cf. H-R tradeoff for BPP=P requires at least EXP ⊄ SIZE(2.1n
)!
Slide15Hardness vs. Randomness Trade-offs[Yao ’82, Blum &
Micali
’84
]
Hard problem exists
Good Pseudo-Random Generator (PRG)
exists.Simulate randomized algorithms det.ly with PRG!
Theorem
[Impagliazzo & Wigderson 1998]
2O(
)-time computable decision problem Hs.t. no 20.1
-size circuit can compute for every
BPP = P
(L is computed in prob. poly-time w/ bounded errors L is computed in det. poly-time)
Slide16Proof Sketch
Construct
PRG
from hard H.
Simulate
rand. algo. w/
p-random bits.
Slide17Proof Sketch
Construct
PRG
from hard H.
Goal: Construct G
H
: {0,1}
O(log m) → {0,1}m
Pr
s[ C(GH(s)) = 1 ] Prr[ C(r) = 1 ]
Pseudo-random!
truly random!
# possible s = 2
O(log m) = poly(m)# possible r = 2m
PointProof:
good distinguisher D for GH
small circuit C
D for H
For every poly-size circuit C,
Slide18Proof Sketch
Simulate rand.
a
lgo
. w/
p-random bits.
Goal: Det.ly simulate rand. algo
. by GH
L
∈BPPx∊L
x
∉LDef
Pr
r[A(
x,r) = Yes] > 2/3|r| = poly(|x|)A(・,
・): poly-time det. TMPrr[A(x,r) = No] > 2/3
Slide19Proof Sketch
Simulate rand.
a
lgo
. w/
p-
random bits.
Goal: Det.ly simulate rand. algo. by G
H
Trivial SimulationEnumerate all possible
-bit strings!
A(x,00…00)=
Yes
A(x,00…01)
=No
…A(x,11…10)
=
Yes
A(x,11…11)=
Yes
#Yes >
x
∊
L
#No >
x
∉
L
Require O(2
m
)=O(2
poly(n)
) time…
Slide20Proof Sketch
Simulate rand.
a
lgo
. w/
p-
random bits.
Goal: Det.ly simulate rand. algo. by G
H
Simulation w/ GH
Enumerate all possible
-bit seeds of GH!
A(x,GH(0…0))
=No
…A(x,G
H(1…1))
=Yes
#Yes >
x
∊
L
#No >
x
∉
L
Require 2
O(log m)
= poly(n) time!
A(x,
・
)
=
circuit C
Slide21Is Circuit Complexity Essential?
Proving “some problem is really hard” is HARD! (e.g. NP
≠
P)
It’s the
ultimate goal
in complexity theory…Can avoid “proving hardness” for derandomization?
NO! Derandomization implies proving hardness!!
BPP=P
Some problem is hard.Theorem [Kabanets
& Impagliazzo ‘03]
prAM NP
Some problem is extremely hard.
Theorem [Gutfreund & Kawachi ‘10,
Aaronson, Aydinlioglu, Buhrman, Hitchcock, & van Melkebeek ‘11]
Slide22BPP
P
NEXP
SIZE
, or
Permanent
A
SIZE
Theorem [Kabanets & Impagliazzo ‘
03]
Resolving “arithmetic-circuit version of NP vs. P“
prAM
NPEXPNP
SIZE
Theorem
[Gutfreund &
Kawachi
‘10,
Aaronson,
Aydinlioglu
,
Buhrman
, Hitchcock, & van
Melkebeek
‘11
]
Slide23Summary
Proving circuit complexity
Derandomization
through Pseudo-Random Generator
BPP = P, AM = NP, and more…
Derandomization Proving circuit complexity
Proving Circuit Complexity
Derandomization