Circuit Complexity and Derandomization
Tokyo Institute of Technology
Akinori Kawachi
Layout
• 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
Deterministic v.s. Randomized Algorithms
for (Decision) Problems
Randomness is useful for real-world computation!
n = input length
Decision problem: PRIME
Input: n-bit number x (0 ≤ x < 2n)
Output: “Yes” if x ∈ PRIME (x is prime)
“No” otherwise
Exponential-time
speed-up!
Elementary Det. algorithm: O(2n/2) time [Eratosthenes, B.C. 2c]
Rand. algorithm: O(n3) time w/ succ. prob. 99% [Miller 1976, Rabin 1980]
Deterministic v.s. Randomized Algorithms
for (Decision) Problems
How much randomness make computation strong?
Decision problem: PRIME
Input: n-bit number x (0 ≤ N < 2n)
Output: “Yes” if x ∈ PRIME (x is prime)
“No” otherwise
Polynomial-time
slow-down
Rand. algorithm: O(n3) time w/
succ. prob. 99% [Miller 1976, Rabin 1980]
Det. algorithm: O(n12) time [Agrawal, Kayal & Saxena 2004 Gödel Prize]
Derandomization Conjecture
Always poly-time derandomization possible?
Conjecture
BPP = P
Randomization yields NO exponential speed-up!
P = {problem: poly-time det. TM computes}
BPP = {problem: poly-time prob. TM computes
w/ bounded errors}
Class BPP
Class BPP (Bounded-error Prob. Poly-time)
L∈BPP
Def
x∊L
x∉L
Prr[A(x,r) = Yes] > 2/3
Prr[A(x,r) = No] > 2/3
r is uniform over {0,1}m
m = |r| = poly(|x|)
A(・,・): poly-time det. TM
Nondeterministic Version
Conjecture
AM = NP
Class AM (Arthur-Merlin Games)
L∈AM
Def
x∊L
x∉L
Prr[∃w: A(x,w,r) = Yes] > 2/3
Prr[∀w: A(x,w,r) = No] > 2/3
|r|,|w| = poly(|x|)
A(・,・,・): poly-time det. TM
Hardness vs. Randomness Trade-offs
[Yao ’82, Blum & Micali ’84]
• Hard problem exists
Similar theorem holds
 Good Pseudo-RandominGenerator
(PRG)
exists.
nondet. version
(AM=NP)
[Klivans & van
Melkebeek
2001]
• Simulate randomized algorithms
det.ly
with PRG!
Theorem [Impagliazzo & Wigderson 1998]
∃2O(𝑛)-time computable decision problem H
s.t. no 20.1𝑛-size circuit can compute for every 𝑛
BPP = P
(L is computed in prob. poly-time w/ bounded errors
 L is computed in det. poly-time)
Circuit
Gate set = {∧, ∨, ￢, 0, 1}
∨
∧
∧
∨
∧
x1
￢
x2
x3
0
Circuit
1
Gate set = {∧, ∨, ￢,
0,
1}
1∨0 = 1
∨
0∧1 = 0
0
1
∧
1
0
1∧0 = 0
∧
1
1∧1 = 1
∨
0
0
∧
Input
= (1,1,0)
1
1
0∨1 = 1
1
￢0 = 1
￢
1
1
0
0
0
Size = 7
Depth = 5
Circuit Complexity
Size of circuits is measure for computational resource!
Definition
s(n)-size circuit family {Cn:{0,1}n→{0,1}}n computes L
Def
x ∈ L  C|x|(x) = 1
x ∉ L  C|x|(x) = 0
& size of Cn ≤ s(n)
Circuit complexity of L
:= min { size of circuit family computing L }
Computational Power of Circuits
Theorem [Lupanov 1970]
Circuit complexity of any problem = O(2n/n)
any (even non-recursive) problem can be computed
by some O(2n/n)-size circuit family.
SIZE(poly) = {problem:
poly-size circuit family can compute}
Theorem [Fisher & Pippenger 1979]
P ⊂ SIZE(poly)
Poly-time TM can be simulated by poly-size circuit family.
NP vs. P and Circuits
Conjecture
NP ≠ P
Some NP problem cannot be computed by any poly-time TM.
Conjecture
NP ⊄ SIZE(poly)
Some NP problem has superpoly circuit complexity.
Note: NP ⊄ SIZE(poly)  NP ≠ P
Proving super-poly circuit complexity in NP
solves NP vs. P!
Current Status
Randomized version
of NEXP (Buhrman, Fortnow, & Thierauf 1998)
Theorem
MA-EXP ⊄ SIZE(poly)
Theorem (Williams 2011)
Const-depth poly-size
w/ Modulo gates
NEXP ⊄ ACC0(poly)
Grand Challenge
NEXP ⊄ SIZE(poly)
Cf. H-R tradeoff for BPP=P requires at least EXP ⊄ SIZE(2.1n)!
Hardness 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(𝑛)-time computable decision problem H
s.t. no 20.1𝑛-size circuit can compute for every 𝑛
BPP = P
(L is computed in prob. poly-time w/ bounded errors
 L is computed in det. poly-time)
Proof Sketch
1. Construct PRG from hard H.
2. Simulate rand. algo. w/ p-random bits.
Proof Sketch
1. Construct PRG from hard H.
Goal: Construct GH: {0,1}O(log m) → {0,1}m
For every poly-size circuit C,
Prs[ C(GH(s)) = 1 ] ≈ Prr[ C(r) = 1 ]
Pseudo-random!
truly random!
Proof: ∃good distinguisher D for GH  ∃small circuit CD for H
Point
# possible s = 2O(log m) = poly(m)
# possible r = 2m
Proof Sketch
2. Simulate rand. algo. w/ p-random bits.
Goal: Det.ly simulate rand. algo. by GH
L∈BPP
Def
x∊L
x∉L
Prr[A(x,r) = Yes] > 2/3
Prr[A(x,r) = No] > 2/3
|r| = poly(|x|)
A(・,・): poly-time det. TM
Proof Sketch
2. Simulate rand. algo. w/ p-random bits.
Goal: Det.ly simulate rand. algo. by GH
Trivial Simulation
Enumerate all possible 𝑚-bit strings!
A(x,00…00) A(x,00…01)
#Yes > 2
𝑚
∙
2
3
#No > 2
𝑚
∙
2
3
x∊L
x∉L
=
=
No
A(x,11…10) A(x,11…11)
Require O(2m)=O(2poly(n)) time…
Yes
Yes
=
=
Yes
…
Proof Sketch
2. Simulate rand. algo. w/ p-random bits.
Goal: Det.ly simulate rand. algo. by GH
Simulation w/ GH
A(x,・) = circuit C
Enumerate all possible 𝑂 log 𝑚 -bit seeds of GH!
A(x,GH(0…0))
…
=
=
A(x,G (1…1))
Require 2O(log m) =H poly(n) time!
No
Yes
#Yes > 3
𝑂
∙
2
4
log 𝑚
#No > 3
𝑂
∙
2
4
log 𝑚
x∊L
x∉L
Is 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!!
Theorem [Kabanets & Impagliazzo ‘03]
BPP=P  Some problem is hard.
Theorem [Gutfreund & Kawachi ‘10,
Aaronson, Aydinlioglu, Buhrman, Hitchcock, & van Melkebeek ‘11]
prAM ⊆ NP  Some problem is extremely hard.
Theorem [Kabanets & Impagliazzo ‘03]
BPP = P
NEXP ⊄ SIZE poly , or
Permanent ∉ ASIZE poly
Resolving
“arithmetic-circuit version of NP vs. P“
Theorem [Gutfreund & Kawachi ‘10,
Aaronson, Aydinlioglu, Buhrman, Hitchcock, & van Melkebeek ‘11]
prAM ⊆ NP
EXPNP ⊄ SIZE 20.1𝑛
Summary
• Proving circuit complexity  Derandomization
– through Pseudo-Random Generator
– BPP = P, AM = NP, and more…
• Derandomization  Proving circuit complexity
Proving Circuit Complexity ≈ Derandomization