circuit complexity and derandomization tokyo institute of technology akinori kawachi
TRANSCRIPT
Circuit Complexity and Derandomization
Tokyo Institute of TechnologyAkinori 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 Algorithmsfor (Decision) Problems
Randomness is useful for real-world computation!
Decision problem: PRIME
Input: n-bit number x (0 x < 2n)
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
Deterministic v.s. Randomized Algorithmsfor (Decision) Problems
How much randomness make computation strong?
Det. algorithm: O(n12) 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
Derandomization Conjecture
BPP = PRandomization 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}
Class BPP
Class BPP (Bounded-error Prob. Poly-time)
L BPP∈x L∊x L∉
DefPrr[A(x,r) = Yes] > 2/3
r is uniform over {0,1}m
m = |r| = poly(|x|)A( ・ , ・ ): poly-time det. TM
Prr[A(x,r) = No] > 2/3
Nondeterministic Version
AM = NPConjecture
Class AM (Arthur-Merlin Games)
L AM∈x L∊x L∉
DefPrr[w: A(x,w,r) = Yes] > 2/3
|r|,|w| = poly(|x|)A( ・ , ・ , ・ ): poly-time det. TM
Prr[w: A(x,w,r) = No] > 2/3
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()-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]
Circuit
x3
∧
x1 x2 0
￢
∨∧
∧
∨
Gate set = { , , ∧ ∨ ￢ , 0, 1}
Gate set = { , , ∧ ∨ ￢ , 0, 1}
Circuit
0
∧
1 1
￢
∨∧
∧
∨
1 1 = 1∧
1
1
1
1
0 0
0
1
10
￢ 0 = 1
1 0 = 0∧ 0 1 = 1∨
0 1 = 0∧0
1 0 = 1∨
1
Input = (1,1,0) 0
Size = 7Depth = 5
Circuit ComplexitySize of circuits is measure for computational resource!
Circuit complexity of L:= min { size of circuit family computing L }
s(n)-size circuit family {Cn:{0,1}n→{0,1}}n computes L
Definition
Defx L C|x|(x) = 1x L C|x|(x) = 0
& size of Cn s(n)
Computational Power of Circuits
Circuit complexity of any problem = O(2n/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}
NP vs. P and Circuits
NP ≠ PConjecture
Some NP problem cannot be computed by any poly-time TM.
NP SIZE(poly)⊄Conjecture
Some NP problem has superpoly circuit complexity.
Note: NP SIZE(poly) ⊄ NP ≠ P
Proving super-poly circuit complexity in NPsolves NP vs. P!
NEXP SIZE(poly)⊄
MA-EXP SIZE(poly)⊄
Current StatusTheorem (Buhrman, Fortnow, & Thierauf 1998)
NEXP ACC⊄ 0(poly)Theorem (Williams 2011)
Randomized version of NEXP
Const-depth poly-size w/ Modulo gates
Grand Challenge
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()-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)
Proof Sketch
1. Construct PRG from hard H.2. Simulate rand. algo. w/ p-random bits.
Proof Sketch1. Construct PRG from hard H.
Goal: Construct GH: {0,1}O(log m) → {0,1}m
Prs[ C(GH(s)) = 1 ] Prr[ C(r) = 1 ]Pseudo-random! truly random!
# possible s = 2O(log m) = poly(m)# possible r = 2m
Point
Proof: good distinguisher D for GH small circuit CD for H
For every poly-size circuit C,
Proof Sketch2. Simulate rand. algo. w/ p-random bits.
Goal: Det.ly simulate rand. algo. by GH
L BPP∈x L∊x L∉
DefPrr[A(x,r) = Yes] > 2/3
|r| = poly(|x|)A( ・ , ・ ): poly-time det. TM
Prr[A(x,r) = No] > 2/3
Proof Sketch2. Simulate rand. algo. w/ p-random bits.
Goal: Det.ly simulate rand. algo. by GH
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(2m)=O(2poly(n)) time…
Proof Sketch2. Simulate rand. algo. w/ p-random bits.
Goal: Det.ly simulate rand. algo. by GH
Simulation w/ GH
Enumerate all possible -bit seeds of GH!
A(x,GH(0…0))=
No
… A(x,GH(1…1))=
Yes
#Yes > x L∊#No > x L∉
Require 2O(log m) = poly(n) time!
A(x, ・ ) = circuit C
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!!
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]
BPP P
NEXP SIZE, orPermanent ASIZE
Theorem [Kabanets & Impagliazzo ‘03]
Resolving “arithmetic-circuit version of NP vs. P“
prAM NP
EXPNP SIZE
Theorem [Gutfreund & Kawachi ‘10, Aaronson, Aydinlioglu, Buhrman, Hitchcock, & van Melkebeek ‘11]
Summary
• Proving circuit complexity Derandomization– through Pseudo-Random Generator– BPP = P, AM = NP, and more…
• Derandomization Proving circuit complexity
Proving Circuit Complexity Derandomization