Circuit Complexity and Derandomization

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 Algorithmsfor (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 xPRIME (x is prime) Exponential-time speed-up! “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]

Deterministic v.s. Randomized Algorithmsfor (Decision) Problems How much randomness make computation strong? Decision problem: PRIME Input: n-bit number x (0 N < 2n) Output: “Yes” if xPRIME (x is prime) Polynomial-time slow-down “No” otherwise Rand. algorithm: O(n3) time w/ succ. prob. 99% [Miller 1976, Rabin 1980] Det. algorithm: O(n12) time [Agrawal, Kayal & Saxena 2004Gödel 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 x∊L Prr[A(x,r) = Yes] > 2/3 Def x∉L 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 x∊L Prr[w: A(x,w,r) = Yes] > 2/3 Def x∉L 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  Good Pseudo-Random Generator (PRG) exists. • Simulate randomized algorithms det.ly with PRG! Similar theorem holds in nondet. version (AM=NP) [Klivans & van Melkebeek 2001] Theorem[Impagliazzo & Wigderson 1998] 2O()-time computabledecision problem H s.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)

Circuit Gate set = {∧, ∨, ￢, 0, 1} ∨ ∧ ∧ ∨ ∧ ￢ x1 x2 x3 0

Circuit 1 Gate set = {∧, ∨, ￢, 0, 1} 1∨0= 1 ∨ 0∧1 = 0 0 1 ∧ 0∨1 = 1 1∧0= 0 1 0 ∧ ∨ ￢0 = 1 1∧1 = 1 1 1 0 0 ∧ ￢ 1 1 Input = (1,1,0) 0 Size = 7 Depth = 5 1 1 0 0

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 x L  C|x|(x) = 1 size of Cn s(n) & Def x L  C|x|(x) = 0 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 anypoly-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 Theorem (Buhrman, Fortnow, & Thierauf 1998) • MA-EXP ⊄ SIZE(poly) Const-depth poly-size w/ Modulo gates Theorem (Williams 2011) 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()-time computabledecision problem H s.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 • Construct PRG from hard H. • Simulate rand. algo. w/ p-random bits.

Proof Sketch • 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 • Simulate rand. algo. w/ p-random bits. Goal: Det.ly simulate rand. algo. by GH L∈BPP x∊L Prr[A(x,r) = Yes] > 2/3 Def x∉L Prr[A(x,r) = No] > 2/3 |r| = poly(|x|) A(・,・): poly-time det. TM

Proof Sketch • Simulate rand. algo. w/ p-random bits. Goal: Det.ly simulate rand. algo. by GH Trivial Simulation Enumerate all possible -bit strings! … Require O(2m)=O(2poly(n)) time… A(x,00…00) A(x,00…01) A(x,11…10) A(x,11…11) = = = = Yes No Yes Yes #Yes > x∊L x∉L #No >

Proof Sketch • 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 -bit seeds of GH! … Require 2O(log m) = poly(n) time! A(x,GH(0…0)) A(x,GH(1…1)) = = No Yes #Yes > x∊L x∉L #No >

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] prAMNP  Some problem is extremely hard.

Theorem[Kabanets & Impagliazzo‘03] BPP P NEXP SIZE, or Permanent ASIZE Resolving “arithmetic-circuit version of NP vs. P“ Theorem[Gutfreund & Kawachi‘10, Aaronson, Aydinlioglu, Buhrman, Hitchcock, & van Melkebeek ‘11] prAM NP EXPNP SIZE

Summary • Proving circuit complexity  Derandomization • through Pseudo-Random Generator • BPP = P, AM = NP, and more… • Derandomization  Proving circuit complexity Proving Circuit Complexity Derandomization

Circuit Complexity and Derandomization