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Circuit Complexity and Derandomization Tokyo Institute of Technology Akinori Kawachi

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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

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Deterministic v.s. Randomized Algorithms for (Decision) Problems Randomness is useful for real-world computation! Decision problem: PRIME “No” otherwise Elementary Det. algorithm: O(2 n/2 ) time [Eratosthenes, B.C. 2c] Rand. algorithm: O(n 3 ) time w/ succ. prob. 99% [Miller 1976, Rabin 1980] Exponential-time speed-up! n = input length

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Deterministic v.s. Randomized Algorithms for (Decision) Problems How much randomness make computation strong? Gödel Prize Det. algorithm: O(n 12 ) time [Agrawal, Kayal & Saxena 2004 Gödel Prize] Rand. algorithm: O(n 3 ) time w/ succ. prob. 99% [Miller 1976, Rabin 1980] “No” otherwise Polynomial-time slow-down Decision problem: PRIME

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Derandomization Conjecture BPP = P NO Randomization yields NO exponential speed-up! derandomization Always poly-time derandomization possible? Conjecture det. P = {problem: poly-time det. TM computes} prob. BPP = {problem: poly-time prob. TM computes w/ bounded errors w/ bounded errors}

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Class BPP BPP Class BPP (Bounded-error Prob. Poly-time) L ∈ BPP x∊Lx∊L x∉Lx∉L Def Pr r [A(x,r) = Yes] > 2/3 r is uniform over {0,1} m m = |r| = poly(|x|) A( ・, ・ ): poly-time det. TM Pr r [A(x,r) = No] > 2/3

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Nondeterministic Version AM = NP Conjecture AM Class AM (Arthur-Merlin Games) L ∈ AM x∊Lx∊L x∉Lx∉L Def |r|,|w| = poly(|x|) A( ・, ・, ・ ): poly-time det. TM

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Hardness vs. Randomness Trade-offs [Yao ’82, Blum & Micali ’84] Hard problem exists Pseudo-Random Generator Good Pseudo-Random Generator (PRG) exists. Simulate randomized algorithms det.ly with PRG! Theorem [Impagliazzo & Wigderson 1998] 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]

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Circuit x3x3 ∧ x1x1 x2x2 0 ￢ ∨∧ ∧ ∨ Gate set = { ∧, ∨, ￢, 0, 1}

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Circuit 0 ∧ 11 ￢ ∨∧ ∧ ∨ 1 ∧ 1 = 1 1 1 1 1 00 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

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Circuit 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 Definition Def &

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Computational 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(2 n /n)-size circuit family. Theorem [Fisher & Pippenger 1979] Poly-time TM can be simulated by poly-size circuit family. SIZE(poly) = {problem: poly-size circuit family can compute}

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NP vs. P and Circuits NP ≠ P Conjecture 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 NP solves NP vs. P!

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NEXP ⊄ SIZE(poly) MA-EXP ⊄ SIZE(poly) Current Status Theorem (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 )!

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Hardness vs. Randomness Trade-offs [Yao ’82, Blum & Micali ’84] Hard problem exists Pseudo-Random Generator Good Pseudo-Random Generator (PRG) exists. Simulate randomized algorithms det.ly with PRG! Theorem [Impagliazzo & Wigderson 1998] BPP = P (L is computed in prob. poly-time w/ bounded errors L is computed in det. poly-time)

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Proof Sketch 1.Construct PRG from hard H. 2.Simulate rand. algo. w/ p-random bits.

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Proof Sketch 1.Construct PRG from hard H. Goal: Construct G H : {0,1} O(log m) → {0,1} m Pseudo-random! truly random! # possible s = 2 O(log m) = poly(m) # possible r = 2 m Point For every poly-size circuit C,

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

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Proof Sketch 2.Simulate rand. algo. w/ p-random bits. Goal: Det.ly simulate rand. algo. by G H Trivial Simulation A(x,00…00) = Yes A(x,00…01) = No … A(x,11…10) = Yes A(x,11…11) = Yes x∊Lx∊L x∉Lx∉L Require O(2 m )=O(2 poly(n) ) time…

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Proof Sketch 2.Simulate rand. algo. w/ p-random bits. Goal: Det.ly simulate rand. algo. by G H Simulation w/ G H A(x,G H (0…0)) = No … A(x,G H (1…1)) = Yes x∊Lx∊L x∉Lx∉L Require 2 O(log m) = poly(n) time! A(x, ・ ) = circuit C

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Is Circuit Complexity Essential? Proving “some problem is really hard” is HARD! (e.g. NP≠P) ultimate goal – 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] Theorem [Gutfreund & Kawachi ‘10, Aaronson, Aydinlioglu, Buhrman, Hitchcock, & van Melkebeek ‘11] Theorem [Gutfreund & Kawachi ‘10, Aaronson, Aydinlioglu, Buhrman, Hitchcock, & van Melkebeek ‘11]

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Theorem [Kabanets & Impagliazzo ‘03] Resolving “arithmetic-circuit version of NP vs. P“ Theorem [Gutfreund & Kawachi ‘10, Aaronson, Aydinlioglu, Buhrman, Hitchcock, & van Melkebeek ‘11] Theorem [Gutfreund & Kawachi ‘10, Aaronson, Aydinlioglu, Buhrman, Hitchcock, & van Melkebeek ‘11]

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Summary Proving circuit complexity Derandomization – through Pseudo-Random Generator – BPP = P, AM = NP, and more… Derandomization Proving circuit complexity

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CS151 Complexity Theory Lecture 7 April 20, 2015.

CS151 Complexity Theory Lecture 7 April 20, 2015.

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