Dispersers and Circuit Lower Bounds Alexander Golovnev New York - - PowerPoint PPT Presentation

Dispersers and Circuit Lower Bounds Alexander Golovnev

New York University

ITCS 2016

depth-3 circuits

Dispersers f: {0, 1}n → {0, 1} is a disperser if f|S ̸≡ const, ∀S = {x ∈ {0, 1}n : p1(x) = . . . = pk(x) = 0}.

• Bit-fixing Disperser
• pi x

xj cj

• parity
• 3 f

2

n [Hås89]

• Projections Disperser
• pi x

xj xk cj

• BCH codes [PSZ97]
• 2

3 f

2n

• n [PSZ97]

x4 x5 x7 x4 x5 x7 x4 x5 x7 x4 x5 x7 x4 x5 x7 x4 x5 x7 depth: 3, bottom fan-in: unbounded x4 x7 x5 x7 x5 x7 x5 x7 x5 x7 x4 x7 depth: 3, bottom fan-in: 2

2

depth-3 circuits

Dispersers f: {0, 1}n → {0, 1} is a disperser if f|S ̸≡ const, ∀S = {x ∈ {0, 1}n : p1(x) = . . . = pk(x) = 0}.

• Bit-fixing Disperser
• pi(x) = xj ⊕ cj
• parity
• Σ3(f) ≥ 2Ω(√n) [Hås89]
• Projections Disperser
• pi x

xj xk cj

• BCH codes [PSZ97]
• 2

3 f

2n

• n [PSZ97]

∨ ∧ ∨ x4 x5 x7 ∨ x4 x5 x7 ∧ ∨ x4 x5 x7 ∨ x4 x5 x7 ∧ ∨ x4 x5 x7 ∨ x4 x5 x7 depth: 3, bottom fan-in: unbounded x4 x7 x5 x7 x5 x7 x5 x7 x5 x7 x4 x7 depth: 3, bottom fan-in: 2

2

depth-3 circuits

Dispersers f: {0, 1}n → {0, 1} is a disperser if f|S ̸≡ const, ∀S = {x ∈ {0, 1}n : p1(x) = . . . = pk(x) = 0}.

• Bit-fixing Disperser
• pi(x) = xj ⊕ cj
• parity
• Σ3(f) ≥ 2Ω(√n) [Hås89]
• Projections Disperser
• pi(x) = xj ⊕ xk ⊕ cj
• BCH codes [PSZ97]
• Σ2

3(f) ≥ 2n−o(n) [PSZ97]

∨ ∧ ∨ x4 x5 x7 ∨ x4 x5 x7 ∧ ∨ x4 x5 x7 ∨ x4 x5 x7 ∧ ∨ x4 x5 x7 ∨ x4 x5 x7 depth: 3, bottom fan-in: unbounded ∨ ∧ ∨ x4 x7 ∨ x5 x7 ∧ ∨ x5 x7 ∨ x5 x7 ∧ ∨ x5 x7 ∨ x4 x7 depth: 3, bottom fan-in: 2

2

general circuits

Dispersers f: {0, 1}n → {0, 1} is a disperser if f|S ̸≡ const, ∀S = {x ∈ {0, 1}n : p1(x) = . . . = pk(x) = 0}.

• Affine Disperser
• pi(x) = ⊕

j∈J xj ⊕ ci

• constructions in P [BK09]
• C(f) ≥ 3.01n [FGHK16]
• Quadratic Disperser
• deg pi

2

• over large fields [Dvi09]
• C f

3 1n [GK16]

x1 x2 x3 x4 ⊕ ∧ ∨ ⊕ ⊕ ⊕ depth: unbounded, fan-in: 2

3

general circuits

Dispersers f: {0, 1}n → {0, 1} is a disperser if f|S ̸≡ const, ∀S = {x ∈ {0, 1}n : p1(x) = . . . = pk(x) = 0}.

• Affine Disperser
• pi(x) = ⊕

j∈J xj ⊕ ci

• constructions in P [BK09]
• C(f) ≥ 3.01n [FGHK16]
• Quadratic Disperser
• deg(pi) ≤ 2
• over large fields [Dvi09]
• C(f) ≥ 3.1n [GK16]

x1 x2 x3 x4 ⊕ ∧ ∨ ⊕ ⊕ ⊕ depth: unbounded, fan-in: 2

3

log-depth circuits

Dispersers f: {0, 1}n → {0, 1} is a disperser if f|S ̸≡ const, ∀S = {x ∈ {0, 1}n : p1(x) = . . . = pk(x) = 0}.

• Varieties of const deg
• deg(pi) ≤ const
• no known constructions
• ω(n)-bound for s.-p. NC1
• Varieties of poly deg
• deg pi

n

• no known constructions
• n -bound for NC1

x1 x2 x3 x4 ⊕ ∧ ∨ ≡

depth: O(log n), fan-in: 2 series-parallel circuit

x2 x3 x4

depth: O log n , fan-in: 2

4

log-depth circuits

Dispersers f: {0, 1}n → {0, 1} is a disperser if f|S ̸≡ const, ∀S = {x ∈ {0, 1}n : p1(x) = . . . = pk(x) = 0}.

• Varieties of const deg
• deg(pi) ≤ const
• no known constructions
• ω(n)-bound for s.-p. NC1
• Varieties of poly deg
• deg(pi) ≤ nε
• no known constructions
• ω(n)-bound for NC1

x1 x2 x3 x4 ⊕ ∧ ∨ ≡

depth: O(log n), fan-in: 2 series-parallel circuit

x2 x3 x4

depth: O(log n), fan-in: 2

⊕ ∧ ∨ ⊕ ⊕ 4