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

Dispersers and Circuit Lower Bounds Alexander Golovnev New York University ITCS 2016

• p i x n [Hås89] 3 f • p i x o n [PSZ97] 3 f depth- 3 circuits x 7 x 5 x 4 x 4 x 7 x 5 x 4 x 7 x 5 depth: 3, bottom fan-in: unbounded x 7 x 7 x 5 x 7 x 5 x 7 x 5 x 7 x 5 x 7 x 4 x 7 depth: 3, bottom fan-in: 2 x 4 x 5 x 5 x 4 • Bit-fixing Disperser x j c j • parity • 2 • Projections Disperser x j x k c j • BCH codes [PSZ97] • 2 2 n x 4 x 5 x 7 x 4 Dispersers x 7 2 f : { 0 , 1 } n → { 0 , 1 } is a disperser if f | S ̸≡ const , ∀ S = { x ∈ { 0 , 1 } n : p 1 ( x ) = . . . = p k ( x ) = 0 } .

• p i x o n [PSZ97] 3 f depth- 3 circuits x 7 x 4 x 5 x 7 x 4 x 5 x 7 x 4 x 5 x 7 depth: 3, bottom fan-in: unbounded x 4 Dispersers x 7 x 5 x 7 x 5 x 7 x 5 x 7 x 5 x 7 x 4 x 7 depth: 3, bottom fan-in: 2 x 5 x 4 2 x j c j • BCH codes [PSZ97] • 2 2 n • Projections Disperser • parity • Bit-fixing Disperser x 4 x 5 x 7 x 4 x 5 x 7 x k f : { 0 , 1 } n → { 0 , 1 } is a disperser if f | S ̸≡ const , ∀ S = { x ∈ { 0 , 1 } n : p 1 ( x ) = . . . = p k ( x ) = 0 } . ∨ ∨ ∨ ∨ ∨ ∨ • p i ( x ) = x j ⊕ c j ∧ ∧ ∧ ∨ • Σ 3 ( f ) ≥ 2 Ω( √ n ) [Hås89]

depth- 3 circuits x 4 x 5 x 7 Dispersers x 4 x 5 x 7 x 4 x 5 x 7 depth: 3, bottom fan-in: unbounded x 7 x 7 x 5 x 7 x 5 x 7 x 5 x 7 x 5 x 7 x 4 x 7 depth: 3, bottom fan-in: 2 x 4 2 x 5 • BCH codes [PSZ97] x 4 • Bit-fixing Disperser x 7 x 5 x 4 • parity x 7 x 5 x 4 • Projections Disperser f : { 0 , 1 } n → { 0 , 1 } is a disperser if f | S ̸≡ const , ∀ S = { x ∈ { 0 , 1 } n : p 1 ( x ) = . . . = p k ( x ) = 0 } . ∨ ∨ ∨ ∨ ∨ ∨ • p i ( x ) = x j ⊕ c j ∧ ∧ ∧ ∨ • Σ 3 ( f ) ≥ 2 Ω( √ n ) [Hås89] ∨ ∨ ∨ ∨ ∨ ∨ • p i ( x ) = x j ⊕ x k ⊕ c j ∧ ∧ ∧ ∨ 3 ( f ) ≥ 2 n − o ( n ) [PSZ97] • Σ 2

general circuits 2 depth: unbounded, fan-in: 2 x 4 x 3 x 2 x 1 3 1 n [GK16] Dispersers • over large fields [Dvi09] • C f • deg p i • Quadratic Disperser • constructions in P [BK09] • Affine Disperser 3 f : { 0 , 1 } n → { 0 , 1 } is a disperser if f | S ̸≡ const , ∀ S = { x ∈ { 0 , 1 } n : p 1 ( x ) = . . . = p k ( x ) = 0 } . • p i ( x ) = ⊕ j ∈ J x j ⊕ c i ⊕ ∧ • C ( f ) ≥ 3 . 01 n [FGHK16] ∨ ⊕ ⊕ ⊕

general circuits • over large fields [Dvi09] depth: unbounded, fan-in: 2 x 4 x 3 x 2 x 1 Dispersers 3 • Quadratic Disperser • constructions in P [BK09] • Affine Disperser f : { 0 , 1 } n → { 0 , 1 } is a disperser if f | S ̸≡ const , ∀ S = { x ∈ { 0 , 1 } n : p 1 ( x ) = . . . = p k ( x ) = 0 } . • p i ( x ) = ⊕ j ∈ J x j ⊕ c i ⊕ ∧ • C ( f ) ≥ 3 . 01 n [FGHK16] ∨ ⊕ ⊕ • deg ( p i ) ≤ 2 ⊕ • C ( f ) ≥ 3 . 1 n [GK16]

log-depth circuits n -bound for NC 1 depth: O log n , fan-in: 2 x 4 x 3 x 2 series-parallel circuit x 4 x 3 x 2 Dispersers x 1 • • no known constructions • Varieties of const deg • no known constructions 4 • deg p i • Varieties of poly deg n f : { 0 , 1 } n → { 0 , 1 } is a disperser if f | S ̸≡ const , ∀ S = { x ∈ { 0 , 1 } n : p 1 ( x ) = . . . = p k ( x ) = 0 } . ⊕ ∧ • deg ( p i ) ≤ const ∨ • ω ( n ) -bound for s.-p. NC 1 ≡ depth: O ( log n ) , fan-in: 2

log-depth circuits x 1 x 4 x 3 x 2 series-parallel circuit Dispersers x 4 x 3 x 2 4 • no known constructions • Varieties of poly deg • no known constructions • Varieties of const deg f : { 0 , 1 } n → { 0 , 1 } is a disperser if f | S ̸≡ const , ∀ S = { x ∈ { 0 , 1 } n : p 1 ( x ) = . . . = p k ( x ) = 0 } . ⊕ ∧ • deg ( p i ) ≤ const ∨ • ω ( n ) -bound for s.-p. NC 1 ≡ depth: O ( log n ) , fan-in: 2 • deg ( p i ) ≤ n ε ⊕ ∧ ⊕ ∨ • ω ( n ) -bound for NC 1 ⊕ depth: O ( log n ) , fan-in: 2