Weighted Gate Elimination Alexander Golovnev New York University - PowerPoint PPT Presentation

Weighted Gate Elimination Alexander Golovnev New York University Alexander S. Kulikov Steklov Institute of Mathematics ITCS 2016

Gate Elimination Lower Bounds for Affine Dispersers Lower Bound for Quadratic Dispersers Open Problems

Outline Gate Elimination Lower Bounds for Affine Dispersers Lower Bound for Quadratic Dispersers Open Problems

Boolean Circuits Inputs: x 1 x 2 x 3 x 1 , . . . , x n , 0 , 1 g 1 = x 1 ⊕ x 2 1 Gates: g 2 = x 2 ∧ x 3 g 1 ∧ g 2 binary ⊕ g 3 = g 1 ∨ g 2 functions g 3 ∨ g 4 Fan-out: g 4 = g 2 ∨ 1 ∨ unbounded g 5 = g 3 ≡ g 4 g 5 Depth: ≡ unbounded

Exponential Bounds Lower Bound Counting shows that almost all functions of n variables have circuit size Ω(2 n / n ) [Shannon 1949]. Upper Bound Any function can be computed by circuits of size (1 + o (1))2 n / n [Lupanov 1958].

New n quadratic dispersers [G, Kulikov 2015] (non-explicit) Explicit Lower Bounds Previous 2 n f ( x ) = ⊕ i < j x i x j [Kloss, Malyshev 1965] f ( x ) = [ ∑ x i ≡ 3 0] 2 n [Schnorr 1974] 2 . 5 n f ( x , a , b ) = x a ⊕ x b [Paul 1977] 2 . 5 n symmetric [Stockmeyer 1977] 3 n f ( x , a , b , c ) = x a x b ⊕ x c [Blum 1984] 3 n affine dispersers [Demenkov, Kulikov 2011] 3 . 011 n affine dispersers [FGHK 2015]

Explicit Lower Bounds Previous 2 n f ( x ) = ⊕ i < j x i x j [Kloss, Malyshev 1965] f ( x ) = [ ∑ x i ≡ 3 0] 2 n [Schnorr 1974] 2 . 5 n f ( x , a , b ) = x a ⊕ x b [Paul 1977] 2 . 5 n symmetric [Stockmeyer 1977] 3 n f ( x , a , b , c ) = x a x b ⊕ x c [Blum 1984] 3 n affine dispersers [Demenkov, Kulikov 2011] 3 . 011 n affine dispersers [FGHK 2015] New 3 . 11 n quadratic dispersers [G, Kulikov 2015] (non-explicit)

GK15 (non-explicit) FGHK15 FGHK15 Explicit Lower Bounds: Pictorially 3 n 2 n S77, P77 n KM65 DK11 B84 S74 1965 1975 1985 1995 2005 2015

GK15 (non-explicit) FGHK15 Explicit Lower Bounds: Pictorially 3 n 2 n S77, P77 FGHK15 n KM65 DK11 B84 S74 1965 1975 1985 1995 2005 2015

FGHK15 Explicit Lower Bounds: Pictorially 3 n GK15 (non-explicit) 2 n S77, P77 FGHK15 n KM65 DK11 B84 S74 1965 1975 1985 1995 2005 2015

Gate Elimination Method To prove, say, a 3 n lower bound for all functions f from a certain class C : ■ show that for any circuit computing f , one can find a substitution eliminating at least 3 gates; ■ show that the resulting subfunction still belongs to C ; ■ proceed by induction.

Gate Elimination: Example x 1 x 2 x 3 x 4 G 1 G 2 ⊕ ∧ G 3 G 4 ⊕ ∨ G 5 ⊕ G 6 ⊕

Gate Elimination: Example x 1 x 2 x 3 x 4 G 1 G 2 ⊕ ∧ G 3 G 4 ⊕ ∨ G 5 ⊕ G 6 ⊕ assign x 1 = 1

Gate Elimination: Example x 2 x 3 x 4 1 G 1 G 2 ⊕ ∧ G 3 G 4 ⊕ ∨ G 5 ⊕ G 6 ⊕ G 5 now computes G 3 ⊕ 1 = ¬ G 3

Gate Elimination: Example x 2 x 3 x 4 G 1 G 2 ⊕ ∧ G 3 G 4 ⊕ ∨ ¬ G 6 ⊕

Gate Elimination: Example x 2 x 3 x 4 G 1 G 2 ⊕ ∧ G 3 G 4 ⊕ ∨ ¬ G 6 ⊕ now we can change the binary function assigned to G 6

Gate Elimination: Example x 2 x 3 x 4 G 1 G 2 ⊕ ∧ G 3 G 4 ⊕ ∨ G 6 ≡

Gate Elimination: Example x 2 x 3 x 4 G 1 G 2 ⊕ ∧ G 3 G 4 ⊕ ∨ G 6 ≡ now assign x 3 = 0

Gate Elimination: Example x 2 x 4 0 G 1 G 2 ⊕ ∧ G 3 G 4 ⊕ ∨ G 6 ≡ G 1 then is equal to x 2

Gate Elimination: Example x 2 x 4 0 G 2 ∧ G 3 G 4 ⊕ ∨ G 6 ≡

Gate Elimination: Example x 2 x 4 0 G 2 ∧ G 3 G 4 ⊕ ∨ G 6 ≡ G 2 = 0

Gate Elimination: Example x 2 x 4 G 2 0 G 3 G 4 ⊕ ∨ G 6 ≡

Gate Elimination: Example x 2 x 4 G 4 ⊕ G 6 ≡

Binary Functions There are 16 Boolean functions. ■ 2 constant functions: 0 , 1 ; ■ 4 degenerate functions: x , x ⊕ 1 , y , y ⊕ 1 ; ■ 2 xor-type functions: x ⊕ y , x ⊕ y ⊕ 1 ; ■ 8 and-type functions: ( x ⊕ a )( y ⊕ b ) ⊕ c where a , b , c ∈ 0 , 1 .

Outline Gate Elimination Lower Bounds for Affine Dispersers Lower Bound for Quadratic Dispersers Open Problems

3 n Lower Bound Theorem If f : { 0 , 1 } n → { 0 , 1 } is an affine disperser for dimension d = o ( n ) , then size ( f ) ≥ 3 n − o ( n ) .

Affine Dispersers ■ A function f : { 0 , 1 } n → { 0 , 1 } is called an affine disperser for dimension d if it is non-constant on any affine subspace of dimension at least d . ■ An affine dispereser for dimension d cannot become constant after any n − d linear substitutions (i.e., substitutions of the form x 2 ⊕ x 3 ⊕ x 9 = 0 ). ■ There exist explicit constructions of affine dispersers for subliner dimension d = o ( n ) (e.g., [Ben-Sasson, Kopparty, 2012]).

XOR-layered Circuits y y x z x z t t x ⊕ y ⊕ ∨ ∧ ∨ ∧ x ⊕ y ⊕ z ⊕ ⊕ ⊕ ∨ ∨ ≡ ≡ inputs ( C ) = 4 inputs ( C ′ ) = 6 size ( C ) = 7 size ( C ′ ) = 5 inputs ( C ′ ) + size ( C ′ ) ≤ inputs ( C ) + size ( C ) .

3 n − o ( n ) Lower Bound Theorem [Demenkov, Kulikov 2011] For a circuit C computing an affine disperser for dimension d : inputs ( C ) + size ( C ) ≥ 4( n − d ) . Corollary size ( f ) ≥ 3 n − o ( n ) for an affine disperser for d = o ( n ) . Proposition The bound is tight: size ( IP ) = n − 1 and IP is an affine disperser for dimension d = n /2 + 1 .

Proof ■ Want to show: inputs ( C ) + size ( C ) ≥ 4( n − d ) . ■ Make n − d affine restrictions each time reducing ( inputs + size ) by at least 4 . ■ Convert C to XOR-layered and take a top-gate A : Case 1 Case 2 L 1 L 2 L 1 L 2 L 1 ← 0 : L 1 ← 0 : ∧ A ∧ A ∆ size = 2 ∆ size = 3 ∆ inp = 2 ∆ inp = 1

Outline Gate Elimination Lower Bounds for Affine Dispersers Lower Bound for Quadratic Dispersers Open Problems

Main Result Theorem Let f : { 0 , 1 } n → { 0 , 1 } be a function that is not constant on any set S ⊆ { 0 , 1 } n of size at least 2 n /100 that can be defined as S = { x : p 1 ( x ) = · · · = p 2 n ( x ) = 0 } , deg ( p i ) ≤ 2 . Then size ( f ) ≥ 3 . 11 n .

Quadratic Dispersers ■ A random function is not constant on any set S of size s that can be defined as S = { x ∈ { 0 , 1 } n : p 1 ( x ) = · · · = p s / n 3 ( x ) = 0 } . ■ We need much weaker dispersers: ( n , 2 n , 2 n /100 ) -dispersers. Even in NP. Even with multiple outputs.

Regular Gate Elimination ■ make a substitution; ■ decrease S by a factor of 2 ; ■ eliminate at least 3 gates; ■ S belongs to the same class; ■ repeat n − o ( n ) times.

Weighted Gate Elimination ■ make a restriction; ■ decrease S by a factor of α ; ■ make sure to eliminate at least 3 log α gates; ■ S belongs to the same class; ■ repeat until S becomes small (e.g., 2 n /100 ).

y y x x Toy Example y x ⊕ ∧ ⊕ ∨ S ⊆ { 0 , 1 } n

y x Toy Example y x ⊕ ∧ ⊕ ∨ xy = 0 y x S ⊆ { 0 , 1 } n ⊕ ∧ ⊕ ∨

Toy Example y x ⊕ ∧ ⊕ ∨ xy = 0 xy = 1 y y x x S ⊆ { 0 , 1 } n ⊕ ∧ ⊕ ⊕ ∧ ⊕ ∨ ∨

Toy Example y x ⊕ ∧ ⊕ ∨ xy = 0 xy = 1 y y x x S ⊆ { 0 , 1 } n ⊕ ∧ ⊕ ⊕ ∧ ⊕ ∨ ∨ S 0 = S | xy =0 S 1 = S | xy =1

Outline Gate Elimination Lower Bounds for Affine Dispersers Lower Bound for Quadratic Dispersers Open Problems

Lower bounds in other models? Connections to algorithms for Circuit-SAT? Open Problems ■ Quadratic dispersers in NP?

Connections to algorithms for Circuit-SAT? Open Problems ■ Quadratic dispersers in NP? ■ Lower bounds in other models?

Open Problems ■ Quadratic dispersers in NP? ■ Lower bounds in other models? ■ Connections to algorithms for Circuit-SAT?

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