Depth 4 lower bounds for elementary symmetric polynomials Nutan - PowerPoint PPT Presentation

Depth 4 lower bounds for elementary symmetric polynomials Nutan Limaye CSE, IITB Joint work with Herv´ e Fournier, Meena Mahajan and Srikanth Srinivasan Workshop on low depth complexity St. Petersburg, Russia, May, 2016

Elementary symmetric polynomials Elemenraty symmetric polynomial of degree D on n variables. S D X Y n ( X ) := x i T ⊆ [ n ]: | T | = D i ∈ T

Elementary symmetric polynomials Elemenraty symmetric polynomial of degree D on n variables. S D X Y n ( X ) := x i T ⊆ [ n ]: | T | = D i ∈ T where, X := { x 1 , x 2 , . . . , x n } .

Elementary symmetric polynomials Elemenraty symmetric polynomial of degree D on n variables. S D X Y n ( X ) := x i T ⊆ [ n ]: | T | = D i ∈ T where, X := { x 1 , x 2 , . . . , x n } . Polynomials which have polynomial sized circuits DET, IMM,

Elementary symmetric polynomials Elemenraty symmetric polynomial of degree D on n variables. S D X Y n ( X ) := x i T ⊆ [ n ]: | T | = D i ∈ T where, X := { x 1 , x 2 , . . . , x n } . Polynomials which have polynomial sized circuits DET, IMM, S D n are all in VP

Elementary symmetric polynomials Elemenraty symmetric polynomial of degree D on n variables. S D X Y n ( X ) := x i T ⊆ [ n ]: | T | = D i ∈ T where, X := { x 1 , x 2 , . . . , x n } . Polynomials which have polynomial sized circuits DET, IMM, S D n are all in VP, i.e. they have poly sized circuits.

Elementary symmetric polynomials Elemenraty symmetric polynomial of degree D on n variables. S D X Y n ( X ) := x i T ⊆ [ n ]: | T | = D i ∈ T where, X := { x 1 , x 2 , . . . , x n } . Polynomials which have polynomial sized circuits DET, IMM, S D n are all in VP, i.e. they have poly sized circuits.

Recall: depth 3, depth 4 formulas Depth 3 formulas X Y X

Recall: depth 3, depth 4 formulas Depth 3 formulas X Y X Depth 4 formulas X Y X Y

Recall: depth 3, depth 4 formulas Depth 3 formulas X Y X Depth 4 formulas X Y X Y Depth 4 formulas with fan-in bounds X Y [ p ] X Y [ q ]

Recall: depth 3, depth 4 formulas Depth 3 formulas X Y X Depth 4 formulas X Y X Y Depth 4 formulas with fan-in bounds X Y [ p ] X Y [ q ] Homogeneous vs. inhomogeneous Degree of all the input polynomials to any P gate is the same.

Known results Small inhomogeneous formulas exist For every D 2 N , there is a depth 3 inhomogeneous formula n of size n O (1) [Ben-Or]. computing S D

Known results Small inhomogeneous formulas exist For every D 2 N , there is a depth 3 inhomogeneous formula n of size n O (1) [Ben-Or]. computing S D Homogeneous depth 3 lower bound Any depth 3 homogeneous formula computing S D n requires size n Ω ( D ) [Nisan & Wigderson, 1997].

Known results Small inhomogeneous formulas exist For every D 2 N , there is a depth 3 inhomogeneous formula n of size n O (1) [Ben-Or]. computing S D Homogeneous depth 3 lower bound Any depth 3 homogeneous formula computing S D n requires size n Ω ( D ) [Nisan & Wigderson, 1997].

Natural questions Depth 4 homogeneous formulas for S D n Does S D n have a depth 4 homogeneous formula of size poly( n , D )?

Natural questions Depth 4 homogeneous formulas for S D n Does S D n have a depth 4 homogeneous formula of size poly( n , D )? Open.

Natural questions Depth 4 homogeneous formulas for S D n Does S D n have a depth 4 homogeneous formula of size poly( n , D )? Open. Strong lower bounds for inhomonegeous depth 3 formulas

Natural questions Depth 4 homogeneous formulas for S D n Does S D n have a depth 4 homogeneous formula of size poly( n , D )? Open. Strong lower bounds for inhomonegeous depth 3 formulas Does there exists an explicit f ( X ) 2 F [ X ] on n variables such that depth 3 inhomogeneous formula computing f requires size n ω (1) ?

Natural questions Depth 4 homogeneous formulas for S D n Does S D n have a depth 4 homogeneous formula of size poly( n , D )? Open. Strong lower bounds for inhomonegeous depth 3 formulas Does there exists an explicit f ( X ) 2 F [ X ] on n variables such that depth 3 inhomogeneous formula computing f requires size n ω (1) ? Open.

Natural questions Depth 4 homogeneous formulas for S D n Does S D n have a depth 4 homogeneous formula of size poly( n , D )? Open. Strong lower bounds for inhomonegeous depth 3 formulas Does there exists an explicit f ( X ) 2 F [ X ] on n variables such that depth 3 inhomogeneous formula computing f requires size n ω (1) ? Open.

Our result Depth 4 homogeneous formulas for S D n Does S D n have a depth 4 homogeneous formula of size poly( n , D )?

Our result Depth 4 homogeneous formulas for S D n Does S D n have a depth 4 homogeneous formula of size poly( n , D )? Open.

Our result Depth 4 homogeneous formulas for S D n Does S D n have a depth 4 homogeneous formula of size poly( n , D )? Open. Our Result: Any depth 4 homogeneous ΣΠΣΠ [ t ] formula computing S D n requires size n Ω ( D / t ) .

Our result Depth 4 homogeneous formulas for S D n Does S D n have a depth 4 homogeneous formula of size poly( n , D )? Open. Our Result: Any depth 4 homogeneous ΣΠΣΠ [ t ] formula computing S D n requires size n Ω ( D / t ) . For D = O (log n / log log n ).

Our result Depth 4 homogeneous formulas for S D n Does S D n have a depth 4 homogeneous formula of size poly( n , D )? Open. Our Result: Any depth 4 homogeneous ΣΠΣΠ [ t ] formula computing S D n requires size n Ω ( D / t ) . For D = O (log n / log log n ).

Proving lower bounds Notation Let p ( X ) be a polynomial over a field F . Let C be an arithmetic circuit of size s .

Proving lower bounds Notation Let p ( X ) be a polynomial over a field F . Let C be an arithmetic circuit of size s . Goal To prove that if C computes p then s � L .

Proving lower bounds Notation Let p ( X ) be a polynomial over a field F . Let C be an arithmetic circuit of size s . Goal To prove that if C computes p then s � L . To prove this

Proving lower bounds Notation Let p ( X ) be a polynomial over a field F . Let C be an arithmetic circuit of size s . Goal To prove that if C computes p then s � L . To prove this Design a function µ : F [ X ] ! R , such that

Proving lower bounds Notation Let p ( X ) be a polynomial over a field F . Let C be an arithmetic circuit of size s . Goal To prove that if C computes p then s � L . To prove this Design a function µ : F [ X ] ! R , such that µ ( C )  U · s

Proving lower bounds Notation Let p ( X ) be a polynomial over a field F . Let C be an arithmetic circuit of size s . Goal To prove that if C computes p then s � L . To prove this Design a function µ : F [ X ] ! R , such that µ ( C )  U · s µ ( p ) > L

Proving lower bounds Notation Let p ( X ) be a polynomial over a field F . Let C be an arithmetic circuit of size s . Goal To prove that if C computes p then s � L . To prove this Design a function µ : F [ X ] ! R , such that µ ( C )  U · s µ ( p ) > L Conclude that s � L / U

Partial derivatives of S D n Notation

Partial derivatives of S D n Notation ∂ k ( p ) For R = { i 1 , i 2 , . . . , i k } ✓ [ n ], let ∂ R ( p ) := ∂ x i 1 ∂ x i 2 ... ∂ x ik

Partial derivatives of S D n Notation ∂ k ( p ) For R = { i 1 , i 2 , . . . , i k } ✓ [ n ], let ∂ R ( p ) := ∂ x i 1 ∂ x i 2 ... ∂ x ik Example Let R = { 1 } . Then, ∂ R ( x 1 + x 2 + x 3 ) = 1

Partial derivatives of S D n Notation ∂ k ( p ) For R = { i 1 , i 2 , . . . , i k } ✓ [ n ], let ∂ R ( p ) := ∂ x i 1 ∂ x i 2 ... ∂ x ik Example Let R = { 1 } . Then, ∂ R ( x 1 + x 2 + x 3 ) = 1 Let R = { 1 , 2 } . Then, ∂ R ( x 1 + x 2 + x 3 ) = 0

Partial derivatives of S D n Notation ∂ k ( p ) For R = { i 1 , i 2 , . . . , i k } ✓ [ n ], let ∂ R ( p ) := ∂ x i 1 ∂ x i 2 ... ∂ x ik Example Let R = { 1 } . Then, ∂ R ( x 1 + x 2 + x 3 ) = 1 Let R = { 1 , 2 } . Then, ∂ R ( x 1 + x 2 + x 3 ) = 0 n ) := ∂ ( S D n ) = S D − 1 Let R = { 1 } . Then, ∂ R ( S D n − 1 ( x 2 , . . . , x n ) ∂ x 1

Partial derivatives of S D n Notation ∂ k ( p ) For R = { i 1 , i 2 , . . . , i k } ✓ [ n ], let ∂ R ( p ) := ∂ x i 1 ∂ x i 2 ... ∂ x ik Example Let R = { 1 } . Then, ∂ R ( x 1 + x 2 + x 3 ) = 1 Let R = { 1 , 2 } . Then, ∂ R ( x 1 + x 2 + x 3 ) = 0 n ) := ∂ ( S D n ) = S D − 1 Let R = { 1 } . Then, ∂ R ( S D n − 1 ( x 2 , . . . , x n ) ∂ x 1 Let ∂ k ( p ) := { ∂ R ( p ) | R ✓ [ n ] , | R | = k } .

Partial derivatives of S D n Notation ∂ k ( p ) For R = { i 1 , i 2 , . . . , i k } ✓ [ n ], let ∂ R ( p ) := ∂ x i 1 ∂ x i 2 ... ∂ x ik Example Let R = { 1 } . Then, ∂ R ( x 1 + x 2 + x 3 ) = 1 Let R = { 1 , 2 } . Then, ∂ R ( x 1 + x 2 + x 3 ) = 0 n ) := ∂ ( S D n ) = S D − 1 Let R = { 1 } . Then, ∂ R ( S D n − 1 ( x 2 , . . . , x n ) ∂ x 1 Let ∂ k ( p ) := { ∂ R ( p ) | R ✓ [ n ] , | R | = k } . The partial derivative measure: µ k ( p ) := dim { span ∂ k ( p ) }