What can functional computation do? ∑ Sym d ( x 1 ,..., x n ) x i 1 ··· x i d = 1 ≤ i 1 < ··· < i d ≤ n If we only care about x ∈ { 0,1 } n , Sym d ( x ) = f ( ℓ ) = a 0 + a 1 ℓ + ··· + a n + 1 ℓ n + 1 ∈ depth-3 powering circuits Syntactic computation of Sym d by such depth-3 powering circuits require n Ω ( d ) size. [Nisan-Wigderson]
Why does the proof break down? Associates a number to every polynomial. ▶ Lower bounds in alg. complexity use some complexity measure. ▶ Often is the rank of collection of linear operators.
Why does the proof break down? Associates a number to every polynomial. ▶ Lower bounds in alg. complexity use some complexity measure. ▶ Often is the rank of collection of linear operators. ▶ For depth-3 powering circuits, dim ∂ = k ( f ) Γ k ( f ) =
Why does the proof break down? Associates a number to every polynomial. ▶ Lower bounds in alg. complexity use some complexity measure. ▶ Often is the rank of collection of linear operators. ▶ For depth-3 powering circuits, dim ∂ = k ( f ) Γ k ( f ) = Γ k ( ℓ d ) = 1 Γ k ( ℓ d 1 + ··· + ℓ d s ) ≤ s
Why does the proof break down? Associates a number to every polynomial. ▶ Lower bounds in alg. complexity use some complexity measure. ▶ Often is the rank of collection of linear operators. ▶ For depth-3 powering circuits, dim ∂ = k ( f ) Γ k ( f ) = Γ k ( ℓ d ) = 1 Γ k ( ℓ d 1 + ··· + ℓ d s ) ≤ s ▶ Partial derivatives don’t behave well with functional computation.
Why does the proof break down? Associates a number to every polynomial. ▶ Lower bounds in alg. complexity use some complexity measure. ▶ Often is the rank of collection of linear operators. ▶ For depth-3 powering circuits, dim ∂ = k ( f ) Γ k ( f ) = Γ k ( ℓ d ) = 1 Γ k ( ℓ d 1 + ··· + ℓ d s ) ≤ s ▶ Partial derivatives don’t behave well with functional computation. ( x 1 + ··· + x n ) n = x 1 ··· x n + (non-multilinear terms)
Why does the proof break down? Associates a number to every polynomial. ▶ Lower bounds in alg. complexity use some complexity measure. ▶ Often is the rank of collection of linear operators. ▶ For depth-3 powering circuits, dim ∂ = k ( f ) Γ k ( f ) = Γ k ( ℓ d ) = 1 Γ k ( ℓ d 1 + ··· + ℓ d s ) ≤ s ▶ Partial derivatives don’t behave well with functional computation. ( x 1 + ··· + x n ) n = x 1 ··· x n + (non-multilinear terms) ≡ x 1 ··· x n + (lower degree terms)
Why does the proof break down? Difgerent partial derivatives have difgerent leading monomials Associates a number to every polynomial. ▶ Lower bounds in alg. complexity use some complexity measure. ▶ Often is the rank of collection of linear operators. ▶ For depth-3 powering circuits, dim ∂ = k ( f ) Γ k ( f ) = Γ k ( ℓ d ) = 1 Γ k ( ℓ d 1 + ··· + ℓ d s ) ≤ s ▶ Partial derivatives don’t behave well with functional computation. ( x 1 + ··· + x n ) n = x 1 ··· x n + (non-multilinear terms) ≡ x 1 ··· x n + (lower degree terms)
Measures for various lower bounds hom- circuits hom- circuits hom- circuits . . . dim ∂ = k ( f ) depth- 3 powering circuits
Measures for various lower bounds hom- circuits hom- circuits . . . dim ∂ = k ( f ) depth- 3 powering circuits dim ∂ = k ( f ) hom- ΣΠΣ circuits
Measures for various lower bounds hom- circuits . . . dim ∂ = k ( f ) depth- 3 powering circuits dim ∂ = k ( f ) hom- ΣΠΣ circuits � hom- ΣΠΣΠ [ d ] circuits dim x = ℓ ∂ = k ( f )
Measures for various lower bounds . . . dim ∂ = k ( f ) depth- 3 powering circuits dim ∂ = k ( f ) hom- ΣΠΣ circuits � hom- ΣΠΣΠ [ d ] circuits dim x = ℓ ∂ = k ( f ) � x = ℓ ∂ = k ( f ) � hom- ΣΠΣΠ circuits dim mult
Measures for various lower bounds . . . dim ∂ = k ( f ) depth- 3 powering circuits dim ∂ = k ( f ) hom- ΣΠΣ circuits � hom- ΣΠΣΠ [ d ] circuits dim x = ℓ ∂ = k ( f ) � x = ℓ ∂ = k ( f ) � hom- ΣΠΣΠ circuits dim mult
Outline 30 35 40 45 50 55 60 Intro Typical syntactic lower bounds Modifications
Outline 30 35 40 45 50 55 60 Intro Typical syntactic lower bounds Modifications
A natural attempt well? If a circuit C ∈ � computes a polynomial P , can the unique multilinear representation P ′ be computed effjciently as
A natural attempt well? If a circuit C ∈ � computes a polynomial P , can the unique multilinear representation P ′ be computed effjciently as No... (unless VP = VNP) ( x 11 y 1 + ··· + x 1 n y n ) ··· ( x n 1 y 1 + ··· + x nn y n )
A natural attempt well? If a circuit C ∈ � computes a polynomial P , can the unique multilinear representation P ′ be computed effjciently as No... (unless VP = VNP) ( x 11 y 1 + ··· + x 1 n y n ) ··· ( x n 1 y 1 + ··· + x nn y n ) = Perm ( X ) · y 1 ··· y n + non-multilinear terms
A natural attempt well? If a circuit C ∈ � computes a polynomial P , can the unique multilinear representation P ′ be computed effjciently as No... (unless VP = VNP) ( x 11 y 1 + ··· + x 1 n y n ) ··· ( x n 1 y 1 + ··· + x nn y n ) ≡ Perm ( X ) · y 1 ··· y n + lower degree terms
A natural attempt well? ...easy to extract homogeneous components. If a circuit C ∈ � computes a polynomial P , can the unique multilinear representation P ′ be computed effjciently as No... (unless VP = VNP) ( x 11 y 1 + ··· + x 1 n y n ) ··· ( x n 1 y 1 + ··· + x nn y n ) ≡ Perm ( X ) · y 1 ··· y n + lower degree terms
A natural attempt well? If a circuit C ∈ � computes a polynomial P , can the unique multilinear representation P ′ be computed effjciently as
A natural attempt well? If a circuit C ∈ � computes a polynomial P of low individual degree, can the unique multilinear representation P ′ be computed effjciently as
A natural attempt well? Not clear, even when dealing with multi-quadratics ... If a circuit C ∈ � computes a polynomial P of low individual degree, can the unique multilinear representation P ′ be computed effjciently as
Lemma Partial evaluations as proxies For polynomials , Think of the polynomial Perm ...
Lemma Partial evaluations as proxies For polynomials , Think of the polynomial Perm ...
Lemma Partial evaluations as proxies For polynomials , Think of the polynomial Perm ... =
Lemma Partial evaluations as proxies For polynomials , Think of the polynomial Perm ... 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 =
Partial evaluations as proxies Think of the polynomial Perm ... 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 = Lemma For nice polynomials P ( y , z ) , P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 = k � � ∂ = k ( P ) ⊆ y
Partial evaluations as proxies Think of the polynomial Perm ... 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 = Lemma For set-multilinear polynomials P ( y , z ) , P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 = k � � ∂ = k ( P ) ⊆ y
Partial evaluations as proxies Lemma For set-multilinear polynomials P ( y , z ) , P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 = k ∂ = k � � ( P ) ⊆ y
Partial evaluations as proxies Lemma For set-multilinear polynomials P ( y , z ) , P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 = k ∂ = k � � ( P ) ⊆ y P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 = k dim ∂ = k � � ( P ) ≤ dim ∴ y
Partial evaluations as proxies ... For arbitrary polynomials, If dim ∂ = k ( P ) is small, does that also imply that y dim { P ( a , z ) } also has to be small-ish?
Partial evaluations as proxies ... For arbitrary polynomials, If dim ∂ = k ( P ) is small, does that also imply that y dim { P ( a , z ) } also has to be small-ish?
Partial evaluations as proxies ... For arbitrary polynomials, If dim ∂ = k ( P ) is small, does that also imply that y dim { P ( a , z ) } also has to be small-ish? ∑� � · a e P ( a + y , z ) = : P z ( a + y ) ∂ y e P z =
Partial evaluations as proxies ... For arbitrary polynomials, If dim ∂ = k ( P ) is small, does that also imply that y dim { P ( a , z ) } also has to be small-ish? · a e + ∑ � � ∑ � � · a e P ( a + y , z ) = : P z ( a + y ) ∂ y e P z ∂ y e P z = | e | 0 ≤ k | e | 0 > k
Partial evaluations as proxies ... For arbitrary polynomials, If dim ∂ = k ( P ) is small, does that also imply that y dim { P ( a , z ) } also has to be small-ish? · a e + ∑ � � ∑ � � · a e P ( a + y , z ) = : P z ( a + y ) ∂ y e P z ∂ y e P z = | e | 0 ≤ k | e | 0 > k
Partial evaluations as proxies ... For arbitrary polynomials, If dim ∂ = k ( P ) is small, does that also imply that y dim { P ( a , z ) } also has to be small-ish? ∑ � � · a e P ( a + y , z ) = : P z ( a + y ) ∂ y e P z = | e | 0 ≤ k
Partial evaluations as proxies ... For arbitrary polynomials, If dim ∂ = k ( P ) is small, does that also imply that y dim { P ( a , z ) } also has to be small-ish? ∑ � � · a e P ( a + y , z ) = : P z ( a + y ) ∂ y e P z = | e | 0 ≤ k ∑ � � · a e If P has ind. degree r , = ∂ y e P z | e | 0 ≤ k | e | 1 ≤ rk
Partial evaluations as proxies ... For arbitrary polynomials, If dim ∂ = k ( P ) is small, does that also imply that y dim { P ( a , z ) } also has to be small-ish? ∑ � � · a e P ( a + y , z ) = : P z ( a + y ) ∂ y e P z = | e | 0 ≤ k ∑ � � · a e If P has ind. degree r , = ∂ y e P z | e | 0 ≤ k | e | 1 ≤ rk ∑ � � y = 0 · a e ∂ y e P z ∴ P ( a , z ) = | e | 0 ≤ k | e | 1 ≤ rk
Partial evaluations as proxies ... Lemma For any polynomial P of individual degree at most r , then �� � P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 = k � � � ∂ = r k ⊆ span ( P ) y y = 0 Proof. ∑ � � · a e P ( a + y , z ) = : P z ( a + y ) ∂ y e P z = | e | 0 ≤ k ∑ � � · a e If P has ind. degree r , = ∂ y e P z | e | 0 ≤ k | e | 1 ≤ rk ∑ � � y = 0 · a e ∂ y e P z ∴ P ( a , z ) = | e | 0 ≤ k | e | 1 ≤ rk
Partial evaluations as proxies ... is also small. Lemma For any polynomial P of individual degree at most r , then �� � P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 = k � � � ∂ = r k ⊆ span ( P ) y y = 0 If dim ∂ = r k ( P ) is small, P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 = k � � then dim
Making the measure ‘functional’ Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y
Making the measure ‘functional’ Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 2 ) � � k ( P ) : = dim ⊂ � [ z ]
Making the measure ‘functional’ Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 2 ) � � k ( P ) : = dim ⊂ � [ z ] k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 3 ) ⊂ � n
Making the measure ‘functional’ Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 2 ) � � k ( P ) : = dim ⊂ � [ z ] k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 3 ) ⊂ � n Observation If P ≡ F , then Γ ( 3 ) k ( P ) = Γ ( 3 ) k ( F ) .
Making the measure ‘functional’ Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 2 ) � � k ( P ) : = dim ⊂ � [ z ] k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 3 ) ⊂ � n
Making the measure ‘functional’ For the nice polynomial, unless which is small For the circuit class, which is huge , is large Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 2 ) � � ⊂ � [ z ] k ( P ) : = dim k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 3 ) ⊂ � n Theorem Let C ∈ � be a size s circuit computing a polynomial P of individual degree at most r . If P ≡ Perm , then s must be large.
is large For the nice polynomial, unless which is small For the circuit class, which is huge , Making the measure ‘functional’ Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 2 ) � � ⊂ � [ z ] k ( P ) : = dim k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 3 ) ⊂ � n Theorem Let C ∈ � be a size s circuit computing a polynomial P of individual degree at most r . If P ≡ Perm , then s must be large.
is large which is huge unless which is small For the circuit class, Making the measure ‘functional’ Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 2 ) � � ⊂ � [ z ] k ( P ) : = dim k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 3 ) ⊂ � n Theorem Let C ∈ � be a size s circuit computing a polynomial P of individual degree at most r . If P ≡ Perm , then s must be large. For the nice polynomial, Perm , Γ ( 3 ) k ( Perm ) = Γ ( 2 ) k ( Perm ) = Γ ( 1 ) k ( Perm )
Making the measure ‘functional’ which is huge unless which is small For the circuit class, is large Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 2 ) � � ⊂ � [ z ] k ( P ) : = dim k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 3 ) ⊂ � n Theorem Let C ∈ � be a size s circuit computing a polynomial P of individual degree at most r . If P ≡ Perm , then s must be large. For the nice polynomial, Perm , Γ ( 3 ) k ( P ) ≤ Γ ( 2 ) Γ ( 3 ) k ( Perm ) = Γ ( 2 ) k ( P ) k ( Perm ) = Γ ( 1 ) k ( Perm )
Making the measure ‘functional’ which is huge unless which is small For the circuit class, is large Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 2 ) � � ⊂ � [ z ] k ( P ) : = dim k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 3 ) ⊂ � n Theorem Let C ∈ � be a size s circuit computing a polynomial P of individual degree at most r . If P ≡ Perm , then s must be large. For the nice polynomial, Perm , Γ ( 3 ) k ( P ) ≤ Γ ( 2 ) Γ ( 3 ) k ( Perm ) = Γ ( 2 ) k ( P ) k ( Perm ) ≤ Γ ( 1 ) = Γ ( 1 ) r k ( P ) k ( Perm )
is large which is huge unless which is small For the circuit class, Making the measure ‘functional’ Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 2 ) � � ⊂ � [ z ] k ( P ) : = dim k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 3 ) ⊂ � n Theorem Let C ∈ � be a size s circuit computing a polynomial P of individual degree at most r . If P ≡ Perm , then s must be large. For the nice polynomial, Perm , Γ ( 3 ) k ( P ) ≤ Γ ( 2 ) Γ ( 3 ) k ( Perm ) = Γ ( 2 ) k ( P ) k ( Perm ) ≤ Γ ( 1 ) = Γ ( 1 ) r k ( P ) k ( Perm )
Making the measure ‘functional’ which is huge which is small For the circuit class, Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 2 ) � � ⊂ � [ z ] k ( P ) : = dim k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 3 ) ⊂ � n Theorem Let C ∈ � be a size s circuit computing a polynomial P of individual degree at most r . If P ≡ Perm , then s must be large. For the nice polynomial, Perm , Γ ( 3 ) k ( P ) ≤ Γ ( 2 ) Γ ( 3 ) k ( Perm ) = Γ ( 2 ) k ( P ) k ( Perm ) ≤ Γ ( 1 ) = Γ ( 1 ) r k ( P ) k ( Perm ) unless s is large
Useful in the case of depth three powering, and homogeneous Function versions of common measures depth three circuits. Dimension of shifted partial derivatives ▶ Dimension of partial derivatives Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 2 ) ⊂ � n
Useful in the case of depth three powering, and homogeneous Function versions of common measures depth three circuits. Dimension of shifted partial derivatives ▶ Dimension of partial derivatives Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 2 ) ⊂ � n
Function versions of common measures Useful in the case of depth three powering, and homogeneous depth three circuits. ▶ Dimension of partial derivatives Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 2 ) ⊂ � n ▶ Dimension of shifted partial derivatives k , ℓ ( P ) : = dim ( y , z ) = ℓ ∂ = k Γ ( 1 ) P ( y , z ) y
Function versions of common measures Useful in the case of depth three powering, and homogeneous depth three circuits. (If you set parameters right) ▶ Dimension of partial derivatives Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 2 ) ⊂ � n ▶ Dimension of shifted partial derivatives k , ℓ ( P ) : = dim ( y , z ) = ℓ ∂ = k Γ ( 1 ) P ( y , z ) y z = ℓ · P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 2 ) � � � � k , ℓ ( P ) : = T T
Function versions of common measures Useful in the case of depth three powering, and homogeneous depth three circuits. Useful for a natural sub-class of hom. depth four circuits. ▶ Dimension of partial derivatives Γ ( 1 ) k ( P ) : = dim ∂ = k P ( y , z ) ⊂ � [ y , z ] y k ( P ) : = dim { P ( a , b ) : a , b ∈ { 0,1 } ∗ , | a | 0 ≤ k } Γ ( 2 ) ⊂ � n ▶ Dimension of shifted partial derivatives k , ℓ ( P ) : = dim ( y , z ) = ℓ ∂ = k Γ ( 1 ) P ( y , z ) y z = ℓ · P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 2 ) � � � � k , ℓ ( P ) : = T T
Tie measure for depth four circuits Messes up evaluations circuit class circuit class Finally, left to check that hard polynomial ▶ Dimension of projected shifted partial derivatives x = ℓ · ∂ = k Γ ( 1 ) � � �� k , ℓ ( P ( y , z )) : = dim mult ( P ) y
Tie measure for depth four circuits Messes up evaluations circuit class circuit class Finally, left to check that hard polynomial ▶ Dimension of projected shifted partial derivatives x = ℓ · ∂ = k Γ ( 1 ) � � �� k , ℓ ( P ( y , z )) : = dim mult ( P ) y x = ℓ · ∂ = k �� � mod � x 2 i = 0 �� = dim ( P ) y
Tie measure for depth four circuits Messes up evaluations circuit class circuit class Finally, left to check that hard polynomial ▶ Dimension of projected shifted partial derivatives x = ℓ · ∂ = k Γ ( 1 ) � � �� k , ℓ ( P ( y , z )) : = dim mult ( P ) y x = ℓ · ∂ = k �� � mod � x 2 i = 0 �� = dim ( P ) y
Tie measure for depth four circuits Messes up evaluations circuit class circuit class Finally, left to check that hard polynomial ▶ Dimension of projected shifted partial derivatives x = ℓ · ∂ = k Γ ( 1 ) � � �� k , ℓ ( P ( y , z )) : = dim mult ( P ) y x = ℓ · ∂ = k �� � mod � x 2 i = 0 �� = dim ( P ) y x = ℓ · ∂ = k Γ ( 2 ) �� � mod � x 2 �� k , ℓ ( P ( y , z )) : = dim ( P ) i = x i y
Tie measure for depth four circuits Messes up evaluations circuit class circuit class Finally, left to check that hard polynomial ▶ Dimension of projected shifted partial derivatives x = ℓ · ∂ = k Γ ( 1 ) � � �� k , ℓ ( P ( y , z )) : = dim mult ( P ) y x = ℓ · ∂ = k �� � mod � x 2 i = 0 �� = dim ( P ) y x = ℓ · ∂ = k Γ ( 2 ) �� � mod � x 2 �� k , ℓ ( P ( y , z )) : = dim ( P ) i = x i y z = ℓ · P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 3 ) �� � � k , ℓ ( P ( y , z )) : = dim
Tie measure for depth four circuits Messes up evaluations circuit class circuit class Finally, left to check that hard polynomial ▶ Dimension of projected shifted partial derivatives x = ℓ · ∂ = k Γ ( 1 ) � � �� k , ℓ ( P ( y , z )) : = dim mult ( P ) y x = ℓ · ∂ = k �� � mod � x 2 i = 0 �� = dim ( P ) y x = ℓ · ∂ = k Γ ( 2 ) �� � mod � x 2 �� k , ℓ ( P ( y , z )) : = dim ( P ) i = x i y z = ℓ · P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 3 ) �� � � k , ℓ ( P ( y , z )) : = dim z = ℓ · P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 4 ) � � � � k , ℓ ( P ( y , z )) T T : = dim
Tie measure for depth four circuits Messes up evaluations Finally, left to check that ▶ Dimension of projected shifted partial derivatives x = ℓ · ∂ = k Γ ( 1 ) � � �� k , ℓ ( P ( y , z )) : = dim mult ( P ) y x = ℓ · ∂ = k �� � mod � x 2 i = 0 �� = dim ( P ) y x = ℓ · ∂ = k Γ ( 2 ) �� � mod � x 2 �� k , ℓ ( P ( y , z )) : = dim ( P ) i = x i y z = ℓ · P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 3 ) �� � � k , ℓ ( P ( y , z )) : = dim z = ℓ · P ( a , z ) : a ∈ { 0,1 } | y | , | a | 0 ≤ k Γ ( 4 ) � � � � k , ℓ ( P ( y , z )) T T : = dim Γ ( 4 ) k , ℓ ( circuit class ) ≤ Γ ( 1 ) r k , ℓ ( circuit class ) ≪ Γ ( 4 ) k , ℓ ( hard polynomial )
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