Functional lower bounds for arithmetic circuits and connections to - - PowerPoint PPT Presentation
Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity Michael Forbes Mrinal Kumar Ramprasad Saptharishi Princeton University Rutgers University T el Aviv University Computational Complexity
Michael Forbes Mrinal Kumar Ramprasad Saptharishi Princeton University Rutgers University T el Aviv University Computational Complexity Conference (CCC 2016) T
x1 x2 x3
∧ ∧ ¬ ∧ ∧ ¬ ∨ ∨ ∨ ∧
f (x1, x2, x3) : {0,1}n → {0,1}
a boolean function
x1 x2 x3
+ + + + + + × × × +
f (x1, x2, x3) ∈ [x]
a polynomial
x1 x2 x3
+ + + + + + × × × +
f (x1, x2, x3) ∈ [x]
a polynomial of low degree
x1 x2 x3
+ + + + + + × × × +
f (x1, x2, x3) ∈ [x]
a polynomial of low degree
NP P VP VNP is simpler to prove than P NP.
VNP NP P VP VP VNP is simpler to prove than P NP.
#P VNP NP P
SAC1
VP VP VNP is simpler to prove than P NP.
#P VNP NP P
SAC1
VP VP ̸= VNP is simpler to prove than P ̸= NP.
Theorem ([Agrawal-Vinay, Koiran, Tavenas])
Can be computed by arithmetic circuits
Can be computed by
(Or) Cannot be computed by arithmetic circuits
size Cannot be computed by
size
Theorem ([Agrawal-Vinay, Koiran, Tavenas])
Can be computed by arithmetic circuits
Can be computed by
(Or) Cannot be computed by arithmetic circuits
size Cannot be computed by
size
Theorem ([Agrawal-Vinay, Koiran, Tavenas])
Can be computed by arithmetic circuits
Can be computed by
(Or) Cannot be computed by arithmetic circuits
Cannot be computed by
A recent surge in optimism in the field.
Someone even conjectured that VP ̸= VNP would be resolved by 2018...
1980 1985 1990 1995 2000 2005 2010 2015
Each is one result in algebraic complexity lower bounds.
Theorem ([KLSS,KS])
There is an explicit polynomial with coeffjcients such that any homogeneous depth- arithmetic circuit computing it must have size . Question: Can they be lifted to boolean circuits?
A recent surge in optimism in the field.
Someone even conjectured that VP ̸= VNP would be resolved by 2018...
1980 1985 1990 1995 2000 2005 2010 2015
Each is one result in algebraic complexity lower bounds.
Theorem ([KLSS,KS])
There is an explicit polynomial f with 0/1 coeffjcients such that any homogeneous depth-4 arithmetic circuit computing it must have size nΩ(
Question: Can they be lifted to boolean circuits?
A recent surge in optimism in the field.
Someone even conjectured that VP ̸= VNP would be resolved by 2018...
1980 1985 1990 1995 2000 2005 2010 2015
Each is one result in algebraic complexity lower bounds.
Theorem ([KLSS,KS])
There is an explicit polynomial f with 0/1 coeffjcients such that any homogeneous depth-4 arithmetic circuit computing it must have size nΩ(
Question: Can they be lifted to boolean circuits?
Theorem ([KLSS,KS])
There is an explicit polynomial f with 0/1 coeffjcients such that any homogeneous depth-4 arithmetic circuit computing it must have size nΩ(
Theorem ([KLSS,KS])
There is an explicit polynomial f with 0/1 coeffjcients such that any homogeneous depth-4 arithmetic circuit computing it must have size nΩ(
One of the key steps in Williams’ proof of NEXP ̸⊂ ACC0:
Theorem ([Yao, BT, AG])
For any boolean function F computed by an ACC0 circuit of size s, there is a univariate polynomial g ∈ [y] and a multilinear polynomial h(x) = ∑ hαxα with 2polylog(s) monomials such that, ∀x ∈ {0,1}n , F (x) = g (h(x)).
Theorem ([KLSS,KS])
There is an explicit polynomial f with 0/1 coeffjcients such that any homogeneous depth-4 arithmetic circuit computing it must have size nΩ(
One of the key steps in Williams’ proof of NEXP ̸⊂ ACC0:
Theorem ([Yao, BT, AG])
For any boolean function F computed by an ACC0 circuit of size s, there is a depth-4 arithmetic circuit C of size 2polylog(s) such that, ∀x ∈ {0,1}n , F (x) = C(x).
Theorem ([KLSS,KS])
There is an explicit polynomial f with 0/1 coeffjcients such that any homogeneous depth-4 arithmetic circuit computing it must have size nΩ(
One of the key steps in Williams’ proof of NEXP ̸⊂ ACC0:
Theorem ([Yao, BT, AG])
For any boolean function F computed by an ACC0 circuit of size s, there is a depth-4 arithmetic circuit C of size 2polylog(s) such that, ∀x ∈ {0,1}n , F (x) = C(x).
Theorem ([KLSS,KS])
There is an explicit polynomial f with 0/1 coeffjcients such that any homogeneous depth-4 arithmetic circuit computing it must have size nΩ(
One of the key steps in Williams’ proof of NEXP ̸⊂ ACC0:
Theorem ([Yao, BT, AG])
For any boolean function F computed by an ACC0 circuit of size s, there is a depth-4 arithmetic circuit C of size 2polylog(s) such that, ∀x ∈ {0,1}n , F (x) = C(x).
Definition
An arithmetic circuit C is said to functionally compute a polynomial F if ∀x ∈ {0,1}n , C(x) = F (x). If computed a multilinear polynomial, then functional computation = syntactic computation. But even if computes a multi-quadratic polynomial, the definition is meaningful.
Definition
An arithmetic circuit C is said to functionally compute a polynomial F if ∀x ∈ {0,1}n , C(x) = F (x).
▶ If C computed a multilinear polynomial, then
functional computation = syntactic computation.
▶ But even if C computes a multi-quadratic polynomial, the definition
is meaningful.
What we know:
There is an explicit n-variate degree d polynomial F with 0/1 coeffjcients such that any homogeneous depth-4 arithmetic circuit computing it must have size nΩ(
What we know:
There is an explicit n-variate degree d polynomial F with 0/1 coeffjcients such that any homogeneous depth-4 arithmetic circuit computing it must have size nΩ(
What we’d have liked to prove:
Let C be a homogeneous depth-4 circuit that computes a polynomial P that functionally computes F , i.e. ∀x ∈ {0,1}n , P(x) = F (x). Then, C must have size nΩ(
What we know:
There is an explicit n-variate degree d polynomial F with 0/1 coeffjcients such that any homogeneous depth-4 arithmetic circuit computing it must have size nΩ(
What we prove:
Let C be a homogeneous depth-4 circuit that computes a polynomial P
∀x ∈ {0,1}n , P(x) = F (x). Then, C must have size nΩr (
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So far T ypical syntactic lower bounds Modifications Q&A
Diagram not to scale
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So far Typical syntactic lower bounds Modifications Q&A
Diagram not to scale
30 35 40 45 50 55 60
So far Typical syntactic lower bounds Modifications Q&A
Diagram not to scale
30 35 40 45 50 55 60
So far Typical syntactic lower bounds Modifications Q&A
Diagram not to scale
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So far Typical syntactic lower bounds Modifications Q&A
Diagram not to scale
Symd(x1,..., xn) = ∑
1≤i1<···<id≤n
xi1 ··· xid
Symd(x1,..., xn) = ∑
1≤i1<···<id≤n
xi1 ··· xid If we only care about x ∈ {0,1}n, ℓ := ∑ xi 1 2 3 4 5 6 7 8 ··· Sym4 1 4 15 35 70 ···
Symd(x1,..., xn) = ∑
1≤i1<···<id≤n
xi1 ··· xid If we only care about x ∈ {0,1}n, Symd(x) = f (ℓ) depth-3 powering circuits
Symd(x1,..., xn) = ∑
1≤i1<···<id≤n
xi1 ··· xid If we only care about x ∈ {0,1}n, Symd(x) = f (ℓ) = a0 + a1ℓ + ··· + an+1ℓn+1 depth-3 powering circuits
Symd(x1,..., xn) = ∑
1≤i1<···<id≤n
xi1 ··· xid If we only care about x ∈ {0,1}n, Symd(x) = f (ℓ) = a0 + a1ℓ + ··· + an+1ℓn+1 ∈ depth-3 powering circuits
Symd(x1,..., xn) = ∑
1≤i1<···<id≤n
xi1 ··· xid If we only care about x ∈ {0,1}n, Symd(x) = f (ℓ) = a0 + a1ℓ + ··· + an+1ℓn+1 ∈ depth-3 powering circuits Syntactic computation of Symd by such depth-3 powering circuits require nΩ(d) size. [Nisan-Wigderson]
▶ Lower bounds in alg. complexity use some complexity measure.
Associates a number to every polynomial.
▶ Often is the rank of collection of linear operators.
▶ Lower bounds in alg. complexity use some complexity measure.
Associates a number to every polynomial.
▶ Often is the rank of collection of linear operators. ▶ For depth-3 powering circuits,
Γk(f ) = dim∂ =k(f )
▶ Lower bounds in alg. complexity use some complexity measure.
Associates a number to every polynomial.
▶ Often is the rank of collection of linear operators. ▶ For depth-3 powering circuits,
Γk(f ) = dim∂ =k(f ) Γk(ℓd) = 1 Γk(ℓd
1 + ··· + ℓd s )
≤ s
▶ Lower bounds in alg. complexity use some complexity measure.
Associates a number to every polynomial.
▶ Often is the rank of collection of linear operators. ▶ For depth-3 powering circuits,
Γk(f ) = dim∂ =k(f ) Γk(ℓd) = 1 Γk(ℓd
1 + ··· + ℓd s )
≤ s
▶ Partial derivatives don’t behave well with functional computation.
▶ Lower bounds in alg. complexity use some complexity measure.
Associates a number to every polynomial.
▶ Often is the rank of collection of linear operators. ▶ For depth-3 powering circuits,
Γk(f ) = dim∂ =k(f ) Γk(ℓd) = 1 Γk(ℓd
1 + ··· + ℓd s )
≤ s
▶ Partial derivatives don’t behave well with functional computation.
(x1 + ··· + xn)n = x1 ··· xn + (non-multilinear terms)
▶ Lower bounds in alg. complexity use some complexity measure.
Associates a number to every polynomial.
▶ Often is the rank of collection of linear operators. ▶ For depth-3 powering circuits,
Γk(f ) = dim∂ =k(f ) Γk(ℓd) = 1 Γk(ℓd
1 + ··· + ℓd s )
≤ s
▶ Partial derivatives don’t behave well with functional computation.
(x1 + ··· + xn)n = x1 ··· xn + (non-multilinear terms) ≡ x1 ··· xn + (lower degree terms)
▶ Lower bounds in alg. complexity use some complexity measure.
Associates a number to every polynomial.
▶ Often is the rank of collection of linear operators. ▶ For depth-3 powering circuits,
Γk(f ) = dim∂ =k(f ) Γk(ℓd) = 1 Γk(ℓd
1 + ··· + ℓd s )
≤ s
▶ Partial derivatives don’t behave well with functional computation.
(x1 + ··· + xn)n = x1 ··· xn + (non-multilinear terms) ≡ x1 ··· xn + (lower degree terms) Difgerent partial derivatives have difgerent leading monomials
depth-3 powering circuits dim ∂ =k(f ) hom- circuits hom- circuits hom- circuits . . .
depth-3 powering circuits dim ∂ =k(f ) hom-ΣΠΣ circuits dim ∂ =k(f ) hom- circuits hom- circuits . . .
depth-3 powering circuits dim ∂ =k(f ) hom-ΣΠΣ circuits dim ∂ =k(f ) hom-ΣΠΣΠ[
dim x=ℓ∂ =k(f ) hom- circuits . . .
depth-3 powering circuits dim ∂ =k(f ) hom-ΣΠΣ circuits dim ∂ =k(f ) hom-ΣΠΣΠ[
dim x=ℓ∂ =k(f ) hom-ΣΠΣΠ circuits dim mult
. .
depth-3 powering circuits dim ∂ =k(f ) hom-ΣΠΣ circuits dim ∂ =k(f ) hom-ΣΠΣΠ[
dim x=ℓ∂ =k(f ) hom-ΣΠΣΠ circuits dim mult
. .
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Intro Typical syntactic lower bounds Modifications
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Intro Typical syntactic lower bounds Modifications
If a circuit C ∈ computes a polynomial P , can the unique multilinear representation P ′ be computed effjciently as well?
If a circuit C ∈ computes a polynomial P , can the unique multilinear representation P ′ be computed effjciently as well? No... (unless VP = VNP) (x11y1 + ··· + x1nyn)···(xn1y1 + ··· + xnnyn)
If a circuit C ∈ computes a polynomial P , can the unique multilinear representation P ′ be computed effjciently as well? No... (unless VP = VNP) (x11y1 + ··· + x1nyn)···(xn1y1 + ··· + xnnyn) = Perm(X) · y1 ···yn + non-multilinear terms
If a circuit C ∈ computes a polynomial P , can the unique multilinear representation P ′ be computed effjciently as well? No... (unless VP = VNP) (x11y1 + ··· + x1nyn)···(xn1y1 + ··· + xnnyn) ≡ Perm(X) · y1 ···yn + lower degree terms
If a circuit C ∈ computes a polynomial P , can the unique multilinear representation P ′ be computed effjciently as well? No... (unless VP = VNP) (x11y1 + ··· + x1nyn)···(xn1y1 + ··· + xnnyn) ≡ Perm(X) · y1 ···yn + lower degree terms ...easy to extract homogeneous components.
If a circuit C ∈ computes a polynomial P , can the unique multilinear representation P ′ be computed effjciently as well?
If a circuit C ∈ computes a polynomial P of low individual degree, can the unique multilinear representation P ′ be computed effjciently as well?
If a circuit C ∈ computes a polynomial P of low individual degree, can the unique multilinear representation P ′ be computed effjciently as well? Not clear, even when dealing with multi-quadratics ...
Think of the polynomial Perm...
Lemma
For polynomials ,
Think of the polynomial Perm...
Lemma
For polynomials ,
Think of the polynomial Perm...
=
Lemma
For polynomials ,
Think of the polynomial Perm...
= 1 1 1
Lemma
For polynomials ,
Think of the polynomial Perm...
= 1 1 1
Lemma
For nice polynomials P(y,z), ∂ =k
y
(P) ⊆
Think of the polynomial Perm...
= 1 1 1
Lemma
For set-multilinear polynomials P(y,z), ∂ =k
y
(P) ⊆
Lemma
For set-multilinear polynomials P(y,z), ∂ =k
y
(P) ⊆
Lemma
For set-multilinear polynomials P(y,z), ∂ =k
y
(P) ⊆
dim∂ =k
y
(P) ≤ dim
For arbitrary polynomials, If dim∂ =k
y
(P) is small, does that also imply that dim{P(a,z)} also has to be small-ish?
For arbitrary polynomials, If dim∂ =k
y
(P) is small, does that also imply that dim{P(a,z)} also has to be small-ish?
For arbitrary polynomials, If dim∂ =k
y
(P) is small, does that also imply that dim{P(a,z)} also has to be small-ish? P(a + y,z) =: Pz(a + y) = ∑ ∂yePz
For arbitrary polynomials, If dim∂ =k
y
(P) is small, does that also imply that dim{P(a,z)} also has to be small-ish? P(a + y,z) =: Pz(a + y) = ∑
|e|0≤k
∑
|e|0>k
For arbitrary polynomials, If dim∂ =k
y
(P) is small, does that also imply that dim{P(a,z)} also has to be small-ish? P(a + y,z) =: Pz(a + y) = ∑
|e|0≤k
∑
|e|0>k
For arbitrary polynomials, If dim∂ =k
y
(P) is small, does that also imply that dim{P(a,z)} also has to be small-ish? P(a + y,z) =: Pz(a + y) = ∑
|e|0≤k
For arbitrary polynomials, If dim∂ =k
y
(P) is small, does that also imply that dim{P(a,z)} also has to be small-ish? P(a + y,z) =: Pz(a + y) = ∑
|e|0≤k
If P has ind. degree r, = ∑
|e|0≤k |e|1≤rk
For arbitrary polynomials, If dim∂ =k
y
(P) is small, does that also imply that dim{P(a,z)} also has to be small-ish? P(a + y,z) =: Pz(a + y) = ∑
|e|0≤k
If P has ind. degree r, = ∑
|e|0≤k |e|1≤rk
∴ P(a,z) = ∑
|e|0≤k |e|1≤rk
Lemma
For any polynomial P of individual degree at most r, then
span
y
(P)
P(a + y,z) =: Pz(a + y) = ∑
|e|0≤k
If P has ind. degree r, = ∑
|e|0≤k |e|1≤rk
∴ P(a,z) = ∑
|e|0≤k |e|1≤rk
Lemma
For any polynomial P of individual degree at most r, then
span
y
(P)
then dim
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z]
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim
Γ (3)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ n
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim
Γ (3)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ n
Observation
If P ≡ F , then Γ (3)
k (P) = Γ (3) k (F ).
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim
Γ (3)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ n
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim
Γ (3)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ 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, , which is huge For the circuit class, which is small unless is large
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim
Γ (3)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ 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, , which is huge For the circuit class, which is small unless is large
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim
Γ (3)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ 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)
which is huge For the circuit class, which is small unless is large
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim
Γ (3)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ 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)
which is huge For the circuit class, Γ (3)
k (P) ≤ Γ (2) k (P)
which is small unless is large
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim
Γ (3)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ 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)
which is huge For the circuit class, Γ (3)
k (P) ≤ Γ (2) k (P)
≤ Γ (1)
r k (P)
which is small unless is large
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim
Γ (3)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ 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)
which is huge For the circuit class, Γ (3)
k (P) ≤ Γ (2) k (P)
≤ Γ (1)
r k (P)
which is small unless is large
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim
Γ (3)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ 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)
which is huge For the circuit class, Γ (3)
k (P) ≤ Γ (2) k (P)
≤ Γ (1)
r k (P)
which is small unless s is large
▶ Dimension of partial derivatives
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ n Useful in the case of depth three powering, and homogeneous depth three circuits. Dimension of shifted partial derivatives
▶ Dimension of partial derivatives
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ n Useful in the case of depth three powering, and homogeneous depth three circuits. Dimension of shifted partial derivatives
▶ Dimension of partial derivatives
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ n Useful in the case of depth three powering, and homogeneous depth three circuits.
▶ Dimension of shifted partial derivatives
Γ (1)
k,ℓ(P) := dim (y,z)=ℓ ∂ =k y
P(y,z)
▶ Dimension of partial derivatives
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ n Useful in the case of depth three powering, and homogeneous depth three circuits.
▶ Dimension of shifted partial derivatives
Γ (1)
k,ℓ(P) := dim (y,z)=ℓ ∂ =k y
P(y,z) Γ (2)
k,ℓ(P) :=
▶ Dimension of partial derivatives
Γ (1)
k (P) := dim ∂ =k y
P(y,z) ⊂ [y,z] Γ (2)
k (P) := dim {P(a,b) : a,b ∈ {0,1}∗ , |a|0 ≤ k}
⊂ n Useful in the case of depth three powering, and homogeneous depth three circuits.
▶ Dimension of shifted partial derivatives
Γ (1)
k,ℓ(P) := dim (y,z)=ℓ ∂ =k y
P(y,z) Γ (2)
k,ℓ(P) :=
▶ Dimension of projected shifted partial derivatives
Γ (1)
k,ℓ(P(y,z))
:= dim
y
(P)
Finally, left to check that
circuit class circuit class hard polynomial
▶ Dimension of projected shifted partial derivatives
Γ (1)
k,ℓ(P(y,z))
:= dim
y
(P)
dim
y
(P)
i = 0
Messes up evaluations
Finally, left to check that
circuit class circuit class hard polynomial
▶ Dimension of projected shifted partial derivatives
Γ (1)
k,ℓ(P(y,z))
:= dim
y
(P)
dim
y
(P)
i = 0
Messes up evaluations
Finally, left to check that
circuit class circuit class hard polynomial
▶ Dimension of projected shifted partial derivatives
Γ (1)
k,ℓ(P(y,z))
:= dim
y
(P)
dim
y
(P)
i = 0
Messes up evaluations Γ (2)
k,ℓ(P(y,z))
:= dim
y
(P)
i = xi
circuit class circuit class hard polynomial
▶ Dimension of projected shifted partial derivatives
Γ (1)
k,ℓ(P(y,z))
:= dim
y
(P)
dim
y
(P)
i = 0
Messes up evaluations Γ (2)
k,ℓ(P(y,z))
:= dim
y
(P)
i = xi
k,ℓ(P(y,z))
:= dim
circuit class circuit class hard polynomial
▶ Dimension of projected shifted partial derivatives
Γ (1)
k,ℓ(P(y,z))
:= dim
y
(P)
dim
y
(P)
i = 0
Messes up evaluations Γ (2)
k,ℓ(P(y,z))
:= dim
y
(P)
i = xi
k,ℓ(P(y,z))
:= dim
k,ℓ(P(y,z))
:= dim
circuit class circuit class hard polynomial
▶ Dimension of projected shifted partial derivatives
Γ (1)
k,ℓ(P(y,z))
:= dim
y
(P)
dim
y
(P)
i = 0
Messes up evaluations Γ (2)
k,ℓ(P(y,z))
:= dim
y
(P)
i = xi
k,ℓ(P(y,z))
:= dim
k,ℓ(P(y,z))
:= dim
Γ (4)
k,ℓ(circuit class) ≤ Γ (1) r k,ℓ(circuit class) ≪ Γ (4) k,ℓ(hard polynomial)
▶ Dimension of projected shifted partial derivatives
Γ (1)
k,ℓ(P(y,z))
:= dim
y
(P)
dim
y
(P)
i = 0
Messes up evaluations Γ (2)
k,ℓ(P(y,z))
:= dim
y
(P)
i = xi
k,ℓ(P(y,z))
:= dim
k,ℓ(P(y,z))
:= dim
Γ (4)
k,ℓ(circuit class) ≤ Γ (1) r k,ℓ(circuit class) ≪ Γ (4) k,ℓ(hard polynomial)
▶ Always worth asking if syntactic lower bounds also extend to the
functional lower bounds. Might have connections to boolean complexity and proof complexity.
▶ Possible to use proxies like additive derivatives (f (x + a) − f (x)),
partial evaluations, or T aylor expansion tricks etc. to lift known examples. The individual degree restriction is annoying... Question: Can we remove the ind. degree restriction and prove functional lower bounds for sums of powers of quadratics? Question: What about approximate functional computation, i.e. agreement on most points on ?
▶ Always worth asking if syntactic lower bounds also extend to the
functional lower bounds. Might have connections to boolean complexity and proof complexity.
▶ Possible to use proxies like additive derivatives (f (x + a) − f (x)),
partial evaluations, or T aylor expansion tricks etc. to lift known examples.
▶ The individual degree restriction is annoying...
Question: Can we remove the ind. degree restriction and prove functional lower bounds for sums of powers of quadratics? Question: What about approximate functional computation, i.e. agreement on most points on ?
▶ Always worth asking if syntactic lower bounds also extend to the
functional lower bounds. Might have connections to boolean complexity and proof complexity.
▶ Possible to use proxies like additive derivatives (f (x + a) − f (x)),
partial evaluations, or T aylor expansion tricks etc. to lift known examples.
▶ The individual degree restriction is annoying... ▶ Question: Can we remove the ind. degree restriction and prove
functional lower bounds for sums of powers of quadratics? Question: What about approximate functional computation, i.e. agreement on most points on ?
▶ Always worth asking if syntactic lower bounds also extend to the
functional lower bounds. Might have connections to boolean complexity and proof complexity.
▶ Possible to use proxies like additive derivatives (f (x + a) − f (x)),
partial evaluations, or T aylor expansion tricks etc. to lift known examples.
▶ The individual degree restriction is annoying... ▶ Question: Can we remove the ind. degree restriction and prove
functional lower bounds for sums of powers of quadratics?
▶ Question: What about approximate functional computation, i.e.
agreement on most points on {0,1}n?
30 35 40 45 50 55 60
Intro Typical syntactic lower bounds Modifications
Questions?
30 35 40 45 50 55 60
Intro Typical syntactic lower bounds Modifications
Questions?
Stick around for more lower bounds in Mrinal’s talks