Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Circuit Complexity Minicourse:
The Shrinkage Exponent of Formulas over
U2
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
A. S. Kulikov
http://logic.pdmi.ras.ru/∼kulikov/
1 / 12
Random Restrictions
I
I
a restriction is an element of {0, 1, ∗}n ; the
interpretation of giving the value ∗ to a variable is that
it remains a variable, while in the other cases the given
constant is substituted as the value of the varaible
n = {ρ ∈ {0, 1, ∗}n : |ρ−1 (∗)| = m} is the set of all
Rm
restrictions which leave exactly m variables unassigned
I
n uniformly at random
choose ρ ∈ Rn
I
it is easy to see that, for any basis, E [L(fρ )] ≤ L(f )
(each leaf has a probability of remainig unassigned)
I
for B2 this inequality is tight (consider the parity
function)
I
for U2 however we expect more, as ∨ and ∧ gates have
the potential to be determined by one input alone
I
in the following we consider the U2 basis
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
2 / 12
Random Restrictions
I
I
a restriction is an element of {0, 1, ∗}n ; the
interpretation of giving the value ∗ to a variable is that
it remains a variable, while in the other cases the given
constant is substituted as the value of the varaible
n = {ρ ∈ {0, 1, ∗}n : |ρ−1 (∗)| = m} is the set of all
Rm
restrictions which leave exactly m variables unassigned
I
n uniformly at random
choose ρ ∈ Rn
I
it is easy to see that, for any basis, E [L(fρ )] ≤ L(f )
(each leaf has a probability of remainig unassigned)
I
for B2 this inequality is tight (consider the parity
function)
I
for U2 however we expect more, as ∨ and ∧ gates have
the potential to be determined by one input alone
I
in the following we consider the U2 basis
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
2 / 12
Random Restrictions
I
I
a restriction is an element of {0, 1, ∗}n ; the
interpretation of giving the value ∗ to a variable is that
it remains a variable, while in the other cases the given
constant is substituted as the value of the varaible
n = {ρ ∈ {0, 1, ∗}n : |ρ−1 (∗)| = m} is the set of all
Rm
restrictions which leave exactly m variables unassigned
I
n uniformly at random
choose ρ ∈ Rn
I
it is easy to see that, for any basis, E [L(fρ )] ≤ L(f )
(each leaf has a probability of remainig unassigned)
I
for B2 this inequality is tight (consider the parity
function)
I
for U2 however we expect more, as ∨ and ∧ gates have
the potential to be determined by one input alone
I
in the following we consider the U2 basis
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
2 / 12
Random Restrictions
I
I
a restriction is an element of {0, 1, ∗}n ; the
interpretation of giving the value ∗ to a variable is that
it remains a variable, while in the other cases the given
constant is substituted as the value of the varaible
n = {ρ ∈ {0, 1, ∗}n : |ρ−1 (∗)| = m} is the set of all
Rm
restrictions which leave exactly m variables unassigned
I
n uniformly at random
choose ρ ∈ Rn
I
it is easy to see that, for any basis, E [L(fρ )] ≤ L(f )
(each leaf has a probability of remainig unassigned)
I
for B2 this inequality is tight (consider the parity
function)
I
for U2 however we expect more, as ∨ and ∧ gates have
the potential to be determined by one input alone
I
in the following we consider the U2 basis
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
2 / 12
Random Restrictions
I
I
a restriction is an element of {0, 1, ∗}n ; the
interpretation of giving the value ∗ to a variable is that
it remains a variable, while in the other cases the given
constant is substituted as the value of the varaible
n = {ρ ∈ {0, 1, ∗}n : |ρ−1 (∗)| = m} is the set of all
Rm
restrictions which leave exactly m variables unassigned
I
n uniformly at random
choose ρ ∈ Rn
I
it is easy to see that, for any basis, E [L(fρ )] ≤ L(f )
(each leaf has a probability of remainig unassigned)
I
for B2 this inequality is tight (consider the parity
function)
I
for U2 however we expect more, as ∨ and ∧ gates have
the potential to be determined by one input alone
I
in the following we consider the U2 basis
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
2 / 12
Random Restrictions
I
I
a restriction is an element of {0, 1, ∗}n ; the
interpretation of giving the value ∗ to a variable is that
it remains a variable, while in the other cases the given
constant is substituted as the value of the varaible
n = {ρ ∈ {0, 1, ∗}n : |ρ−1 (∗)| = m} is the set of all
Rm
restrictions which leave exactly m variables unassigned
I
n uniformly at random
choose ρ ∈ Rn
I
it is easy to see that, for any basis, E [L(fρ )] ≤ L(f )
(each leaf has a probability of remainig unassigned)
I
for B2 this inequality is tight (consider the parity
function)
I
for U2 however we expect more, as ∨ and ∧ gates have
the potential to be determined by one input alone
I
in the following we consider the U2 basis
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
2 / 12
Random Restrictions
I
I
a restriction is an element of {0, 1, ∗}n ; the
interpretation of giving the value ∗ to a variable is that
it remains a variable, while in the other cases the given
constant is substituted as the value of the varaible
n = {ρ ∈ {0, 1, ∗}n : |ρ−1 (∗)| = m} is the set of all
Rm
restrictions which leave exactly m variables unassigned
I
n uniformly at random
choose ρ ∈ Rn
I
it is easy to see that, for any basis, E [L(fρ )] ≤ L(f )
(each leaf has a probability of remainig unassigned)
I
for B2 this inequality is tight (consider the parity
function)
I
for U2 however we expect more, as ∨ and ∧ gates have
the potential to be determined by one input alone
I
in the following we consider the U2 basis
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
2 / 12
Random Restrictions
I
I
a restriction is an element of {0, 1, ∗}n ; the
interpretation of giving the value ∗ to a variable is that
it remains a variable, while in the other cases the given
constant is substituted as the value of the varaible
n = {ρ ∈ {0, 1, ∗}n : |ρ−1 (∗)| = m} is the set of all
Rm
restrictions which leave exactly m variables unassigned
I
n uniformly at random
choose ρ ∈ Rn
I
it is easy to see that, for any basis, E [L(fρ )] ≤ L(f )
(each leaf has a probability of remainig unassigned)
I
for B2 this inequality is tight (consider the parity
function)
I
for U2 however we expect more, as ∨ and ∧ gates have
the potential to be determined by one input alone
I
in the following we consider the U2 basis
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
2 / 12
Shrinkage Exponent
I
the shrinkage exponent, Γ, is the supremum of all values
for which
Γ
n [L(fρ )] = O( L(f ) + 1)
Eρ∈Rn
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
I
Γ ≥ 1.5 (Subbotovskaya, 61)
I
Γ ≤ 2 (Khrapchenko, 71)
I
a function f such that L(f ) = Ω(nΓ+1−o(1) ) (Andreev,
87)
I
Γ ≥ 1.55 (Nisan and Impagliazzo, 93)
I
Γ ≥ 1.63 (Paterson and Zwick, 93)
I
Γ = 2 (Håstad, 98)
3 / 12
Shrinkage Exponent
I
the shrinkage exponent, Γ, is the supremum of all values
for which
Γ
n [L(fρ )] = O( L(f ) + 1)
Eρ∈Rn
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
I
Γ ≥ 1.5 (Subbotovskaya, 61)
I
Γ ≤ 2 (Khrapchenko, 71)
I
a function f such that L(f ) = Ω(nΓ+1−o(1) ) (Andreev,
87)
I
Γ ≥ 1.55 (Nisan and Impagliazzo, 93)
I
Γ ≥ 1.63 (Paterson and Zwick, 93)
I
Γ = 2 (Håstad, 98)
3 / 12
Shrinkage Exponent
I
the shrinkage exponent, Γ, is the supremum of all values
for which
Γ
n [L(fρ )] = O( L(f ) + 1)
Eρ∈Rn
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
I
Γ ≥ 1.5 (Subbotovskaya, 61)
I
Γ ≤ 2 (Khrapchenko, 71)
I
a function f such that L(f ) = Ω(nΓ+1−o(1) ) (Andreev,
87)
I
Γ ≥ 1.55 (Nisan and Impagliazzo, 93)
I
Γ ≥ 1.63 (Paterson and Zwick, 93)
I
Γ = 2 (Håstad, 98)
3 / 12
Shrinkage Exponent
I
the shrinkage exponent, Γ, is the supremum of all values
for which
Γ
n [L(fρ )] = O( L(f ) + 1)
Eρ∈Rn
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
I
Γ ≥ 1.5 (Subbotovskaya, 61)
I
Γ ≤ 2 (Khrapchenko, 71)
I
a function f such that L(f ) = Ω(nΓ+1−o(1) ) (Andreev,
87)
I
Γ ≥ 1.55 (Nisan and Impagliazzo, 93)
I
Γ ≥ 1.63 (Paterson and Zwick, 93)
I
Γ = 2 (Håstad, 98)
3 / 12
Shrinkage Exponent
I
the shrinkage exponent, Γ, is the supremum of all values
for which
Γ
n [L(fρ )] = O( L(f ) + 1)
Eρ∈Rn
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
I
Γ ≥ 1.5 (Subbotovskaya, 61)
I
Γ ≤ 2 (Khrapchenko, 71)
I
a function f such that L(f ) = Ω(nΓ+1−o(1) ) (Andreev,
87)
I
Γ ≥ 1.55 (Nisan and Impagliazzo, 93)
I
Γ ≥ 1.63 (Paterson and Zwick, 93)
I
Γ = 2 (Håstad, 98)
3 / 12
Shrinkage Exponent
I
the shrinkage exponent, Γ, is the supremum of all values
for which
Γ
n [L(fρ )] = O( L(f ) + 1)
Eρ∈Rn
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
I
Γ ≥ 1.5 (Subbotovskaya, 61)
I
Γ ≤ 2 (Khrapchenko, 71)
I
a function f such that L(f ) = Ω(nΓ+1−o(1) ) (Andreev,
87)
I
Γ ≥ 1.55 (Nisan and Impagliazzo, 93)
I
Γ ≥ 1.63 (Paterson and Zwick, 93)
I
Γ = 2 (Håstad, 98)
3 / 12
Shrinkage Exponent
I
the shrinkage exponent, Γ, is the supremum of all values
for which
Γ
n [L(fρ )] = O( L(f ) + 1)
Eρ∈Rn
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
I
Γ ≥ 1.5 (Subbotovskaya, 61)
I
Γ ≤ 2 (Khrapchenko, 71)
I
a function f such that L(f ) = Ω(nΓ+1−o(1) ) (Andreev,
87)
I
Γ ≥ 1.55 (Nisan and Impagliazzo, 93)
I
Γ ≥ 1.63 (Paterson and Zwick, 93)
I
Γ = 2 (Håstad, 98)
3 / 12
Shrinkage Exponent
I
the shrinkage exponent, Γ, is the supremum of all values
for which
Γ
n [L(fρ )] = O( L(f ) + 1)
Eρ∈Rn
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
I
Γ ≥ 1.5 (Subbotovskaya, 61)
I
Γ ≤ 2 (Khrapchenko, 71)
I
a function f such that L(f ) = Ω(nΓ+1−o(1) ) (Andreev,
87)
I
Γ ≥ 1.55 (Nisan and Impagliazzo, 93)
I
Γ ≥ 1.63 (Paterson and Zwick, 93)
I
Γ = 2 (Håstad, 98)
3 / 12
Subbotovskaya’s Lower Bound
Theorem (Subbotovskaya, 61)
If L(f ) > 1, then
n
[L(fρ )] ≤ (1 −
Eρ∈Rn−1
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
1.5
)L(f ) .
n
Proof.
I let x be any variable; by x i we denote the i-th leaf
marked by x or x, by gxi the gate to which xi enters and
by Nxi the other subformula that enters gxi ; let also nx
denotesP
the number of leaves marked by x or x (thus,
L(f ) = nx )
I since the formula is optimal, all Nxi are disjoint
(otherwise, Nxi ⊂ Nxj , but in this case we can simplify
the formula by assigning a value to xi in Nxj )
I assigning either 0 or 1 to x deletes the leaf x i and
assigning the complement value deletes both x i and Nxi
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
4 / 12
Subbotovskaya’s Lower Bound
Theorem (Subbotovskaya, 61)
If L(f ) > 1, then
n
[L(fρ )] ≤ (1 −
Eρ∈Rn−1
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
1.5
)L(f ) .
n
Proof.
I let x be any variable; by x i we denote the i-th leaf
marked by x or x, by gxi the gate to which xi enters and
by Nxi the other subformula that enters gxi ; let also nx
denotesP
the number of leaves marked by x or x (thus,
L(f ) = nx )
I since the formula is optimal, all Nxi are disjoint
(otherwise, Nxi ⊂ Nxj , but in this case we can simplify
the formula by assigning a value to xi in Nxj )
I assigning either 0 or 1 to x deletes the leaf x i and
assigning the complement value deletes both x i and Nxi
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
4 / 12
Subbotovskaya’s Lower Bound
Theorem (Subbotovskaya, 61)
If L(f ) > 1, then
n
[L(fρ )] ≤ (1 −
Eρ∈Rn−1
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
1.5
)L(f ) .
n
Proof.
I let x be any variable; by x i we denote the i-th leaf
marked by x or x, by gxi the gate to which xi enters and
by Nxi the other subformula that enters gxi ; let also nx
denotesP
the number of leaves marked by x or x (thus,
L(f ) = nx )
I since the formula is optimal, all Nxi are disjoint
(otherwise, Nxi ⊂ Nxj , but in this case we can simplify
the formula by assigning a value to xi in Nxj )
I assigning either 0 or 1 to x deletes the leaf x i and
assigning the complement value deletes both x i and Nxi
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
4 / 12
Subbotovskaya’s Lower Bound
Theorem (Subbotovskaya, 61)
If L(f ) > 1, then
n
[L(fρ )] ≤ (1 −
Eρ∈Rn−1
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
1.5
)L(f ) .
n
Proof.
I let x be any variable; by x i we denote the i-th leaf
marked by x or x, by gxi the gate to which xi enters and
by Nxi the other subformula that enters gxi ; let also nx
denotesP
the number of leaves marked by x or x (thus,
L(f ) = nx )
I since the formula is optimal, all Nxi are disjoint
(otherwise, Nxi ⊂ Nxj , but in this case we can simplify
the formula by assigning a value to xi in Nxj )
I assigning either 0 or 1 to x deletes the leaf x i and
assigning the complement value deletes both x i and Nxi
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
4 / 12
Subbotovskaya’s Lower Bound
Theorem (Subbotovskaya, 61)
If L(f ) > 1, then
n
[L(fρ )] ≤ (1 −
Eρ∈Rn−1
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
1.5
)L(f ) .
n
Proof.
I let x be any variable; by x i we denote the i-th leaf
marked by x or x, by gxi the gate to which xi enters and
by Nxi the other subformula that enters gxi ; let also nx
denotesP
the number of leaves marked by x or x (thus,
L(f ) = nx )
I since the formula is optimal, all Nxi are disjoint
(otherwise, Nxi ⊂ Nxj , but in this case we can simplify
the formula by assigning a value to xi in Nxj )
I assigning either 0 or 1 to x deletes the leaf x i and
assigning the complement value deletes both x i and Nxi
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
4 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Proof (Cont’d)
A. S. Kulikov
I
(L(f ) − L(f |x=0 )) + (L(f ) − L(f |x=1 )) ≥ 3nx
I
1 X
E [L(fρ )] =
2n
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
X
L(f |xi =b )
Andreev’s Lower
Bound
i∈[1..n] b∈{0,1}
= L(f ) −
1 X
2n
X
(L(f ) − L(f |xi =b ))
i∈[1..n] b∈{0,1}
= (1 −
1.5
)L(f )
n
5 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Proof (Cont’d)
A. S. Kulikov
I
(L(f ) − L(f |x=0 )) + (L(f ) − L(f |x=1 )) ≥ 3nx
I
1 X
E [L(fρ )] =
2n
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
X
L(f |xi =b )
Andreev’s Lower
Bound
i∈[1..n] b∈{0,1}
= L(f ) −
1 X
2n
X
(L(f ) − L(f |xi =b ))
i∈[1..n] b∈{0,1}
= (1 −
1.5
)L(f )
n
5 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Proof (Cont’d)
A. S. Kulikov
I
(L(f ) − L(f |x=0 )) + (L(f ) − L(f |x=1 )) ≥ 3nx
I
1 X
E [L(fρ )] =
2n
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
X
L(f |xi =b )
Andreev’s Lower
Bound
i∈[1..n] b∈{0,1}
= L(f ) −
1 X
2n
X
(L(f ) − L(f |xi =b ))
i∈[1..n] b∈{0,1}
= (1 −
1.5
)L(f )
n
5 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Corollary (Subbotovskaya, 61)
A. S. Kulikov
n
Eρ∈Rn−1
[L(fρ ) − 1/3] ≤ (1 −
1.5
)(L(f ) − 1/3) .
n
Proof.
I
if L(f ) > 1, then the corollary follows from the previous
lemma
I
if L(f ) = 1, then E [L(fρ )] = 1 − 1/n and
(2/3 − 1/n) ≤ 2/3(1 − 3/2n)
I
if L(f ) = 0, then −1/3 ≤ −2/3(1 − 3/2n)
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
6 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Corollary (Subbotovskaya, 61)
A. S. Kulikov
n
Eρ∈Rn−1
[L(fρ ) − 1/3] ≤ (1 −
1.5
)(L(f ) − 1/3) .
n
Proof.
I
if L(f ) > 1, then the corollary follows from the previous
lemma
I
if L(f ) = 1, then E [L(fρ )] = 1 − 1/n and
(2/3 − 1/n) ≤ 2/3(1 − 3/2n)
I
if L(f ) = 0, then −1/3 ≤ −2/3(1 − 3/2n)
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
6 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Corollary (Subbotovskaya, 61)
A. S. Kulikov
n
Eρ∈Rn−1
[L(fρ ) − 1/3] ≤ (1 −
1.5
)(L(f ) − 1/3) .
n
Proof.
I
if L(f ) > 1, then the corollary follows from the previous
lemma
I
if L(f ) = 1, then E [L(fρ )] = 1 − 1/n and
(2/3 − 1/n) ≤ 2/3(1 − 3/2n)
I
if L(f ) = 0, then −1/3 ≤ −2/3(1 − 3/2n)
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
6 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Corollary (Subbotovskaya, 61)
A. S. Kulikov
n
Eρ∈Rn−1
[L(fρ ) − 1/3] ≤ (1 −
1.5
)(L(f ) − 1/3) .
n
Proof.
I
if L(f ) > 1, then the corollary follows from the previous
lemma
I
if L(f ) = 1, then E [L(fρ )] = 1 − 1/n and
(2/3 − 1/n) ≤ 2/3(1 − 3/2n)
I
if L(f ) = 0, then −1/3 ≤ −2/3(1 − 3/2n)
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
6 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Corollary (Subbotovskaya, 61)
A. S. Kulikov
n
Eρ∈Rn−1
[L(fρ ) − 1/3] ≤ (1 −
1.5
)(L(f ) − 1/3) .
n
Proof.
I
if L(f ) > 1, then the corollary follows from the previous
lemma
I
if L(f ) = 1, then E [L(fρ )] = 1 − 1/n and
(2/3 − 1/n) ≤ 2/3(1 − 3/2n)
I
if L(f ) = 0, then −1/3 ≤ −2/3(1 − 3/2n)
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
6 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Lemma
A. S. Kulikov
For any a ≥ 1 and m ≤ n,
n
Y
(1 −
k=m+1
Shrinkage
Exponent
m a
a
)≤
.
k
n
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
Proof.
I
I
(1 − a/k) ≤ (1 − 1/k)a
Qn
k=m+1 (1 − 1/k) = m/n
7 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Lemma
A. S. Kulikov
For any a ≥ 1 and m ≤ n,
n
Y
(1 −
k=m+1
Shrinkage
Exponent
m a
a
)≤
.
k
n
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
Proof.
I
I
(1 − a/k) ≤ (1 − 1/k)a
Qn
k=m+1 (1 − 1/k) = m/n
7 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Lemma
A. S. Kulikov
For any a ≥ 1 and m ≤ n,
n
Y
(1 −
k=m+1
Shrinkage
Exponent
m a
a
)≤
.
k
n
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
Proof.
I
I
(1 − a/k) ≤ (1 − 1/k)a
Qn
k=m+1 (1 − 1/k) = m/n
7 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Lemma
A. S. Kulikov
For any a ≥ 1 and m ≤ n,
n
Y
(1 −
k=m+1
Shrinkage
Exponent
m a
a
)≤
.
k
n
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
Proof.
I
I
(1 − a/k) ≤ (1 − 1/k)a
Qn
k=m+1 (1 − 1/k) = m/n
7 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Theorem (Subbotovskaya, 61)
1.5
n [L(fρ )] ≤ L(f ) + 1/3 .
Eρ∈Rn
A. S. Kulikov
Shrinkage
Exponent
Proof.
I
I
construct a random sequence of functions
fn = f , fn−1 , . . . , fm (m = n), where fi is obtained from
fi+1 by assigning a random value to a random variable
E [L(fk ) − 1/3] ≤ (1 −
1.5
k+1 )E [L(fk+1 )
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
− 1/3]
I
n
Y
E [L(fm ) − 1/3] ≤
k=m+1
1.5
≤
1.5
1−
k
!
(L(f ) − 1/3)
(L(f ) − 1/3)
8 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Theorem (Subbotovskaya, 61)
1.5
n [L(fρ )] ≤ L(f ) + 1/3 .
Eρ∈Rn
A. S. Kulikov
Shrinkage
Exponent
Proof.
I
I
construct a random sequence of functions
fn = f , fn−1 , . . . , fm (m = n), where fi is obtained from
fi+1 by assigning a random value to a random variable
E [L(fk ) − 1/3] ≤ (1 −
1.5
k+1 )E [L(fk+1 )
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
− 1/3]
I
n
Y
E [L(fm ) − 1/3] ≤
k=m+1
1.5
≤
1.5
1−
k
!
(L(f ) − 1/3)
(L(f ) − 1/3)
8 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Theorem (Subbotovskaya, 61)
1.5
n [L(fρ )] ≤ L(f ) + 1/3 .
Eρ∈Rn
A. S. Kulikov
Shrinkage
Exponent
Proof.
I
I
construct a random sequence of functions
fn = f , fn−1 , . . . , fm (m = n), where fi is obtained from
fi+1 by assigning a random value to a random variable
E [L(fk ) − 1/3] ≤ (1 −
1.5
k+1 )E [L(fk+1 )
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
− 1/3]
I
n
Y
E [L(fm ) − 1/3] ≤
k=m+1
1.5
≤
1.5
1−
k
!
(L(f ) − 1/3)
(L(f ) − 1/3)
8 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Theorem (Subbotovskaya, 61)
1.5
n [L(fρ )] ≤ L(f ) + 1/3 .
Eρ∈Rn
A. S. Kulikov
Shrinkage
Exponent
Proof.
I
I
construct a random sequence of functions
fn = f , fn−1 , . . . , fm (m = n), where fi is obtained from
fi+1 by assigning a random value to a random variable
E [L(fk ) − 1/3] ≤ (1 −
1.5
k+1 )E [L(fk+1 )
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
− 1/3]
I
n
Y
E [L(fm ) − 1/3] ≤
k=m+1
1.5
≤
1.5
1−
k
!
(L(f ) − 1/3)
(L(f ) − 1/3)
8 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Theorem (Subbotovskaya, 61)
1.5
n [L(fρ )] ≤ L(f ) + 1/3 .
Eρ∈Rn
A. S. Kulikov
Shrinkage
Exponent
Proof.
I
I
construct a random sequence of functions
fn = f , fn−1 , . . . , fm (m = n), where fi is obtained from
fi+1 by assigning a random value to a random variable
E [L(fk ) − 1/3] ≤ (1 −
1.5
k+1 )E [L(fk+1 )
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
− 1/3]
I
n
Y
E [L(fm ) − 1/3] ≤
k=m+1
1.5
≤
1.5
1−
k
!
(L(f ) − 1/3)
(L(f ) − 1/3)
8 / 12
Ω(n1.5 ) Bound for Parity
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Theorem
A. S. Kulikov
L(parity ) = Ω(n1.5 ) .
Proof.
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
I
f = parity n
I
n
Eρ∈Rn−1
[L(fρ )] ≤ (1 −
I
in particular, there is a variable xi and a value b such
that L(f |xi =b ) ≤ (1 − 1.5
n )L(f )
I
since f |xi =b is either parity n or parity n , then
1.5
n )L(f )
L(fn−1 ) ≤ (1 −
I
Shrinkage
Exponent
1.5
)L(fn )
n
hence L(parity n ) ≥ n1.5 L(parity 1 ) = n1.5
9 / 12
Ω(n1.5 ) Bound for Parity
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Theorem
A. S. Kulikov
L(parity ) = Ω(n1.5 ) .
Proof.
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
I
f = parity n
I
n
Eρ∈Rn−1
[L(fρ )] ≤ (1 −
I
in particular, there is a variable xi and a value b such
that L(f |xi =b ) ≤ (1 − 1.5
n )L(f )
I
since f |xi =b is either parity n or parity n , then
1.5
n )L(f )
L(fn−1 ) ≤ (1 −
I
Shrinkage
Exponent
1.5
)L(fn )
n
hence L(parity n ) ≥ n1.5 L(parity 1 ) = n1.5
9 / 12
Ω(n1.5 ) Bound for Parity
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Theorem
A. S. Kulikov
L(parity ) = Ω(n1.5 ) .
Proof.
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
I
f = parity n
I
n
Eρ∈Rn−1
[L(fρ )] ≤ (1 −
I
in particular, there is a variable xi and a value b such
that L(f |xi =b ) ≤ (1 − 1.5
n )L(f )
I
since f |xi =b is either parity n or parity n , then
1.5
n )L(f )
L(fn−1 ) ≤ (1 −
I
Shrinkage
Exponent
1.5
)L(fn )
n
hence L(parity n ) ≥ n1.5 L(parity 1 ) = n1.5
9 / 12
Ω(n1.5 ) Bound for Parity
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Theorem
A. S. Kulikov
L(parity ) = Ω(n1.5 ) .
Proof.
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
I
f = parity n
I
n
Eρ∈Rn−1
[L(fρ )] ≤ (1 −
I
in particular, there is a variable xi and a value b such
that L(f |xi =b ) ≤ (1 − 1.5
n )L(f )
I
since f |xi =b is either parity n or parity n , then
1.5
n )L(f )
L(fn−1 ) ≤ (1 −
I
Shrinkage
Exponent
1.5
)L(fn )
n
hence L(parity n ) ≥ n1.5 L(parity 1 ) = n1.5
9 / 12
Ω(n1.5 ) Bound for Parity
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Theorem
A. S. Kulikov
L(parity ) = Ω(n1.5 ) .
Proof.
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
I
f = parity n
I
n
Eρ∈Rn−1
[L(fρ )] ≤ (1 −
I
in particular, there is a variable xi and a value b such
that L(f |xi =b ) ≤ (1 − 1.5
n )L(f )
I
since f |xi =b is either parity n or parity n , then
1.5
n )L(f )
L(fn−1 ) ≤ (1 −
I
Shrinkage
Exponent
1.5
)L(fn )
n
hence L(parity n ) ≥ n1.5 L(parity 1 ) = n1.5
9 / 12
Ω(n1.5 ) Bound for Parity
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Theorem
A. S. Kulikov
L(parity ) = Ω(n1.5 ) .
Proof.
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
I
f = parity n
I
n
Eρ∈Rn−1
[L(fρ )] ≤ (1 −
I
in particular, there is a variable xi and a value b such
that L(f |xi =b ) ≤ (1 − 1.5
n )L(f )
I
since f |xi =b is either parity n or parity n , then
1.5
n )L(f )
L(fn−1 ) ≤ (1 −
I
Shrinkage
Exponent
1.5
)L(fn )
n
hence L(parity n ) ≥ n1.5 L(parity 1 ) = n1.5
9 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Andreev’s Function
A. S. Kulikov
An (x, y ) = YPlog n−1 Ln/ log n x
i=0
j=1
ij
Shrinkage
Exponent
2i
|x| = |y | = n. The bits of x are organized into log n rows of
n/ log n bits each. The function first computes the parity of
the bits in each row and then uses the obtained log n-bit
number to index into the array y .
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
Theorem (Andreev, 85)
L(An ) = Ω(n2.5−o(1) ) .
10 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Andreev’s Function
A. S. Kulikov
An (x, y ) = YPlog n−1 Ln/ log n x
i=0
j=1
ij
Shrinkage
Exponent
2i
|x| = |y | = n. The bits of x are organized into log n rows of
n/ log n bits each. The function first computes the parity of
the bits in each row and then uses the obtained log n-bit
number to index into the array y .
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
Theorem (Andreev, 85)
L(An ) = Ω(n2.5−o(1) ) .
10 / 12
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
Andreev’s Function
A. S. Kulikov
An (x, y ) = YPlog n−1 Ln/ log n x
i=0
j=1
ij
Shrinkage
Exponent
2i
|x| = |y | = n. The bits of x are organized into log n rows of
n/ log n bits each. The function first computes the parity of
the bits in each row and then uses the obtained log n-bit
number to index into the array y .
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
Theorem (Andreev, 85)
L(An ) = Ω(n2.5−o(1) ) .
10 / 12
Proof
I
I
I
I
I
I
I
using counting arguments, Shannon showed that there
exists a function f on log n variables whose formula
complexity is at least n/ log log n
let y ∗ be the truth table for f and consider the function
A∗ (x) = A(x, y ∗ )
hit A∗ with a random restriction that leaves only
m = 2 log n log log n variables
when n is large enough, with probability > 1/2 there
will be at least one free variable in each row of x and in
this case L(Aρ ) ≥ L(f ) ≥ n/ log log n
consider the probability that none of the variables in the
first row remains unrestricted
the probability that one of unassigned variables is not in
the first row is (1 − 1/ log n)
the probability that none of them is in the first row is at
most
1
1 m
− m
) < 2 log n =
(1 −
log n
log2 n
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
11 / 12
Proof
I
I
I
I
I
I
I
using counting arguments, Shannon showed that there
exists a function f on log n variables whose formula
complexity is at least n/ log log n
let y ∗ be the truth table for f and consider the function
A∗ (x) = A(x, y ∗ )
hit A∗ with a random restriction that leaves only
m = 2 log n log log n variables
when n is large enough, with probability > 1/2 there
will be at least one free variable in each row of x and in
this case L(Aρ ) ≥ L(f ) ≥ n/ log log n
consider the probability that none of the variables in the
first row remains unrestricted
the probability that one of unassigned variables is not in
the first row is (1 − 1/ log n)
the probability that none of them is in the first row is at
most
1
1 m
− m
(1 −
) < 2 log n =
log n
log2 n
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
11 / 12
Proof
I
I
I
I
I
I
I
using counting arguments, Shannon showed that there
exists a function f on log n variables whose formula
complexity is at least n/ log log n
let y ∗ be the truth table for f and consider the function
A∗ (x) = A(x, y ∗ )
hit A∗ with a random restriction that leaves only
m = 2 log n log log n variables
when n is large enough, with probability > 1/2 there
will be at least one free variable in each row of x and in
this case L(Aρ ) ≥ L(f ) ≥ n/ log log n
consider the probability that none of the variables in the
first row remains unrestricted
the probability that one of unassigned variables is not in
the first row is (1 − 1/ log n)
the probability that none of them is in the first row is at
most
1
1 m
− m
(1 −
) < 2 log n =
log n
log2 n
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
11 / 12
Proof
I
I
I
I
I
I
I
using counting arguments, Shannon showed that there
exists a function f on log n variables whose formula
complexity is at least n/ log log n
let y ∗ be the truth table for f and consider the function
A∗ (x) = A(x, y ∗ )
hit A∗ with a random restriction that leaves only
m = 2 log n log log n variables
when n is large enough, with probability > 1/2 there
will be at least one free variable in each row of x and in
this case L(Aρ ) ≥ L(f ) ≥ n/ log log n
consider the probability that none of the variables in the
first row remains unrestricted
the probability that one of unassigned variables is not in
the first row is (1 − 1/ log n)
the probability that none of them is in the first row is at
most
1
1 m
− m
(1 −
) < 2 log n =
log n
log2 n
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
11 / 12
Proof
I
I
I
I
I
I
I
using counting arguments, Shannon showed that there
exists a function f on log n variables whose formula
complexity is at least n/ log log n
let y ∗ be the truth table for f and consider the function
A∗ (x) = A(x, y ∗ )
hit A∗ with a random restriction that leaves only
m = 2 log n log log n variables
when n is large enough, with probability > 1/2 there
will be at least one free variable in each row of x and in
this case L(Aρ ) ≥ L(f ) ≥ n/ log log n
consider the probability that none of the variables in the
first row remains unrestricted
the probability that one of unassigned variables is not in
the first row is (1 − 1/ log n)
the probability that none of them is in the first row is at
most
1 m
1
− m
(1 −
) < 2 log n =
log n
log2 n
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
11 / 12
Proof
I
I
I
I
I
I
I
using counting arguments, Shannon showed that there
exists a function f on log n variables whose formula
complexity is at least n/ log log n
let y ∗ be the truth table for f and consider the function
A∗ (x) = A(x, y ∗ )
hit A∗ with a random restriction that leaves only
m = 2 log n log log n variables
when n is large enough, with probability > 1/2 there
will be at least one free variable in each row of x and in
this case L(Aρ ) ≥ L(f ) ≥ n/ log log n
consider the probability that none of the variables in the
first row remains unrestricted
the probability that one of unassigned variables is not in
the first row is (1 − 1/ log n)
the probability that none of them is in the first row is at
most
1 m
1
− m
(1 −
) < 2 log n =
log n
log2 n
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
11 / 12
Proof
I
I
I
I
I
I
I
using counting arguments, Shannon showed that there
exists a function f on log n variables whose formula
complexity is at least n/ log log n
let y ∗ be the truth table for f and consider the function
A∗ (x) = A(x, y ∗ )
hit A∗ with a random restriction that leaves only
m = 2 log n log log n variables
when n is large enough, with probability > 1/2 there
will be at least one free variable in each row of x and in
this case L(Aρ ) ≥ L(f ) ≥ n/ log log n
consider the probability that none of the variables in the
first row remains unrestricted
the probability that one of unassigned variables is not in
the first row is (1 − 1/ log n)
the probability that none of them is in the first row is at
most
1 m
1
− m
(1 −
) < 2 log n =
log n
log2 n
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
11 / 12
Proof
I
I
I
I
I
I
I
using counting arguments, Shannon showed that there
exists a function f on log n variables whose formula
complexity is at least n/ log log n
let y ∗ be the truth table for f and consider the function
A∗ (x) = A(x, y ∗ )
hit A∗ with a random restriction that leaves only
m = 2 log n log log n variables
when n is large enough, with probability > 1/2 there
will be at least one free variable in each row of x and in
this case L(Aρ ) ≥ L(f ) ≥ n/ log log n
consider the probability that none of the variables in the
first row remains unrestricted
the probability that one of unassigned variables is not in
the first row is (1 − 1/ log n)
the probability that none of them is in the first row is at
most
1 m
1
− m
(1 −
) < 2 log n =
log n
log2 n
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
Andreev’s Lower
Bound
11 / 12
Proof (Cont’d)
I
the probability that some row has no free variables is at
most
1
log n
log2 n
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
I
with probability at least 1/2 every row contains an
unassigned variable
Andreev’s Lower
Bound
I
m 1.5
1
1
n/ log log n ≤ Eρ∈Rmn [L(A∗ )] ≤
L(A∗ ) +
2
n
3
I
∗
L(A) ≥ L(A ) = Ω
n2.5
log1.5 n(log log n)2.5
12 / 12
Proof (Cont’d)
I
the probability that some row has no free variables is at
most
1
log n
log2 n
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
I
with probability at least 1/2 every row contains an
unassigned variable
Andreev’s Lower
Bound
I
m 1.5
1
1
n/ log log n ≤ Eρ∈Rmn [L(A∗ )] ≤
L(A∗ ) +
2
n
3
I
∗
L(A) ≥ L(A ) = Ω
n2.5
log1.5 n(log log n)2.5
12 / 12
Proof (Cont’d)
I
the probability that some row has no free variables is at
most
1
log n
log2 n
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
I
with probability at least 1/2 every row contains an
unassigned variable
Andreev’s Lower
Bound
I
m 1.5
1
1
n/ log log n ≤ Eρ∈Rmn [L(A∗ )] ≤
L(A∗ ) +
2
n
3
I
∗
L(A) ≥ L(A ) = Ω
n2.5
log1.5 n(log log n)2.5
12 / 12
Proof (Cont’d)
I
the probability that some row has no free variables is at
most
1
log n
log2 n
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
I
with probability at least 1/2 every row contains an
unassigned variable
Andreev’s Lower
Bound
I
m 1.5
1
1
n/ log log n ≤ Eρ∈Rmn [L(A∗ )] ≤
L(A∗ ) +
2
n
3
I
∗
L(A) ≥ L(A ) = Ω
n2.5
log1.5 n(log log n)2.5
12 / 12
Proof (Cont’d)
I
the probability that some row has no free variables is at
most
1
log n
log2 n
Circuit Complexity
Minicourse:
The Shrinkage
Exponent of
Formulas over U2
A. S. Kulikov
Shrinkage
Exponent
Subbotovskaya’s
Lower Bound
I
with probability at least 1/2 every row contains an
unassigned variable
Andreev’s Lower
Bound
I
m 1.5
1
1
n/ log log n ≤ Eρ∈Rmn [L(A∗ )] ≤
L(A∗ ) +
2
n
3
I
∗
L(A) ≥ L(A ) = Ω
n2.5
log1.5 n(log log n)2.5
12 / 12