Correlation polytope (3) b 0 S 1 � 1 | a ∩ b | = 0 0 R 1 S ab = a 1 0 0 | a ∩ b | = 1 0 0 1 1 ◮ disjoint pairs Q 0 := { ( a, b ) : | a ∩ b | = 0 } ◮ forbidden pairs Q 1 := { ( a, b ) : | a ∩ b | = 1 } Theorem (Razborov ’91) Any rectangle R has µ 0 ( R ) ≤ (1 + ε ) µ 1 ( R ) + 2 − Θ( n ) . ◮ Define µ 0 ( R ) := | R ∩ Q 0 | uniform measure | Q 0 |
Correlation polytope (3) b 0 S 1 � 1 | a ∩ b | = 0 0 R 1 S ab = a 1 0 0 | a ∩ b | = 1 0 0 1 1 ◮ disjoint pairs Q 0 := { ( a, b ) : | a ∩ b | = 0 } ◮ forbidden pairs Q 1 := { ( a, b ) : | a ∩ b | = 1 } Theorem (Razborov ’91) Any rectangle R has µ 0 ( R ) ≤ (1 + ε ) µ 1 ( R ) + 2 − Θ( n ) . ◮ Define µ 0 ( R ) := | R ∩ Q 0 | uniform measure | Q 0 | ◮ Applying Razborov +2 − Θ( n ) ≤ 2 − Θ( n ) µ 0 ( R ) ≤ (1 + ε ) µ 1 ( R ) � �� � =0
The setting ◮ We consider tuples a, b ⊆ [4 n − 1] with | a | = | b | = n a b n symbols 4 n − 1 symbols
The setting ◮ We consider tuples a, b ⊆ [4 n − 1] with | a | = | b | = n a b n symbols 4 n − 1 symbols ◮ Define Q 0 = { ( a, b ) : | a | = | b | = n and | a ∩ b | = 0 } Q 1 = { ( a, b ) : | a | = | b | = n and | a ∩ b | = 1 }
The setting ◮ We consider tuples a, b ⊆ [4 n − 1] with | a | = | b | = n a b n symbols 4 n − 1 symbols ◮ Define Q 0 = { ( a, b ) : | a | = | b | = n and | a ∩ b | = 0 } Q 1 = { ( a, b ) : | a | = | b | = n and | a ∩ b | = 1 } ◮ A rectangle is of the form R = A × B
The setting ◮ We consider tuples a, b ⊆ [4 n − 1] with | a | = | b | = n a b n symbols 4 n − 1 symbols ◮ Define Q 0 = { ( a, b ) : | a | = | b | = n and | a ∩ b | = 0 } Q 1 = { ( a, b ) : | a | = | b | = n and | a ∩ b | = 1 } ◮ A rectangle is of the form R = A × B ◮ Measure of rectangle: µ 0 ( R ) = Pr ( a,b ) ∈ Q 0 [( a, b ) ∈ R ]
Example rectangles
Example rectangles Example 1: ◮ Partition [4 n − 1] = T A ˙ ∪ T B with | T A | ≈ | T B | T A T B
Example rectangles Example 1: ◮ Partition [4 n − 1] = T A ˙ ∪ T B with | T A | ≈ | T B | a b T A T B ◮ Take A := { a ⊆ T A } and B := { b ⊆ T B } → R := A × B
Example rectangles Example 1: ◮ Partition [4 n − 1] = T A ˙ ∪ T B with | T A | ≈ | T B | a b T A T B ◮ Take A := { a ⊆ T A } and B := { b ⊆ T B } → R := A × B ◮ Then µ 1 ( R ) = 0 and µ 0 ( R ) = 2 − Θ( n )
Example rectangles Example 1: ◮ Partition [4 n − 1] = T A ˙ ∪ T B with | T A | ≈ | T B | a b T A T B ◮ Take A := { a ⊆ T A } and B := { b ⊆ T B } → R := A × B ◮ Then µ 1 ( R ) = 0 and µ 0 ( R ) = 2 − Θ( n ) Example 2: ◮ Fix a symbol i . i
Example rectangles Example 1: ◮ Partition [4 n − 1] = T A ˙ ∪ T B with | T A | ≈ | T B | a b T A T B ◮ Take A := { a ⊆ T A } and B := { b ⊆ T B } → R := A × B ◮ Then µ 1 ( R ) = 0 and µ 0 ( R ) = 2 − Θ( n ) Example 2: ◮ Fix a symbol i . i a b ◮ Let A := { a : i ∈ a } and B := { b : i ∈ b } → R := A × B
Example rectangles Example 1: ◮ Partition [4 n − 1] = T A ˙ ∪ T B with | T A | ≈ | T B | a b T A T B ◮ Take A := { a ⊆ T A } and B := { b ⊆ T B } → R := A × B ◮ Then µ 1 ( R ) = 0 and µ 0 ( R ) = 2 − Θ( n ) Example 2: ◮ Fix a symbol i . i a b ◮ Let A := { a : i ∈ a } and B := { b : i ∈ b } → R := A × B ◮ The measures are � 1 � µ 1 ( R ) = Θ and µ 0 ( R ) = 0 n
Partitions A partition is a tuple T = ( T A , T B , i ) T A T B i 2 n − 1 symbols 2 n − 1 symbols
Partitions A partition is a tuple T = ( T A , T B , i ) T A T B i 2 n − 1 symbols 2 n − 1 symbols Observation: We can generate a uniform random ( a, b ) ∈ Q 0 as follows:
Partitions A partition is a tuple T = ( T A , T B , i ) T A T B i 2 n − 1 symbols 2 n − 1 symbols Observation: We can generate a uniform random ( a, b ) ∈ Q 0 as follows: 1. Take a random partition T
Partitions A partition is a tuple T = ( T A , T B , i ) T A T B i a b 2 n − 1 symbols 2 n − 1 symbols Observation: We can generate a uniform random ( a, b ) ∈ Q 0 as follows: 1. Take a random partition T 2. Take a ⊆ T A and b ⊆ T B
Partitions A partition is a tuple T = ( T A , T B , i ) T A T B i a b 2 n − 1 symbols 2 n − 1 symbols Observation: We can generate a uniform random ( a, b ) ∈ Q 0 as follows: 1. Take a random partition T 2. Take a ⊆ T A and b ⊆ T B Hence µ 0 ( R ) = Pr [ a ∈ A, b ∈ B ] ( a,b ) ∈ Q 0
Partitions A partition is a tuple T = ( T A , T B , i ) T A T B i a b 2 n − 1 symbols 2 n − 1 symbols Observation: We can generate a uniform random ( a, b ) ∈ Q 0 as follows: 1. Take a random partition T 2. Take a ⊆ T A and b ⊆ T B Hence � � µ 0 ( R ) = Pr [ a ∈ A, b ∈ B ] = E a ⊆ T A [ a ∈ A ] · Pr Pr b ⊆ T B [ b ∈ B ] ( a,b ) ∈ Q 0 T
Partitions (2) Observation: We can generate a uniform random ( a, b ) ∈ Q 1 as follows:
Partitions (2) T A T B i Observation: We can generate a uniform random ( a, b ) ∈ Q 1 as follows: 1. Take a random partition
Partitions (2) T A T B i a b Observation: We can generate a uniform random ( a, b ) ∈ Q 1 as follows: 1. Take a random partition 2. Take a ⊆ T A ∪ { i } : i ∈ a and b ⊆ T B ∪ { i } : i ∈ b
Partitions (2) T A T B i a b Observation: We can generate a uniform random ( a, b ) ∈ Q 1 as follows: 1. Take a random partition 2. Take a ⊆ T A ∪ { i } : i ∈ a and b ⊆ T B ∪ { i } : i ∈ b Hence � � µ 1 ( R ) = Pr [ a ∈ A, b ∈ B ] = E Pr [ a ∈ A ] · Pr [ b ∈ B ] ( a,b ) ∈ Q 1 T a ⊆ T A ∪{ i } b ⊆ T B ∪{ i } i ∈ a i ∈ b
Easy case I - Small partitions µ 0 ( R ) ≤ (1 + ε ) µ 1 ( R ) + 2 − Θ( n ) Goal:
Easy case I - Small partitions µ 0 ( R ) ≤ (1 + ε ) µ 1 ( R ) + 2 − Θ( n ) Goal: Measure: � � µ 0 ( R ) = E a ⊆ T A [ a ∈ A ] · Pr Pr b ⊆ T B [ b ∈ B ] T
Easy case I - Small partitions µ 0 ( R ) ≤ (1 + ε ) µ 1 ( R ) + 2 − Θ( n ) Goal: Measure: � � ≤ 2 − Θ( n ) µ 0 ( R ) = E a ⊆ T A [ a ∈ A ] Pr · Pr b ⊆ T B [ b ∈ B ] T � �� � � �� � either ≤ 2 − Θ( n ) or ≤ 2 − Θ( n )
Easy case I - Small partitions µ 0 ( R ) ≤ (1 + ε ) µ 1 ( R ) + 2 − Θ( n ) Goal: Assumption: Suppose that all partitions T have ◮ either Pr a ⊆ T A ∪{ i } [ a ∈ A ] ≤ 2 − Θ( n ) ◮ or Pr b ⊆ T B ∪{ i } [ b ∈ B ] ≤ 2 − Θ( n ) Measure: � � ≤ 2 − Θ( n ) µ 0 ( R ) = E a ⊆ T A [ a ∈ A ] Pr · Pr b ⊆ T B [ b ∈ B ] T � �� � � �� � either ≤ 2 − Θ( n ) or ≤ 2 − Θ( n )
Easy case II - Good partitions
Easy case II - Good partitions Assumption: Suppose that all partitions T have ◮ Pr a ⊆ T A [ a ∈ A ] = (1 ± ε ) · Pr a ⊆ T A ∪{ i } : i ∈ a [ a ∈ A ] ◮ and Pr b ⊆ T B [ b ∈ B ] = (1 ± ε ) · Pr b ⊆ T B ∪{ i } : i ∈ b [ b ∈ B ] T A T B i a b 2 n − 1 symbols 2 n − 1 symbols Then � � µ 0 ( R ) = a ⊆ T A [ a ∈ A ] · Pr Pr b ⊆ T B [ b ∈ B ] E T
Easy case II - Good partitions Assumption: Suppose that all partitions T have ◮ Pr a ⊆ T A [ a ∈ A ] = (1 ± ε ) · Pr a ⊆ T A ∪{ i } : i ∈ a [ a ∈ A ] ◮ and Pr b ⊆ T B [ b ∈ B ] = (1 ± ε ) · Pr b ⊆ T B ∪{ i } : i ∈ b [ b ∈ B ] T A T B i a b 2 n − 1 symbols 2 n − 1 symbols Then � � µ 0 ( R ) = a ⊆ T A [ a ∈ A ] · Pr Pr b ⊆ T B [ b ∈ B ] E T � � = (1 ± O ( ε )) · E Pr [ a ∈ A ] · Pr [ b ∈ B ] T a ⊆ T A ∪{ i } b ⊆ T B ∪{ i } i ∈ a i ∈ b
Easy case II - Good partitions Assumption: Suppose that all partitions T have ◮ Pr a ⊆ T A [ a ∈ A ] = (1 ± ε ) · Pr a ⊆ T A ∪{ i } : i ∈ a [ a ∈ A ] ◮ and Pr b ⊆ T B [ b ∈ B ] = (1 ± ε ) · Pr b ⊆ T B ∪{ i } : i ∈ b [ b ∈ B ] T A T B i a b 2 n − 1 symbols 2 n − 1 symbols Then � � µ 0 ( R ) = a ⊆ T A [ a ∈ A ] · Pr Pr b ⊆ T B [ b ∈ B ] E T � � = (1 ± O ( ε )) · E Pr [ a ∈ A ] · Pr [ b ∈ B ] T a ⊆ T A ∪{ i } b ⊆ T B ∪{ i } i ∈ a i ∈ b = (1 ± O ( ε )) · µ 1 ( R )
An example for a bad partition Example: Consider a partition T and rectangle R = A × B with A := { a ⊆ T A } and B := { b ⊆ T B } T A T B i a b 2 n − 1 symbols 2 n − 1 symbols
An example for a bad partition Example: Consider a partition T and rectangle R = A × B with A := { a ⊆ T A } and B := { b ⊆ T B } T A T B i a b 2 n − 1 symbols 2 n − 1 symbols Then a ⊆ T A ∪{ i } [ a ∈ A : i / Pr ∈ a ] = 1 a ∈ T A ∪{ i } [ a ∈ A | i ∈ a ] = 0 Pr � �� � � �� � contribution to µ 0 ( R ) contribution to µ 1 ( R )
Fraction of bad partitions
Fraction of bad partitions ( a, b ) root Phase I: Pick T i ( a, b ) ( a, b ) T ′ ( a, b ) T ′′ ( a, b ) T T A T B
Fraction of bad partitions ( a, b ) root Phase I: Pick T i ( a, b ) ( a, b ) T ′ ( a, b ) T ′′ ( a, b ) T T A T B Phase II: Pick ( a, b ) i ( a ′ , b ′ ) a ( a, b ) ( a, b ) ( a, b ) b T A T B | a ∩ b | = 0
Fraction of bad partitions ( a, b ) root Phase I: Pick T i ( a, b ) ( a, b ) T ′ ( a, b ) T ′′ ( a, b ) T T A T B Phase II: Pick ( a, b ) i ≤ ε fraction ( a ′ , b ′ ) a ( a, b ) ( a, b ) good or ( a, b ) b bad small T A T B | a ∩ b | = 0
Fraction of bad partitions ( a, b ) root Phase I: Pick T i ( a, b ) ( a, b ) T ′ ( a, b ) T ′′ ( a, b ) T T A T B Phase II: Pick ( a, b ) i ≤ ε fraction ( a ′ , b ′ ) a ( a, b ) ( a, b ) good or ( a, b ) b bad small T A T B | a ∩ b | = 0 ◮ Suffices to show: Lemma For any disjoint pair ( a, b ), take a random partition T with a ⊆ T A , b ⊆ T B . Then Pr[ T is bad] ≤ ε.
Pseudo-random behaviour of large set systems Imagine the following setting:
b b b b b b Pseudo-random behaviour of large set systems Imagine the following setting: ◮ n elements
b b b b b b Pseudo-random behaviour of large set systems Imagine the following setting: ◮ n elements ◮ set system S with 2 (1 − o (1)) n sets
b b b b b b Pseudo-random behaviour of large set systems Imagine the following setting: ◮ n elements ◮ set system S with 2 (1 − o (1)) n sets Questions: ◮ Is it possible that ≥ 1 % of elements are in no set at all?
b b b b b b Pseudo-random behaviour of large set systems Imagine the following setting: ◮ n elements ◮ set system S with 2 (1 − o (1)) n sets Questions: ◮ Is it possible that ≥ 1 % of elements are in no set at all? NO! The 0 . 99 n active elements form at most 2 0 . 99 n sets
b b b b b b Pseudo-random behaviour of large set systems Imagine the following setting: ◮ n elements ◮ set system S with 2 (1 − o (1)) n sets Questions: ◮ Is it possible that ≥ 1 % of elements are in no set at all? NO! The 0 . 99 n active elements form at most 2 0 . 99 n sets ◮ Is it possible that ≥ 1 % elements are in ≤ 49 % of sets?
b b b b b b Pseudo-random behaviour of large set systems Imagine the following setting: ◮ n elements ◮ set system S with 2 (1 − o (1)) n sets Questions: ◮ Is it possible that ≥ 1 % of elements are in no set at all? NO! The 0 . 99 n active elements form at most 2 0 . 99 n sets ◮ Is it possible that ≥ 1 % elements are in ≤ 49 % of sets? NO!
b b b b b b Pseudo-random behaviour of large set systems Imagine the following setting: ◮ n elements ◮ set system S with 2 (1 − o (1)) n sets Questions: ◮ Is it possible that ≥ 1 % of elements are in no set at all? NO! The 0 . 99 n active elements form at most 2 0 . 99 n sets ◮ Is it possible that ≥ 1 % elements are in ≤ 49 % of sets? NO! Proof: ◮ Take a random set from S
b b b b b b Pseudo-random behaviour of large set systems Imagine the following setting: ◮ n elements ◮ set system S with 2 (1 − o (1)) n sets Questions: ◮ Is it possible that ≥ 1 % of elements are in no set at all? NO! The 0 . 99 n active elements form at most 2 0 . 99 n sets ◮ Is it possible that ≥ 1 % elements are in ≤ 49 % of sets? NO! Proof: ◮ Take a random set from S ◮ Denote char. vector as x ∈ { 0 , 1 } n
b b b b b b Pseudo-random behaviour of large set systems Imagine the following setting: ◮ n elements ◮ set system S with 2 (1 − o (1)) n sets Questions: ◮ Is it possible that ≥ 1 % of elements are in no set at all? NO! The 0 . 99 n active elements form at most 2 0 . 99 n sets ◮ Is it possible that ≥ 1 % elements are in ≤ 49 % of sets? NO! Proof: ◮ Take a random set from S ◮ Denote char. vector as x ∈ { 0 , 1 } n log |S| = H ( x )
b b b b b b Pseudo-random behaviour of large set systems Imagine the following setting: ◮ n elements ◮ set system S with 2 (1 − o (1)) n sets Questions: ◮ Is it possible that ≥ 1 % of elements are in no set at all? NO! The 0 . 99 n active elements form at most 2 0 . 99 n sets ◮ Is it possible that ≥ 1 % elements are in ≤ 49 % of sets? NO! Proof: ◮ Take a random set from S ◮ Denote char. vector as x ∈ { 0 , 1 } n n subadd � log |S| = H ( x ) ≤ H ( x i ) i =1
b b b b b b Pseudo-random behaviour of large set systems Imagine the following setting: ◮ n elements ◮ set system S with 2 (1 − o (1)) n sets Questions: ◮ Is it possible that ≥ 1 % of elements are in no set at all? NO! The 0 . 99 n active elements form at most 2 0 . 99 n sets ◮ Is it possible that ≥ 1 % elements are in ≤ 49 % of sets? NO! Proof: entropy 1 ◮ Take a random set from S ◮ Denote char. vector as x ∈ { 0 , 1 } n n subadd � log |S| = H ( x ) ≤ H ( x i ) ≤ n − Ω( n ) p 0 i =1 0 0 . 5 1 . 0
b b b b b b Pseudo-random behaviour of large set systems Imagine the following setting: ◮ n elements ◮ set system S with 2 (1 − o (1)) n sets
b b b b b b Pseudo-random behaviour of large set systems Imagine the following setting: ◮ n elements ◮ set system S with 2 (1 − o (1)) n sets Lemma If |S| ≥ 2 (1 − Θ( ε 3 )) n , then a (1 − ε )-fraction of elements i lies in a ( 1 2 ± ε )-fraction of sets.
b b b b b b Pseudo-random behaviour of large set systems Imagine the following setting: ◮ n elements ◮ set system S with 2 (1 − o (1)) n sets Lemma If |S| ≥ 2 (1 − Θ( ε 3 )) n , then a (1 − ε )-fraction of elements i lies in a ( 1 2 ± ε )-fraction of sets. For such an i : · Pr S ⊆ [ n ] [ S ∈ S ] S ⊆ [ n ] [ S ∈ S | i ∈ S ] Pr = S ⊆ [ n ] [ i ∈ S | S ∈ S ] Pr S ⊆ [ n ] [ i ∈ S ] Pr � �� � ∈ 1 � �� � 2 ± ε =1 / 2 = (1 ± O ( ε )) · Pr S ⊆ [ n ] [ S ∈ S ]
Fraction of bad partitions (2)
Fraction of bad partitions (2) T A ∪ { i } T B 2 n symbols Claim: Fix T B ⊇ b ∗ .
Fraction of bad partitions (2) T A ∪ { i } T B i 2 n symbols Take i ∈ T B \ a ∗ at random. Claim: Fix T B ⊇ b ∗ .
Fraction of bad partitions (2) T A ∪ { i } T B i 2 n symbols Take i ∈ T B \ a ∗ at random. Claim: Fix T B ⊇ b ∗ . ⇒ Pr[ T bad for a ’s] ≤ ε .
Fraction of bad partitions (2) T A ∪ { i } T B i 2 n symbols Take i ∈ T B \ a ∗ at random. Claim: Fix T B ⊇ b ∗ . ⇒ Pr[ T bad for a ’s] ≤ ε . � 2 n � = 2 (2 − o (1)) n . ◮ Observe: n
Fraction of bad partitions (2) T A ∪ { i } T B a i 2 n symbols Take i ∈ T B \ a ∗ at random. Claim: Fix T B ⊇ b ∗ . ⇒ Pr[ T bad for a ’s] ≤ ε . � 2 n � = 2 (2 − o (1)) n . ◮ Observe: n ◮ Let A T := { a ∈ A : a ⊆ T A ∪ { i }}
Fraction of bad partitions (2) T A ∪ { i } T B a i 2 n symbols Take i ∈ T B \ a ∗ at random. Claim: Fix T B ⊇ b ∗ . ⇒ Pr[ T bad for a ’s] ≤ ε . � 2 n � = 2 (2 − o (1)) n . ◮ Observe: n ◮ Let A T := { a ∈ A : a ⊆ T A ∪ { i }} ◮ Assume | A T | ≥ 2 (2 − o (1)) n (otherwise T is small)
Fraction of bad partitions (2) T A ∪ { i } T B a i 2 n symbols Take i ∈ T B \ a ∗ at random. Claim: Fix T B ⊇ b ∗ . ⇒ Pr[ T bad for a ’s] ≤ ε . � 2 n � = 2 (2 − o (1)) n . ◮ Observe: n ◮ Let A T := { a ∈ A : a ⊆ T A ∪ { i }} ◮ Assume | A T | ≥ 2 (2 − o (1)) n (otherwise T is small) ◮ From previous slide: A (1 − ε )-fraction i is in ≈ 1 2 fraction of a ∈ A T
Fraction of bad partitions (2) T A ∪ { i } T B a i 2 n symbols Take i ∈ T B \ a ∗ at random. Claim: Fix T B ⊇ b ∗ . ⇒ Pr[ T bad for a ’s] ≤ ε . � 2 n � = 2 (2 − o (1)) n . ◮ Observe: n ◮ Let A T := { a ∈ A : a ⊆ T A ∪ { i }} ◮ Assume | A T | ≥ 2 (2 − o (1)) n (otherwise T is small) ◮ From previous slide: A (1 − ε )-fraction i is in ≈ 1 2 fraction of a ∈ A T ◮ Equivalent to a ⊆ T A ∪{ i } [ a ∈ A | i ∈ a ] = (1 ± O ( ε )) · Pr a ⊆ T A ∪{ i } [ a ∈ A | i / Pr ∈ a ]
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