disjoint X,Y ⊆ {0,1} n Theorem : R = { ( x, y, i ) | x 2 X, y 2 Y, x i 6 = y i } C = { ( x, y ) | x ∈ X, y ∈ Y, d H ( x, y ) = 1 } | C | 2 partition# of R ≥ | X || Y | | C | 2 R cannot be partitioned into monochromatic rectangles < | X || Y | D ( R ) = Ω (2 log | C | − log | X | − log | Y | ) X : all x ∈ {0,1} n with parity 1 for R ⊕ Y : all y ∈ {0,1} n with parity 0 | X | = | Y | = 2 n − 1 | C | = n 2 n − 1 D ( R ⊕ ) = Ω (log n )
disjoint X,Y ⊆ {0,1} n Theorem : R = { ( x, y, i ) | x 2 X, y 2 Y, x i 6 = y i } C = { ( x, y ) | x ∈ X, y ∈ Y, d H ( x, y ) = 1 } | C | 2 partition# of R ≥ | X || Y | R 1 , R 2 , ..., R t : optimal partition of R into monochromatic rectangles t t let then m i = | R i ∩ C | X X | X || Y | = | R i | | C | = m i i =1 i =1 in any monochromatic rectangle: i C C ( x , y ) ∈ C can only appear in C C distinct rows and columns | R i | ≥ m 2 i
disjoint X,Y ⊆ {0,1} n Theorem : R = { ( x, y, i ) | x 2 X, y 2 Y, x i 6 = y i } C = { ( x, y ) | x ∈ X, y ∈ Y, d H ( x, y ) = 1 } | C | 2 partition# of R ≥ | X || Y | R 1 , R 2 , ..., R t : optimal partition of R into monochromatic rectangles let then m i = | R i ∩ C | t t X X | R i | ≥ m 2 | X || Y | = | R i | | C | = m i i i =1 i =1 ! 2 t t t X X X | C | 2 = m 2 | R i | = t | X || Y | m i i ≤ t ≤ t i =1 i =1 i =1 | C | 2 (Cauchy-Schwarz) t ≥ | X || Y |
R ✏ + � ( R ) = R Pub ( R ) + O (log n + log δ − 1 ) ✏ transform any public-coin protocol P to P’ which uses only O(log n +log (1/ δ )) public random bits y ∈ { 0 , 1 } n x ∈ { 0 , 1 } n public random bits r ∼ Σ (of any length) ( 1 if P is wrong on inputs x, y and random bits r Z ( x, y, r ) = 0 otherwise ∀ legal x, y, E r ∼ Σ [ Z ( x, y, r )] ≤ ✏ Goal: ∃ r 1 , r 2 , ..., r t such that for uniform i ∈ [ n ] ∀ legal x, y, E i [ Z ( x, y, r i )] ≤ ✏ + � i is new random bits, { r 1 , r 2 , ..., r t } is hard-wired into protocol P’
R ✏ + � ( R ) = R Pub ( R ) + O (log n + log δ − 1 ) ✏ ( 1 if P is wrong on inputs x, y and random bits r Z ( x, y, r ) = 0 otherwise ∀ legal x, y, E r ∼ Σ [ Z ( x, y, r )] ≤ ✏ Goal: ∃ r 1 , r 2 , ..., r t such that for uniform i ∈ [ n ] ∀ legal x, y, E i [ Z ( x, y, r i )] ≤ ✏ + � sample r 1 , r 2 , ..., r t i.i.d according to ∑ t E i [ Z ( x, y, r i )] = 1 ∀ particular legal x , y, X Z ( x, y, r i ) t i =1 Chernoff " # t ≤ e − 2 δ 2 t X r 1 ,...,r t [ E i [ Z ( x, y, r i )] > ✏ + � ] = Pr Pr Z ( x, y, r i ) > ( ✏ + � ) t bound: r 1 ,...,r t i =1 choose t=O( n / δ 2 ) < 2 − 2 n union bound: r 1 ,...,r t [ ∃ x, y, E i [ Z ( x, y, r i )] > ✏ + � ] < 1 Pr r 1 ,...,r t [ ∀ x, y, E i [ Z ( x, y, r i )] > ✏ + � ] > 0 Pr
R ✏ + � ( R ) = R Pub ( R ) + O (log n + log δ − 1 ) ✏ transform any public-coin protocol P to P’ which uses only O(log n +log δ -1 ) public random bits y ∈ { 0 , 1 } n x ∈ { 0 , 1 } n public random bits r ∼ Σ (of any length) find such random bits r 1 , r 2 , ..., r t , t=O( n / δ 2 ) : ∀ legal inputs x , y Pr i [ P is wrong on x, y with random bits r i ] ≤ ✏ + � Alice and Bob know { r 1 , r 2 , ..., r t } without communication P’ : run P ( x,y,r i ) where uniform i is new public random bits
FORK Relation i : x i = y i FORK ⊂ Σ ` × Σ ` × { 1 , . . . , ` − 1 } x i +1 6 = y i +1 y ∈ Σ ` x ∈ Σ ` alphabet Σ ={1,2, ..., w } output : such an index i that x i = y i and x i +1 ≠ y i +1
FORK Relation i : x i = y i FORK ⊂ Σ ` × Σ ` × { 0 , 1 , . . . , ` } x i +1 6 = y i +1 1 1 2 1 = = = = x 0 x 1 · · · x ` x ` +1 y 0 y 1 · · · y ` y ` +1 x 1 x 2 · · · x ` ∈ Σ ` y 1 y 2 · · · y ` ∈ Σ ` alphabet Σ ={1,2, ..., w } output : such an index i that x i = y i and x i +1 ≠ y i +1 output 0 if x = y and l if x ≠ y entry-wise Alice: 1 1 2 3 1 2 1 3 w =3 Bob: l =6 2 1 3 2 1 2 2 3 correct answers i = 0 4 6
FORK Relation i : x i = y i FORK ⊂ Σ ` × Σ ` × { 0 , 1 , . . . , ` } x i +1 6 = y i +1 1 1 2 1 = = = = x 0 x 1 · · · x ` x ` +1 y 0 y 1 · · · y ` y ` +1 x 1 x 2 · · · x ` ∈ Σ ` y 1 y 2 · · · y ` ∈ Σ ` alphabet Σ ={1,2, ..., w } How? D (FORK) = O (log ` log w ) binary search to maintain an ( i , j ) such that i < j , x i = y i and x j ≠ y j starting with i =0, j = l by exchanging a character in Σ in each round
FORK : | Σ |= w, for ∀ x, y ∈ Σ l , find i that x i = y i and x i +1 ≠ y i +1 ( α , l ) -protocol: successfully solves FORK for ∀ x,y ∈ S for an S ⊆ Σ l of size at least | S | ≥α w l a protocol for FORK is a (1, l ) -protocol Lemma : ∃ c -bit ( α , l ) -protocol for FORK ∃ (c-1) -bit ( α /2, l ) -protocol for FORK P : successfully solves FORK for ∀ x,y ∈ S with | S | ≥α w l WLOG : Alice sends the 1 st bit a ∈ {0,1} choose a larger S a = { x ∈ S | Alice sends a } run P without Alice sending the 1 st bit correct for (under the assumption that Alice sent a ) ∀ x,y ∈ S a
FORK : | Σ |= w, for ∀ x, y ∈ Σ l , find i that x i = y i and x i +1 ≠ y i +1 ( α , l ) -protocol: successfully solves FORK for ∀ x,y ∈ S for an S ⊆ Σ l of size at least | S | ≥α w l a protocol for FORK is a (1, l ) -protocol Lemma : ∃ c -bit ( α , l ) -protocol for FORK ∃ (c-1) -bit ( α /2, l ) -protocol for FORK D (FORK) = Ω (log w ) Why not bigger? How? the subproblem should be nontrivial α < 1/ w may trivialize the problem
FORK : | Σ |= w, for ∀ x, y ∈ Σ l , find i that x i = y i and x i +1 ≠ y i +1 ( α , l ) -protocol: successfully solves FORK for ∀ x,y ∈ S for an S ⊆ Σ l of size at least | S | ≥α w l a protocol for FORK is a (1, l ) -protocol Lemma : ∃ c -bit ( α , l ) -protocol for FORK ∃ (c-1) -bit ( α /2, l ) -protocol for FORK Amplification Lemma : for FORK, for α > 100 w √ ↵ ∃ c -bit ( α , l ) -protocol ∃ c -bit -protocol 2 , ` ( 2 ) D (FORK) = Ω (log ` log w )
a protocol for FORK is a (1, l ) -protocol then it must also be a (1/ w 1/3 , l ) -protocol Lemma : ∃ c -bit ( α , l ) -protocol for FORK ∃ (c-1) -bit ( α /2, l ) -protocol for FORK � 4 � ∃ ( c − Ω (log w ))-bit -protocol w 2 / 3 , ` Amplification Lemma : for FORK, for α > 100 w √ ↵ ∃ c -bit ( α , l ) -protocol ∃ c -bit -protocol 2 , ` ( 2 ) w 1 / 3 , ` � 1 � ∃ ( c − Ω(log w ))-bit -protocol 2 repeat for O(log l ) times c > Ω (log ` log w ) � 1 � ∃ ( c − Ω (log ` log w ))-bit w 1 / 3 , 2 -protocol
FORK : | Σ |= w, for ∀ x, y ∈ Σ l , find i that x i = y i and x i +1 ≠ y i +1 ( α , l ) -protocol: successfully solves FORK for ∀ x,y ∈ S for an S ⊆ Σ l of size at least | S | ≥α w l Lemma : ∃ c -bit ( α , l ) -protocol for FORK ∃ (c-1) -bit ( α /2, l ) -protocol for FORK Amplification Lemma : for FORK, for α > 100 w √ ↵ ∃ c -bit ( α , l ) -protocol ∃ c -bit -protocol 2 , ` ( 2 ) [Gringi, Sipser ’91] D (FORK) = Ω (log ` log w )
FORK : | Σ |= w, for ∀ x, y ∈ Σ l , find i that x i = y i and x i +1 ≠ y i +1 ( α , l ) -protocol: successfully solves FORK for ∀ x,y ∈ S for an S ⊆ Σ l of size at least | S | ≥α w l Amplification Lemma : for FORK, for α > 100 w √ ↵ ∃ c -bit ( α , l ) -protocol ∃ c -bit -protocol 2 , ` ( 2 ) P P’ P : solve inputs from S ⊆ Σ l P’ : use protocol P to solve inputs from a denser S’ ⊆ Σ l/2 y ∈ S 0 ⊆ Σ `/ 2 x ∈ S 0 ⊆ Σ ` / 2 f ( x ) ∈ S ⊆ Σ ` g ( y ) ∈ S ⊆ Σ ` FORK( f ( x ), g ( y )) answers FORK( x , y ) i that f ( x ) i = g ( y ) i , f ( x ) i +1 ≠ g ( y ) i +1 tells us j that x j = y j , x j +1 ≠ y j +1
FORK : | Σ |= w, for ∀ x, y ∈ Σ l , find i that x i = y i and x i +1 ≠ y i +1 ( α , l ) -protocol: successfully solves FORK for ∀ x,y ∈ S for an S ⊆ Σ l of size at least | S | ≥α w l Amplification Lemma : for FORK, for α > 100 w √ ↵ ∃ c -bit ( α , l ) -protocol ∃ c -bit -protocol 2 , ` ( 2 ) P P’ P : solve inputs from S ⊆ Σ l P’ : use protocol P to solve inputs from a denser S’ ⊆ Σ l/2 f ( x ) , g ( y ) ∈ S ( = = = = ( ( x, y ∈ S 0 extension u ∃ u ∈ Σ l/2 : many elements z ∈ S is in form z = ( u,x )
FORK : | Σ |= w, for ∀ x, y ∈ Σ l , find i that x i = y i and x i +1 ≠ y i +1 ( α , l ) -protocol: successfully solves FORK for ∀ x,y ∈ S for an S ⊆ Σ l of size at least | S | ≥α w l Amplification Lemma : for FORK, for α > 100 w √ ↵ ∃ c -bit ( α , l ) -protocol ∃ c -bit -protocol 2 , ` ( 2 ) P P’ P : solve inputs from S ⊆ Σ l P’ : use protocol P to solve inputs from a denser S’ ⊆ Σ l/2 f ( x ) , g ( y ) ∈ S ( ≠ ≠ ≠ ≠ ( ( x, y ∈ S 0 extension ∃ large S’ ⊆ Σ l/2 : any x,y ∈ S’ can be extended to ( x , x ’), ( y , y ’) ∈ S ( x , F ( x )), ( y , G ( y )) ∈ S such that are entry-wise different x’, y’ F ( x ), G ( y )
FORK : | Σ |= w, for ∀ x, y ∈ Σ l , find i that x i = y i and x i +1 ≠ y i +1 ( α , l ) -protocol: successfully solves FORK for ∀ x,y ∈ S for an S ⊆ Σ l of size at least | S | ≥α w l Amplification Lemma : for FORK, for α > 100 w √ ↵ ∃ c -bit ( α , l ) -protocol ∃ c -bit -protocol 2 , ` ( 2 ) S ⊆ Σ ` and | S | ≥ α w ` ∃ u ∈ Σ l/2 : many elements z ∈ S is in form z = ( u,x ) ⇢ or ∃ large S’ ⊆ Σ l/2 : any x,y ∈ S’ can be extended to ( x , F ( x )), ( y , G ( y )) ∈ S such that are entry-wise different F ( x ), G ( y ) √ α ` “ many ” = “ large ” = 2 w 2
S ⊆ Σ ` and : √ α ` | S | ≥ α w ` “ many ” = “ large ” = 2 w 2 ∃ u ∈ Σ l/2 : many elements z ∈ S is in form z = ( u,x ) ⇢ or ∃ large S’ ⊆ Σ l/2 : any x,y ∈ S’ can be extended to ( x , F ( x )), ( y , G ( y )) ∈ S such that are entry-wise different F ( x ), G ( y ) Boolean matrix S : Σ ` / 2 ( 1 if ( u, v ) ∈ S ∀ u, v ∈ Σ `/ 2 , S ( u, v ) = 0 otherwise S is α - dense (of 1 -entries) ≥ p α ∃ a row u that is - dense ⇢ 2 or p α ∃ - fraction of rows - dense ≥ α 2 Σ ` / 2 2 “Either one row is very dense , or there are many rows that are pretty dense .” By contradiction: < p ↵ < p α rows are -dense all rows are -dense and 2 w ` / 2 ≥ α 2 2 2 + p α p α contradiction! density of S < α 2 = α 2
S ⊆ Σ ` and : √ α ` | S | ≥ α w ` “ many ” = “ large ” = 2 w 2 ∃ u ∈ Σ l/2 : many elements z ∈ S is in form z = ( u,x ) ⇢ or ∃ large S’ ⊆ Σ l/2 : any x,y ∈ S’ can be extended to ( x , F ( x )), ( y , G ( y )) ∈ S such that are entry-wise different F ( x ), G ( y ) Boolean matrix S : Σ ` / 2 ( 1 if ( u, v ) ∈ S ∀ u, v ∈ Σ `/ 2 , S ( u, v ) = 0 otherwise S is α - dense (of 1 -entries) ≥ p α ∃ a row u that is - dense ⇢ 2 or p α ∃ - fraction of rows - dense ≥ α 2 Σ ` / 2 2 |{ ( u, x ) ∈ S }| ≥ p ↵ we still need ` ∃ u ∈ Σ ` / 2 : ⇢ 2 w 2 or ∃ ≥ p ↵ 2 w ` / 2 many x ∈ Σ ` / 2 : ` |{ ( x, u ) ∈ S }| ≥ ↵ 2 w 2
∃ ≥ p ↵ , S ⊆ Σ ` 2 w ` / 2 many x ∈ Σ ` / 2 : ` |{ ( x, u ) ∈ S }| ≥ ↵ 2 w 2 p ↵ 2 w ` / 2 such that: ∃ S’ ⊆ Σ l/2 of size | S 0 | ≥ any x,y ∈ S’ can be extended to ( x , F ( x )), ( y , G ( y )) ∈ S such that F ( x ), G ( y ) are entry-wise different Goal: nonempty subsets: find F 1 , F 2 , . . . , F ` / 2 ⊂ Σ and their compliments: F 1 , F 2 , . . . , F `/ 2 ⊂ Σ 2 w ` / 2 many x ∈ Σ l/2 √ ↵ such that for ≥ such that ( x, u ) ∈ S ∃ u ∈ F 1 × · · · × F `/ 2 ( F ( x )= u ) and such that ( x, v ) ∈ S ( G ( x )= v ) ∃ v ∈ F 1 × · · · × F `/ 2 any u ∈ F 1 × · · · × F `/ 2 and any v ∈ F 1 × · · · × F `/ 2 must be entry-wise different: 8 1 i ` u i 6 = v i 2 ,
∃ ≥ p ↵ , S ⊆ Σ ` 2 w ` / 2 many x ∈ Σ ` / 2 : ` |{ ( x, u ) ∈ S }| ≥ ↵ 2 w 2 p ↵ 2 w ` / 2 such that: ∃ S’ ⊆ Σ l/2 of size | S 0 | ≥ any x,y ∈ S’ can be extended to ( x , F ( x )), ( y , G ( y )) ∈ S such that F ( x ), G ( y ) are entry-wise different independently random: F 1 , F 2 , . . . , F ` / 2 ⊂ Σ and their compliments: F 1 , F 2 , . . . , F `/ 2 ⊂ Σ ✓ Σ ◆ is sampled uniformly and each F i ∈ independently at random w/ 2 ` for any “good” x that |{ ( x, u ) ∈ S }| ≥ α 2 w 2 ] >? Pr [ such that ( x, u ) ∈ S ∃ u ∈ F 1 × · · · × F `/ 2 and such that ( x, v ) ∈ S ∃ v ∈ F 1 × · · · × F `/ 2
∃ ≥ p ↵ , S ⊆ Σ ` 2 w ` / 2 many x ∈ Σ ` / 2 : ` |{ ( x, u ) ∈ S }| ≥ ↵ 2 w 2 p ↵ 2 w ` / 2 such that: ∃ S’ ⊆ Σ l/2 of size | S 0 | ≥ any x,y ∈ S’ can be extended to ( x , F ( x )), ( y , G ( y )) ∈ S such that F ( x ), G ( y ) are entry-wise different independently random: F 1 , F 2 , . . . , F ` / 2 ⊂ Σ and their compliments: F 1 , F 2 , . . . , F `/ 2 ⊂ Σ ✓ Σ ◆ is sampled uniformly and each F i ∈ independently at random w/ 2 ` for any “good” x that |{ ( x, u ) ∈ S }| ≥ α 2 w 2 ] >? Pr [ x is “ really good ”
∃ ≥ p ↵ , S ⊆ Σ ` 2 w ` / 2 many x ∈ Σ ` / 2 : ` |{ ( x, u ) ∈ S }| ≥ ↵ 2 w 2 p ↵ 2 w ` / 2 such that: ∃ S’ ⊆ Σ l/2 of size | S 0 | ≥ any x,y ∈ S’ can be extended to ( x , F ( x )), ( y , G ( y )) ∈ S such that F ( x ), G ( y ) are entry-wise different independently random: F 1 , F 2 , . . . , F ` / 2 ⊂ Σ and their compliments: F 1 , F 2 , . . . , F `/ 2 ⊂ Σ ✓ Σ ◆ is sampled uniformly and each F i ∈ independently at random w/ 2 ` for any “good” x that |{ ( x, u ) ∈ S }| ≥ α 2 w 2 Why? Pr[ 8 u 2 F 1 ⇥ · · · ⇥ F `/ 2 , ( x, u ) 62 S ] ⌘ w + 2 < 2e − α w/ 4 ⇣ 1 − α < 2 2 Pr[ 8 v 2 F 1 ⇥ · · · ⇥ F `/ 2 , ( x, v ) 62 S ] x is “really good”: ∃ u ∈ F 1 × · · · × F `/ 2 , ( x, u ) ∈ S and ∃ v ∈ F 1 × · · · × F `/ 2 , ( x, v ) ∈ S
∃ ≥ p ↵ , S ⊆ Σ ` 2 w ` / 2 many x ∈ Σ ` / 2 : ` |{ ( x, u ) ∈ S }| ≥ ↵ 2 w 2 p ↵ 2 w ` / 2 such that: ∃ S’ ⊆ Σ l/2 of size | S 0 | ≥ any x,y ∈ S’ can be extended to ( x , F ( x )), ( y , G ( y )) ∈ S such that F ( x ), G ( y ) are entry-wise different independently random: F 1 , F 2 , . . . , F ` / 2 ⊂ Σ and their compliments: F 1 , F 2 , . . . , F `/ 2 ⊂ Σ ✓ Σ ◆ is sampled uniformly and each F i ∈ independently at random w/ 2 ` for any “good” x that |{ ( x, u ) ∈ S }| ≥ α 2 w 2 Pr[ x is really good ] > 1 − 2e − α w/ 4 E [# of really good x ] ≥ (1 − 2e αw/ 4 ) p α √ α (for ) α > 100 ≥ 2 2 w x is “really good”: ∃ u ∈ F 1 × · · · × F `/ 2 , ( x, u ) ∈ S and ∃ v ∈ F 1 × · · · × F `/ 2 , ( x, v ) ∈ S
FORK : | Σ |= w, for ∀ x, y ∈ Σ l , find i that x i = y i and x i +1 ≠ y i +1 ( α , l ) -protocol: successfully solves FORK for ∀ x,y ∈ S for an S ⊆ Σ l of size at least | S | ≥α w l Amplification Lemma : for FORK, for α > 100 w √ ↵ ∃ c -bit ( α , l ) -protocol ∃ c -bit -protocol 2 , ` ( 2 ) P P’ P : solve inputs from S ⊆ Σ l P’ : use protocol P to solve inputs from a denser S’ ⊆ Σ l/2 ∈ S ∈ S ( ( ≠ ≠ ≠ ≠ = = = = ( ( ( ( x, y ∈ S 0 extension F ( x ) , G ( y ) x, y ∈ S 0 extension u
FORK : | Σ |= w, for ∀ x, y ∈ Σ l , find i that x i = y i and x i +1 ≠ y i +1 ( α , l ) -protocol: successfully solves FORK for ∀ x,y ∈ S for an S ⊆ Σ l of size at least | S | ≥α w l Amplification Lemma : for FORK, for α > 100 w √ ↵ ∃ c -bit ( α , l ) -protocol ∃ c -bit -protocol 2 , ` ( 2 ) P P’ P : solve inputs from S ⊆ Σ l P’ : use protocol P to solve inputs from a denser S’ ⊆ Σ l/2 either: y ∈ S 0 ⊆ Σ `/ 2 x ∈ S 0 ⊆ Σ ` / 2 ( u, x ) ∈ S ( u, y ) ∈ S or: x ∈ S 0 ⊆ Σ ` / 2 y ∈ S 0 ⊆ Σ `/ 2 ( x, F ( x )) ∈ S ( y, G ( y )) ∈ S F ( x ), G ( y ) are entry-wise different
FORK : | Σ |= w, for ∀ x, y ∈ Σ l , find i that x i = y i and x i +1 ≠ y i +1 ( α , l ) -protocol: successfully solves FORK for ∀ x,y ∈ S for an S ⊆ Σ l of size at least | S | ≥α w l Lemma : ∃ c -bit ( α , l ) -protocol for FORK ∃ (c-1) -bit ( α /2, l ) -protocol for FORK Amplification Lemma : for FORK, for α > 100 w √ ↵ ∃ c -bit ( α , l ) -protocol ∃ c -bit -protocol 2 , ` ( 2 ) [Gringi, Spser ’91] D (FORK) = Ω (log ` log w )
Direct Sum • Direct product: The probability of success of performing k independent tasks decreases in k . • Yao’s XOR lemma, the parallel repetition theorem of Ran Raz ... • Direct sum: The amount of resources needed to perform k independent tasks grows with k . • direct sum problems in CC
Direct Sum Settings f : X f × Y f → { 0 , 1 } g : X g × Y g → { 0 , 1 } f ( x f , y f ) g ( x g , y g ) x f ∈ X f y f ∈ Y f x g ∈ X g y g ∈ Y g ( X F = X f × X g F : X F × Y F → { 0 , 1 } 2 with Y F = Y f × Y g F (( x f , x g ) , ( y f , y g )) = ( f ( x f , y f ) , g ( x g , y g )) subproblems are independent : inputs are arbitrary over ∀ (( x f , x g ) , ( y f , y g )) ∈ ( X f × X g ) × ( Y f × Y g ) over over µ F = µ f × µ g µ f µ g X g × Y g X f × Y f
Direct Sum Settings f : X f × Y f → { 0 , 1 } g : X g × Y g → { 0 , 1 } f ( x f , y f ) g ( x g , y g ) x f ∈ X f y f ∈ Y f x g ∈ X g y g ∈ Y g ( X F = X f × X g F : X F × Y F → { 0 , 1 } 2 with Y F = Y f × Y g F (( x f , x g ) , ( y f , y g )) = ( f ( x f , y f ) , g ( x g , y g )) communication complexity: CC ( f, g ) , CC ( F ) for deterministic, randomized, nondeterministic protocols...
Direct Sum Settings f : X f × Y f → { 0 , 1 } g : X g × Y g → { 0 , 1 } f ( x f , y f ) ∧ g ( x g , y g ) x f ∈ X f y f ∈ Y f x g ∈ X g y g ∈ Y g ( X F = X f × X g with F : X F × Y F → { 0 , 1 } Y F = Y f × Y g F (( x f , x g ) , ( y f , y g )) = f ( x f , y f ) ∧ g ( x g , y g ) communication complexity: CC ( f ∧ g ) , CC ( F ) for deterministic, randomized, nondeterministic protocols...
Direct Sum Settings f : X × Y → { 0 , 1 } f ( x 1 , y 1 ) . . . f ( x k , y k ) y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ f k : X k × Y k → { 0 , 1 } k f k ( ~ y ) = ( f ( x 1 , y 1 ) , . . . , f ( x k , y k )) x, ~ communication complexity: CC ( f k )
Direct Sum Problems • Question I: Can CC ( f k ) ≪ k · CC ( f ) ? • Question II: Can CC ( ⋀ k f ) ≪ k · CC ( f ) ? • “Can we solve several problems simultaneously in a way that is substantially better than to solve each of the problems separately?” • Answer(?) to QI: possibly “no” for all functions. • Contemporary tool: Information Complexity
Randomized Protocols f ( x 1 , y 1 ) f : X × Y → { 0 , 1 } . . . f k : X k × Y k → { 0 , 1 } k f ( x k , y k ) y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ • Individually correct: each output ( x i , y i ) is correct with probability > 2/3. • Simultaneously correct: all output ( x i , y i ) are correct simultaneously with probability > 2/3. direct product (conjecture): The probability of simultaneous success is < (2/3) Ω ( k ) with any communication cost ≪ O( k · CC ( f )) . examples : parallel repetition theorem, Yao XOR lemma
EQ : X × Y → { 0 , 1 } X = Y = { 0 , 1 } n y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ where z i indicates whether x i = y i EQ k ( ~ y ) = ~ x, ~ z R Pub (EQ) = O (1) by checking whether h x, r i = h y, r i where r is a shared random Boolean vector X ! and h x, r i := x ( i ) r ( i ) mod 2 i is the inner-product over GF(2)
EQ : X × Y → { 0 , 1 } X = Y = { 0 , 1 } n y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ where z i indicates whether x i = y i EQ k ( ~ y ) = ~ x, ~ z R Pub (EQ) = O (1) recall: Theorem : R ( f ) = O ( R Pub ( f ) + log n )
EQ : X × Y → { 0 , 1 } X = Y = { 0 , 1 } n y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ where z i indicates whether x i = y i EQ k ( ~ y ) = ~ x, ~ z R ( f ) = O ( R Pub ( f ) + log n ) R Pub (EQ) = O (1) repeat the protocol on every each instance: 1/3 k error instance ( x i , y i ) for O(log k ) times 1 Pr[ output ( x i , y i ) = 1 | x i 6 = y i ] 3 k union bound R Pub (EQ k ) = O ( k log k ) all k instances: 1/3 error R (EQ k ) = O ( k log k + log n ) y ] 1 Pr[ 9 i, output ( x i , y i ) = 1 | ~ x 6 = ~ 3
EQ : X × Y → { 0 , 1 } X = Y = { 0 , 1 } n y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ where z i indicates whether x i = y i EQ k ( ~ y ) = ~ x, ~ z R (EQ k ) = O ( k log k + log n ) recall: Theorem : R (EQ) = Θ (log n ) consider k = log n : R (EQ k ) = O (log n log log n ) ⌧ k · R (EQ) = Θ((log n ) 2 )
Randomized Protocols f ( x 1 , y 1 ) f : X × Y → { 0 , 1 } X = Y = { 0 , 1 } n . . . f k : X k × Y k → { 0 , 1 } k f ( x k , y k ) y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ Observations : R ( f k ) ≤ k · R ( f ) individually correct: simultaneously correct: R ( f k ) = O ( k log k · R ( f )) individual: apply the protocol independently on k instances simultaneous: repeat O(log k ) times for every instance individual error ≤ 1/3 k , then apply union bound
Randomized Protocols f ( x 1 , y 1 ) f : X × Y → { 0 , 1 } X = Y = { 0 , 1 } n . . . f k : X k × Y k → { 0 , 1 } k f ( x k , y k ) y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ Observations : individually correct: R Pub ( f k ) ≤ k · R Pub ( f ) simultaneously correct: R Pub ( f k ) = O ( k log k · R Pub ( f )) recall: Theorem : R ( f ) = O ( R Pub ( f ) + log n )
Randomized Protocols f ( x 1 , y 1 ) f : X × Y → { 0 , 1 } X = Y = { 0 , 1 } n . . . f k : X k × Y k → { 0 , 1 } k f ( x k , y k ) y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ R ( f k ) = O R Pub ( f k ) + log kn � � ( simultaneous k log k · R Pub ( f ) + log n � � ≤ O correctness ) when and R Pub ( f ) ⌧ log n R ( f ) = Ω (log n ) this gives an acceleration over for small k k · R ( f )
Randomized Protocols f ( x 1 , y 1 ) f : X × Y → { 0 , 1 } X = Y = { 0 , 1 } n . . . f k : X k × Y k → { 0 , 1 } k f ( x k , y k ) y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ Observations : individually correct: R Pub ( f k ) ≤ k · R Pub ( f ) simultaneously correct: R Pub ( f k ) = O ( k log k · R Pub ( f ))
List-Non-Equality problem: ^ LNE k,n ( ~ y ) = x, ~ x i 6 = y i X = Y = { 0 , 1 } n i y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ R Pub (LNE k,n ) =? R (LNE k,n ) =? 1st trial: run the inner-product protocol on every ( x i , y i ) each x i , ≠ y i is missed with probability 1/3 Pr[ miss one of x i 6 = y i ] = 1 � (2 / 3) k 2nd trial: run the protocol on every ( x i , y i ) for Θ (log k ) times every x i ≠ y i is missed with probability < 1/3 k cost = O( k log k ) 3ird trial: make every x i ≠ y i missed with probability < 1/3 k and every ( x i , y i ) repeated for O(1) times on average!
List-Non-Equality problem: ^ LNE k,n ( ~ y ) = x, ~ x i 6 = y i X = Y = { 0 , 1 } n i y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ for i =1 to k repeat the IP protocol on ( x i , y i ) until detecting x i ≠ y i ; break and return 0 at any time if overall repetitions > C k ; return 1 ; communication complexity: O(C k ) always correct ∃ i, x i = y i ( C -1) k failures in Ck independent trials 8 i, x i 6 = y i each trial succeeds with prob. ≥ 1/2 Chernoff: C =3 , exponentially small probability
List-Non-Equality problem: ^ LNE k,n ( ~ y ) = x, ~ x i 6 = y i X = Y = { 0 , 1 } n i y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ for i =1 to k repeat the IP protocol on ( x i , y i ) until detecting x i ≠ y i ; break and return 0 at any time if overall repetitions > 3 k ; return 1 ; communication complexity: O( k ) always correct ∃ i, x i = y i incorrect with exp(- Ω ( k )) prob. 8 i, x i 6 = y i R Pub (LNE k,n ) = O ( k ) R (LNE k,n ) = O ( k + log n )
List-Non-Equality problem: ^ LNE k,n ( ~ y ) = x, ~ x i 6 = y i X = Y = { 0 , 1 } n i y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ Las Vegas: for i =1 to k repeat for ≤ t times the IP protocol on ( x i , y i ) until detecting x i ≠ y i ; if a ( x i , y i ) has been repeated for t times Alice sends Bob x i to see whether x i = y i and if so break and return 0; return 1 ; always correct if terminates the first costs O( t + n ) bits x i = y i 0 1 t A =O(1) each x i 6 = y i expectedly costs O j 2 − j + n 2 − t X @ j =1 when t = n
List-Non-Equality problem: ^ LNE k,n ( ~ y ) = x, ~ x i 6 = y i X = Y = { 0 , 1 } n i y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ Las Vegas: for i =1 to k repeat for ≤ t times the IP protocol on ( x i , y i ) until detecting x i ≠ y i ; if a ( x i , y i ) has been repeated for n times Alice sends Bob x i to see whether x i = y i and if so break and return 0; return 1 ; always correct if terminates communication cost in expectation: O( k + n ) R Pub (LNE k,n ) = O ( k + n ) R 0 (LNE k,n ) = O ( k + n ) 0
List-Non-Equality problem: ^ LNE k,n ( ~ y ) = x, ~ x i 6 = y i X = Y = { 0 , 1 } n i y = ( y 1 , . . . , y k ) ∈ Y k x = ( x 1 , . . . , x k ) ∈ X k ~ ~ LNE k,n = ∧ k EQ n Monte Carlo: R (LNE k,n ) = O ( k + log n ) Las Vegas: R 0 (LNE k,n ) = O ( k + n ) while: R (EQ) = Θ (log n ) R 0 (EQ) = Θ(log n )
Nondeterministic Protocols N 1 ( f ) : complexity of optimally certifying positive instances of f μ is a probability distribution over 1 s of f : μ is a distribution over { ( x, y ) | f ( x, y ) = 1 } Definition The rectangle size bound of f is 1 B ∗ ( f ) := µ over 1s min max µ ( R ) R where R ranges over all 1 -monochromatic rectangles. Theorem log 2 B ∗ ( f ) ≤ N 1 ( f ) ≤ log 2 B ∗ ( f ) + log 2 n
1 B ∗ ( f ) := max min µ ( R ) µ over 1s R : 1-rect. Theorem log 2 B ∗ ( f ) ≤ N 1 ( f ) ≤ log 2 B ∗ ( f ) + log 2 n N 1 ( f ) = log 2 C 1 ( f ) C 1 ( f ) : #of monochromatic rectangles to cover 1 s of f optimal cover: C = { R 1 , R 2 , . . . , R C 1 ( f ) } for any distribution μ over 1 s of f : 1 X ≤ C 1 ( f ) max R ∈C µ ( R ) min µ ( R ) ≤ C 1 ( f ) 1 ≤ µ ( R ) R ∈C R ∈ C B ∗ ( f ) ≤ C 1 ( f ) the other direction: build up a rectangle cover greedily by always taking the largest rectangle in a uniform μ over remaining 1 s C 1 ( f ) ≤ O ( nB ∗ ( f ))
1 B ∗ ( f ) := max min µ ( R ) µ over 1s R : 1-rect. Theorem log 2 B ∗ ( f ) ≤ N 1 ( f ) ≤ log 2 B ∗ ( f ) + log 2 n B ∗ ( f ∧ g ) ≥ B ∗ ( f ) · B ∗ ( g ) N 1 ( ∧ k f ) ≥ log B ∗ ( ∧ k f ) ≥ k log B ∗ ( f ) ≥ k ( N 1 ( f ) − log n ) by symmetry: N 0 ( ∨ k f ) ≥ k ( N 0 ( f ) − log n ) N ( f k ) ≥ max( N 1 ( ∧ k f ) , N 0 ( ∨ k f )) ≥ k ( N ( f ) − log n ) complexity of optimal nondeterministic protocol for f N ( f ) :
1 B ∗ ( f ) := max min µ ( R ) µ over 1s R : 1-rect. B ∗ ( f ∧ g ) ≥ B ∗ ( f ) · B ∗ ( g ) suppose optimums are achieved by: 1 1 B ∗ ( f ) = min µ f ( R ) , B ∗ ( g ) = min µ g ( R ) R R 1 1 for all 1 -rectangles R B ∗ ( f ) ≤ µ f ( R ) , B ∗ ( g ) ≤ µ g ( R ) Goal: find a distribution μ over 1 s of f ∧ g such that ∀ 1 -rectangles R in f ∧ g , µ ( R ) ≤ µ f ( R f ) µ ( R g ) for some 1 -rectangles R f in f and R g in g 1 1 B ∗ ( f ∧ g ) ≥ µ ( R f ) µ ( R g ) ≥ B ∗ ( f ) · B ∗ ( g ) µ ( R ) ≥
given μ f over 1 s of f, and μ g over 1 s of g Goal: find a distribution μ over 1 s of f ∧ g such that ∀ 1 -rectangles R in f ∧ g , µ ( R ) ≤ µ f ( R f ) µ ( R g ) for some 1 -rectangles R f in f and R g in g define μ over inputs of f ∧ g as: µ (( x f , x g ) , ( y f , y g )) = µ f ( x f , y f ) µ g ( x g , y g ) μ is a distribution over 1 s of f ∧ g ( R f = { ( x f , y f ) | (( x f , ∗ ) , ( y f , ∗ )) ∈ R } ∀ 1- rectangle R in f ∧ g , projections R g = { ( x g , y g ) | (( ∗ , x g ) , ( ∗ , y g )) ∈ R } (because of ∧ ) are 1 -rectangles in f and g R f × R g = { (( x f , x g ) , ( y f , y g )) | (( x f , y f ) ∈ R f , ( x g , y g ) ∈ R g } is a 1 -rectangle in f ∧ g and R ⊆ R f × R g µ ( R ) ≤ µ ( R f × R g ) ≤ µ ( R f ) · µ ( R g )
1 B ∗ ( f ) := max min µ ( R ) µ over 1s R : 1-rect. B ∗ ( f ∧ g ) ≥ B ∗ ( f ) · B ∗ ( g ) key property in the proof: given μ f over 1 s of f, and μ g over 1 s of g find a distribution μ over 1 s of f ∧ g such that ∀ 1 -rectangles R in f ∧ g , µ ( R ) ≤ µ f ( R f ) µ ( R g ) for some 1 -rectangles R f in f and R g in g consequence: N ( f k ) ≥ k ( N ( f ) − log n )
Deterministic Protocols complexity of optimal deterministic protocol for f D ( f ) : CC D ( f k ) k · CC D ( f ) vs. Theorem: D ( f ) ≤ O ( N ( f ) 2 ) D ( f k ) ≥ N ( f k ) ≥ k ( N ( f ) − log n ) ⇣ ⇣p ⌘⌘ D ( f ) − log n ≥ Ω k
rank ( f ∧ g ) = rank ( f ) rank ( g ) communication matrix: M f ∧ g = M f ⊗ M g Kronecker product a 11 B a 1 n B · · · . . ... A ⊗ B = . . . . a m 1 B a mn B · · · A ⊗ B (( i, k ) , ( j, l )) = a ij b kl rank ( A ⊗ B ) = rank ( A ) rank ( B )
rank ( f ∧ g ) = rank ( f ) rank ( g ) so ^ LNE k,n = ∧ k EQ LNE k,n ( ~ y ) = x, ~ x i 6 = y i i rank (LNE k,n ) = rank (EQ) k = (2 n ) k D (LNE k,n ) ≥ log rank (LNE k,n ) = kn = n 2 ( 1 -sided error with R (LNE k,n ) = O ( k + log n ) false negative) recall: R 0 (LNE k,n ) = O ( k + n ) =O( n ) N 1 (LNE k,n ) ≤ R (LNE k,n ) = O ( k + log n ) N 0 (LNE k,n ) ≤ O (log k + n ) (Alice sends ( i , x i ) with x i = y i to Bob) when k = n N (LNE k,n ) = O ( n )
rank ( f ∧ g ) = rank ( f ) rank ( g ) there is a function (LNE) such that D ( f ) = Ω ( N 0 ( f ) N 1 ( f )) D ( f ) = Ω( R 0 ( f ) 2 ) (both achieve largest possible gaps)
Disjointness DISJ : X × Y → { 0 , 1 } S ∩ T = ∅ ? S ⊆ [ n ] T ⊆ [ n ] X = Y = 2 [ n ] ( if S ∩ T = ∅ 1 DISJ( S, T ) = 0 otherwise
Disjointness DISJ : X × Y → { 0 , 1 } n ^ NAND( x i , y i ) i x ∈ { 0 , 1 } n y ∈ { 0 , 1 } n X = Y = { 0 , 1 } n ( 1 ∀ i, x i y i = 0 DISJ( x, y ) = 0 otherwise n n ^ ^ DISJ( x, y ) = x i ∨ ¯ ¯ y i = NAND( x i , y i ) i =1 i
D (DISJ) = Ω ( n ) by fooling set Theorem: [Kalyanasundaram, Schnitger’92] [Razborov’92] [Bar-Yossef, Jayram, Kumar, Sivakumar’02] R (DISJ) = Ω( n ) Theorem: [Babai, Frankl, Simon’02] The deterministic communication complexity on distributional inputs: D µ (DISJ) = O ( √ n log n ) for all product distributions μ .
D (DISJ) = Ω ( n ) by fooling set Theorem: [Kalyanasundaram, Schnitger’92] [Razborov’92] [Bar-Yossef, Jayram, Kumar, Sivakumar’02] R (DISJ) = Ω( n ) idea: R (DISJ) = R ( ∧ n NAND) ≥ Ω ( n ) R (NAND)? [Bar-Yossef, Jayram, Kumar, Sivakumar’02] R (DISJ) ≥ IC µ (DISJ) = IC µ ( ∧ n NAND) ≥ Ω ( n ) IC µ (NAND)
Information Theory entropy: 1 X H ( X ) = P ( x ) log P ( x ) x conditional entropy: X H ( X | Y ) = P ( y ) H ( X | Y = y ) y mutual information: I ( X ; Y ) = H ( X ) − H ( X | Y ) = H ( Y ) − H ( Y | X ) conditional mutual information: I ( X ; Y | Z ) = H ( X | Z ) − H ( X | Y Z ) = I ( X ; Y Z ) − I ( X ; Z )
private-coin randomized protocol π : ( X , Y ) is sampled according to μ X Y communication transcript Π = Π ( X , Y , r A , r B ) mutual info: I ( XY ; Π ) = H ( XY ) − H ( XY | Π ) the amount of info. about inputs one can get by seeing the contents of communications
Definition The ( external ) information cost of a protocol π is IC µ ( π ) = IC ext µ ( π ) = I ( XY ; Π ) Definition: The information complexity of f is IC µ ( f ) = inf π IC µ ( π ) where π ranges over all private-coin randomized protocols for f with bounded-error on all inputs IC µ ( f ) optimizes over the same protocols as R ( f ) input distribution μ is only used to generate Π
X ranges over s values 0 ≤ H ( X ) ≤ log s subadditivity: H ( X, Y ) ≤ H ( X ) + H ( Y ) equality is achieved if and only if X , Y are independent H ( X, Y | Z ) ≤ H ( X | Z ) + H ( Y | Z ) equality is achieved if and only if X , Y are conditionally independent given Z data processing inequality: if X , Z are conditionally independent given Y I ( X ; Y | Z ) ≤ I ( X ; Y )
IC µ ( f ) = inf π I ( XY ; Π ) where π ranges over all private-coin randomized protocols for f with bounded-error on all inputs ∀ µ, R ( f ) ≥ IC µ ( f ) π : optimal private-coin protocol for f ≥ IC µ ( f ) R ( f ) = CC ( π ) ≥ H ( Π ) ≥ I ( XY ; Π ) X ranges over s values 0 ≤ H ( X ) ≤ log s
are mutually independent Z = ( Z 1 , . . . , Z n ) I ( Z ; Π ) ≥ I ( Z 1 ; Π ) + · · · I ( Z n ; Π ) n ^ NAND( x i , y i ) i x ∈ { 0 , 1 } n y ∈ { 0 , 1 } n each ( X i , Y i ) is distributed independently according to μ : Pr[( x i , y i ) = (0 , 0)] = 1 2 Pr[( x i , y i ) = (0 , 1)] = Pr[( x i , y i ) = (1 , 0)] = 1 4 ( X , Y ) follows the product distribution μ n
n X I ( XY ; Π ) ≥ I ( X i Y i ; Π ) i =1 π : optimal private-coin protocol for DISJ n comm. transcript Π = Π ( X , Y , r A , r B ) ^ NAND( x i , y i ) i X Y each ( X i , Y i ) is distributed independently according to μ : Pr[( x i , y i ) = (0 , 0)] = 1 2 Pr[( x i , y i ) = (0 , 1)] = Pr[( x i , y i ) = (1 , 0)] = 1 4 ( X , Y ) follows the product distribution μ n all possible inputs have DISJ( X , Y )=1 (Is this a problem?)
subadditivity data processing n n X X I ( XY ; Π ) ≥ I ( X i Y i ; Π ) ≥ I ( X i Y i ; Π | D ) i =1 i =1 π : optimal private-coin protocol for DISJ n comm. transcript Π = Π ( X , Y , r A , r B ) ^ NAND( x i , y i ) i X Y each ( X i , Y i ) is distributed independently according to μ : ( sample uniform X i ∈ { 0 , 1 } uniformly random if D i =0 Y i = 0 “switches” D i ∈ {0,1} ( X i = 0 if D i =1 D = ( D 1 , . . . , D n ) Y i ∈ { 0 , 1 } uniformly random X i , Y i are conditionally independent given D i !
I ( X i Y i ; Π | D ) ≥ IC µ (NAND | D i ) ( sample uniform X i ∈ { 0 , 1 } uniformly random if D i =0 Y i = 0 “switches” D i ∈ {0,1} ( X i = 0 if D i =1 D = ( D 1 , . . . , D n ) Y i ∈ { 0 , 1 } uniformly random B NAND( A , B ) A Y 2 X 2 Y n X n for i =1 : I ( X i Y i ; Π | D ) = E d 2 ,...d n [ I ( X i Y i ; Π | D 1 , D 2 = d 2 , . . . , D n = d n )] X i , Y i are independent for i >1 fix any particular D 2 = d 2 , . . . , D n = d n Alice and Bob can sample X i , Y i with private coins so that NAND( A , B ) is solved by Π ( AX 2 ...X n , BY 2 ...Y n ) I ( X 1 Y 1 ; Π | D 1 , D 2 = d 2 , . . . , D n = d n ) ≥ IC µ (NAND | D 1 )
I ( X i Y i ; Π | D ) ≥ IC µ (NAND | D i ) B NAND( A , B ) A Y 2 X 2 Y n X n for i =1 : I ( X i Y i ; Π | D ) = E d 2 ,...d n [ I ( X i Y i ; Π | D 1 , D 2 = d 2 , . . . , D n = d n )] X i , Y i are independent for i >1 fix any particular D 2 = d 2 , . . . , D n = d n Alice and Bob can sample X i , Y i with private coins so that NAND( A , B ) is solved by Π ( AX 2 ...X n , BY 2 ...Y n ) this gives a private-coin protocol θ for NAND with bounded error on all inputs such that I ( AB ; Θ | D 1 ) = I ( X 1 Y 1 ; Π | D 1 , D 2 = d 2 , . . . , D n = d n ) I ( X i Y i ; Π | D 1 , D 2 = d 2 , . . . , D n = d n ) ≥ IC µ (NAND | D 1 )
R ( f ) ≥ IC µ ( f ) n n X X I ( XY ; Π ) ≥ I ( X i Y i ; Π ) ≥ I ( X i Y i ; Π | D ) i =1 i =1 I ( X i Y i ; Π | D ) ≥ IC µ (NAND | D i ) R (DISJ) ≥ IC µ (DISJ) = I ( XY ; Π ) ≥ n · IC µ (NAND | D i ) ( X , Y ) is sampled according to μ X Y comm. transcript Π = Π ( X , Y , r A , r B )
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