Toward the KRW conjecture Cubic Formula Lower Bounds via Communication Complexity Irit Dinur Or Meir Irit Dinur, Or Meir Toward the KRW conjecture
Formulas (de-Morgan) Formulas are circuits with fan-out 1 . Cannot store intermediate results. The formula complexity L ( f ) is the size of the smallest formula for f . Irit Dinur, Or Meir Toward the KRW conjecture
Formulas Would like: Explicit f with L ( f ) = n ω (1) . Irit Dinur, Or Meir Toward the KRW conjecture
Formulas Would like: Explicit f with L ( f ) = n ω (1) . State-of-the-art: Andreev’s function [A87] has complexity ˜ Ω( n 3 ) [H98]. Irit Dinur, Or Meir Toward the KRW conjecture
Formulas Would like: Explicit f with L ( f ) = n ω (1) . State-of-the-art: Andreev’s function [A87] has complexity ˜ Ω( n 3 ) [H98]. Proof based on shrinkage [S61, IN93, PZ93. H98, T14]. We now have an optimal analysis of shrinkage. Irit Dinur, Or Meir Toward the KRW conjecture
Formulas Would like: Explicit f with L ( f ) = n ω (1) . State-of-the-art: Andreev’s function [A87] has complexity ˜ Ω( n 3 ) [H98]. Proof based on shrinkage [S61, IN93, PZ93. H98, T14]. We now have an optimal analysis of shrinkage. So shrinkage is unlikely to get us any further. Irit Dinur, Or Meir Toward the KRW conjecture
Composition [KRW91]: we need to understand composition. Irit Dinur, Or Meir Toward the KRW conjecture
Composition [KRW91]: we need to understand composition. Let f : { 0 , 1 } m → { 0 , 1 } , g : { 0 , 1 } n → { 0 , 1 } . The composition f ⋄ g : { 0 , 1 } m × n → { 0 , 1 } is X m n Irit Dinur, Or Meir Toward the KRW conjecture
Composition [KRW91]: we need to understand composition. Let f : { 0 , 1 } m → { 0 , 1 } , g : { 0 , 1 } n → { 0 , 1 } . The composition f ⋄ g : { 0 , 1 } m × n → { 0 , 1 } is g g X a m .......... g n Irit Dinur, Or Meir Toward the KRW conjecture
Composition [KRW91]: we need to understand composition. Let f : { 0 , 1 } m → { 0 , 1 } , g : { 0 , 1 } n → { 0 , 1 } . The composition f ⋄ g : { 0 , 1 } m × n → { 0 , 1 } is g g f X a m (f g)(X) .......... g n Irit Dinur, Or Meir Toward the KRW conjecture
The KRW conjecture f g L (f) g f X m a (f g)(X) .......... g g ......... L (g) L (g) g ......... X 1 X m n ‘ Clearly, L ( f ⋄ g ) ≤ L ( f ) · L ( g ) . Irit Dinur, Or Meir Toward the KRW conjecture
The KRW conjecture f g L (f) g f X m a (f g)(X) .......... g g ......... L (g) L (g) g ......... X 1 X m n ‘ Clearly, L ( f ⋄ g ) ≤ L ( f ) · L ( g ) . KRW conjecture*: ∀ f, g : L ( f ⋄ g ) ≈ L ( f ) · L ( g ) . Irit Dinur, Or Meir Toward the KRW conjecture
The KRW conjecture — Prior Work KRW conjecture*: ∀ f, g : L ( f ⋄ g ) ≈ L ( f ) · L ( g ) . Implies super-polynomial formula lower bounds. Irit Dinur, Or Meir Toward the KRW conjecture
The KRW conjecture — Prior Work KRW conjecture*: ∀ f, g : L ( f ⋄ g ) ≈ L ( f ) · L ( g ) . Implies super-polynomial formula lower bounds. [KRW91] defined the universal relation U . Like a function, but simpler. Suggested to prove the conjecture for U ⋄ U . Irit Dinur, Or Meir Toward the KRW conjecture
The KRW conjecture — Prior Work KRW conjecture*: ∀ f, g : L ( f ⋄ g ) ≈ L ( f ) · L ( g ) . Implies super-polynomial formula lower bounds. [KRW91] defined the universal relation U . Like a function, but simpler. Suggested to prove the conjecture for U ⋄ U . Conjecture was proved for U ⋄ U by [EIRS91] Alternative proof by [HW93]. Recently: [GMWW14] proved conjecture for f ⋄ U . Irit Dinur, Or Meir Toward the KRW conjecture
Our results We prove conjecture for f ⋄ g where g is parity: O ( √ m ) ˜ L ( f ⋄ g ) ≥ L ( f ) · L ( g ) / 2 Irit Dinur, Or Meir Toward the KRW conjecture
Our results We prove conjecture for f ⋄ g where g is parity: O ( √ m ) ˜ L ( f ⋄ g ) ≥ L ( f ) · L ( g ) / 2 Also, we give a structural result: Irit Dinur, Or Meir Toward the KRW conjecture
Our results We prove conjecture for f ⋄ g where g is parity: O ( √ m ) ˜ L ( f ⋄ g ) ≥ L ( f ) · L ( g ) / 2 Also, we give a structural result: KRW conjecture: naive formula is optimal. Irit Dinur, Or Meir Toward the KRW conjecture
Our results We prove conjecture for f ⋄ g where g is parity: O ( √ m ) ˜ L ( f ⋄ g ) ≥ L ( f ) · L ( g ) / 2 Also, we give a structural result: KRW conjecture: naive formula is optimal. Our result: naive formula is essentially the only optimal formula. Irit Dinur, Or Meir Toward the KRW conjecture
Our results We prove conjecture for f ⋄ g where g is parity: O ( √ m ) ˜ L ( f ⋄ g ) ≥ L ( f ) · L ( g ) / 2 Also, we give a structural result: KRW conjecture: naive formula is optimal. Our result: naive formula is essentially the only optimal formula. Actually, the lower bound for f ⋄ g already follows from [H98]. However, our proof is very different, and seems to be more generalizable for other g ’s. Irit Dinur, Or Meir Toward the KRW conjecture
Our results We prove conjecture for f ⋄ g where g is parity: O ( √ m ) ˜ L ( f ⋄ g ) ≥ L ( f ) · L ( g ) / 2 Also, we give a structural result: KRW conjecture: naive formula is optimal. Our result: naive formula is essentially the only optimal formula. Actually, the lower bound for f ⋄ g already follows from [H98]. However, our proof is very different, and seems to be more generalizable for other g ’s. Also: new proof of the state-of-the-art cubic lower bound of [H98]. Irit Dinur, Or Meir Toward the KRW conjecture
Introduction 1 Background 2 Proof Strategy 3 New tools 4 Irit Dinur, Or Meir Toward the KRW conjecture
Outline Introduction 1 Background 2 Proof Strategy 3 New tools 4 Irit Dinur, Or Meir Toward the KRW conjecture
Karchmer-Wigderson Relations [KW90] Relate L ( f ) to complexity of a communication problem KW f . Irit Dinur, Or Meir Toward the KRW conjecture
Karchmer-Wigderson Relations [KW90] Relate L ( f ) to complexity of a communication problem KW f . The KW relation KW f is defined as follows: Alice gets x ∈ f − 1 (0) . Bob gets y ∈ f − 1 (1) . Irit Dinur, Or Meir Toward the KRW conjecture
Karchmer-Wigderson Relations [KW90] Relate L ( f ) to complexity of a communication problem KW f . The KW relation KW f is defined as follows: Alice gets x ∈ f − 1 (0) . Bob gets y ∈ f − 1 (1) . Clearly, x � = y , so ∃ i s.t. x i � = y i . Want to find such i . Irit Dinur, Or Meir Toward the KRW conjecture
Karchmer-Wigderson Relations [KW90] Relate L ( f ) to complexity of a communication problem KW f . The KW relation KW f is defined as follows: Alice gets x ∈ f − 1 (0) . Bob gets y ∈ f − 1 (1) . Clearly, x � = y , so ∃ i s.t. x i � = y i . Want to find such i . Want to talk as little as possible. Irit Dinur, Or Meir Toward the KRW conjecture
Karchmer-Wigderson Relations [KW90] Relate L ( f ) to complexity of a communication problem KW f . The KW relation KW f is defined as follows: Alice gets x ∈ f − 1 (0) . Bob gets y ∈ f − 1 (1) . Clearly, x � = y , so ∃ i s.t. x i � = y i . Want to find such i . Want to talk as little as possible. Only deterministic protocols! Irit Dinur, Or Meir Toward the KRW conjecture
Karchmer-Wigderson Relations [KW90] Relate L ( f ) to complexity of a communication problem KW f . The KW relation KW f is defined as follows: Alice gets x ∈ f − 1 (0) . Bob gets y ∈ f − 1 (1) . Clearly, x � = y , so ∃ i s.t. x i � = y i . Want to find such i . Want to talk as little as possible. Only deterministic protocols! This talk: Assume C ( KW f ) = log L ( f ) . Irit Dinur, Or Meir Toward the KRW conjecture
Karchmer-Wigderson Relations [KW90] Relate L ( f ) to complexity of a communication problem KW f . The KW relation KW f is defined as follows: Alice gets x ∈ f − 1 (0) . Bob gets y ∈ f − 1 (1) . Clearly, x � = y , so ∃ i s.t. x i � = y i . Want to find such i . Want to talk as little as possible. Only deterministic protocols! This talk: Assume C ( KW f ) = log L ( f ) . KRW conjecture: C ( KW f ⋄ g ) ≈ C ( KW f ) + C ( KW g ) . Irit Dinur, Or Meir Toward the KRW conjecture
KRW and KW Can we use KW relations to attack the KRW conjecture? How does KW f ⋄ g look like? Recall: f ⋄ g maps { 0 , 1 } m × n to { 0 , 1 } . Bob Alice X Y m m n n Irit Dinur, Or Meir Toward the KRW conjecture
KRW and KW Can we use KW relations to attack the KRW conjecture? How does KW f ⋄ g look like? Recall: f ⋄ g maps { 0 , 1 } m × n to { 0 , 1 } . Bob Alice g g g g X Y m a b m .......... .......... g g n n Irit Dinur, Or Meir Toward the KRW conjecture
KRW and KW Can we use KW relations to attack the KRW conjecture? How does KW f ⋄ g look like? Recall: f ⋄ g maps { 0 , 1 } m × n to { 0 , 1 } . Bob Alice g g g g f f X Y m a b m 1 .......... 0 .......... g g n n Irit Dinur, Or Meir Toward the KRW conjecture
KRW and KW Can we use KW relations to attack the KRW conjecture? How does KW f ⋄ g look like? Recall: f ⋄ g maps { 0 , 1 } m × n to { 0 , 1 } . Bob Alice g g g g f f X Y m a b m 1 .......... 0 .......... g g n n Irit Dinur, Or Meir Toward the KRW conjecture
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