Equality Alone Does not Simulate Randomness Marc Vinyals Tata - PowerPoint PPT Presentation

Equality Alone Does not Simulate Randomness Marc Vinyals Tata Institute of Fundamental Research Mumbai, India Joint work with Arkadev Chattopadhyay and Shachar Lovett 34th Computational Complexity Conference

Overview Proof Sketch Hierarchy Deterministic Communication P m 1 Alice Bob m 2 m 3 m 4 x f ( x , y ) ? y Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 1 / 14

Overview Proof Sketch Hierarchy Deterministic Communication P Alice Bob x 6 | x + y ? y Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 1 / 14

Overview Proof Sketch Hierarchy Deterministic Communication P x ( mod 2 ) Alice Bob x 6 | x + y ? y Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 1 / 14

Overview Proof Sketch Hierarchy Deterministic Communication P x ( mod 2 ) Alice Bob No / Maybe, y ( mod 3 ) x 6 | x + y ? y Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 1 / 14

Overview Proof Sketch Hierarchy Deterministic Communication P x ( mod 2 ) Alice Bob No / Maybe, y ( mod 3 ) Yes / No x 6 | x + y ? y Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 1 / 14

Overview Proof Sketch Hierarchy Deterministic Communication P Alice Bob x x = y ? y Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 1 / 14

Overview Proof Sketch Hierarchy Deterministic Communication P Alice Bob x Yes / No x x = y ? y ◮ Equality needs n + 1 bits. Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 1 / 14

Overview Proof Sketch Hierarchy Randomized Communication BPP Alice Bob x , r Pr r [ error ] < 1 / 3 y , r Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 2 / 14

Overview Proof Sketch Hierarchy Randomized Communication BPP Sample p among first Θ ( n ) primes Alice Bob x ( mod p ) Yes / No x , r x = y ? y , r Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 2 / 14

Overview Proof Sketch Hierarchy Randomized Communication BPP Alice Bob x , r y , r ◮ Can solve equality with O ( log n ) bits. Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 2 / 14

Overview Proof Sketch Hierarchy Randomized Communication BPP Alice Bob x , r y , r ◮ Can solve equality with O ( 1 ) bits. Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 2 / 14

Overview Proof Sketch Hierarchy Randomized Communication BPP Alice Bob x , r y , r ◮ Can solve equality with O ( 1 ) bits. ◮ Greater-than ◮ x ≥ y ? ◮ O ( log n ) bits. Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 2 / 14

Overview Proof Sketch Hierarchy Randomized Communication BPP Alice Bob x , r y , r ◮ Can solve equality with O ( 1 ) bits. ◮ Greater-than ◮ x ≥ y ? ◮ O ( log n ) bits. ◮ Small-set disjointness ◮ x ∩ y = � ?, promise | x | , | y | ≤ k ◮ O ( k ) bits. Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 2 / 14

Overview Proof Sketch Hierarchy Randomized Communication BPP Alice Bob x , r y , r ◮ Can solve equality with O ( 1 ) bits. ◮ Greater-than ◮ x ≥ y ? ◮ O ( log n ) bits. ◮ Small-set disjointness ◮ x ∩ y = � ?, promise | x | , | y | ≤ k ◮ O ( k ) bits. ◮ Hashing / Equality is enough to efficiently solve all of these. Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 2 / 14

Overview Proof Sketch Hierarchy Communication with EQ Oracle P EQ [Babai, Frankl, Simon ’86] Alice Oracle Bob = x , r y , r ◮ Send f ( x ) , g ( y ) to oracle ◮ Both parties see answer ◮ Cost number of calls Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 3 / 14

Overview Proof Sketch Hierarchy Communication with EQ Oracle P EQ [Babai, Frankl, Simon ’86] � 1 / n 2 π 2 / 6 Alice Oracle Bob = x , r y , r ◮ Send f ( x ) , g ( y ) to oracle ◮ Both parties see answer ◮ Cost number of calls Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 3 / 14

Overview Proof Sketch Hierarchy Communication with EQ Oracle P EQ [Babai, Frankl, Simon ’86] � 1 / n 2 π 2 / 6 Alice Oracle Bob 1 1 = x , r y , r ◮ Send f ( x ) , g ( y ) to oracle ◮ Both parties see answer ◮ Cost number of calls Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 3 / 14

Overview Proof Sketch Hierarchy Communication with EQ Oracle P EQ [Babai, Frankl, Simon ’86] � 1 / n 2 π 2 / 6 Alice Oracle Bob 1 1 = e π π e 0 0 x , r y , r ◮ Send f ( x ) , g ( y ) to oracle ◮ Both parties see answer ◮ Cost number of calls Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 3 / 14

Overview Proof Sketch Hierarchy BPP vs P EQ Question function, is P EQ cost ≃ BPP cost? For every Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 4 / 14

Overview Proof Sketch Hierarchy BPP vs P EQ Question function, is P EQ cost ≃ BPP cost? For every ◮ Known false for partial functions ◮ e.g. Maj ( x ⊕ y ) , promise x ⊕ y has either 2 n / 3 0s or 2 n / 3 1s. ◮ 2 -bit BPP protocol ◮ Sample i ∈ [ n ] ◮ Send x i ◮ Answer x i ⊕ y i ◮ P EQ cost Ω ( n ) [Papakonstantinou, Scheder, Song ’14] . Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 4 / 14

Overview Proof Sketch Hierarchy BPP vs P EQ Question For every total function, is P EQ cost ≃ BPP cost? ◮ Known false for partial functions ◮ e.g. Maj ( x ⊕ y ) , promise x ⊕ y has either 2 n / 3 0s or 2 n / 3 1s. Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 4 / 14

Overview Proof Sketch Hierarchy BPP vs P EQ Question For every total function, is P EQ cost ≃ BPP cost? ◮ Known false for partial functions ◮ e.g. Maj ( x ⊕ y ) , promise x ⊕ y has either 2 n / 3 0s or 2 n / 3 1s. Our result: No. Theorem There is a total function with BPP cost O ( log n ) and P EQ cost Ω ( n ) . Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 4 / 14

Overview Proof Sketch Hierarchy Integer Inner Product Parameters t small constant, n growing, N = 2 n / t − 1 Input t integers in [ − N , N ] Alice x = x 1 ,..., x t Bob y = y 1 ,..., y t � if x 1 y 1 + ··· + x t y t = 0 1 Output IIP ( x , y ) = � 〈 x , y 〉 = 0 � = 0 otherwise Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 5 / 14

Overview Proof Sketch Hierarchy Upper Bound t small constant, n growing, N = 2 n / t − 1 IIP ( x , y ) = � x 1 y 1 + ··· + x t y t = 0 � Protocol ◮ Sample p among first Θ ( n ) primes ◮ Send x 1 ( mod p ) ,..., x t ( mod p ) ◮ Answer 〈 x , y 〉 ≡ 0 ( mod p ) Cost t log p = O ( log n ) Correct with probability 3 / 4 Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 6 / 14

Overview Proof Sketch Hierarchy P GT Lower Bound Alice Oracle Bob ≥ x y ◮ Prove for P GT . ◮ Can simulate EQ with 2 calls to GT . Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 7 / 14

Overview Proof Sketch Hierarchy P GT Lower Bound Alice Oracle Bob ≥ x y ◮ Prove for P GT . ◮ Can simulate EQ with 2 calls to GT . ◮ Cannot use BPP techniques. Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 7 / 14

Overview Proof Sketch Hierarchy Rectangle Partitions Alice Bob 6 | x + y ? x y 0 6 2 8 4 10 1 7 3 9 5 11 ◮ Each bit splits inputs into 2 rectangles. 0 ◮ After c bits have 2 c rectangles. 6 2 8 4 10 1 7 3 9 5 11 Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 8 / 14

Overview Proof Sketch Hierarchy Rectangle Partitions x ( mod 2 ) Alice Bob 6 | x + y ? x y 0 6 2 8 4 10 1 7 3 9 5 11 ◮ Each bit splits inputs into 2 rectangles. 0 ◮ After c bits have 2 c rectangles. 6 2 8 4 10 1 7 3 9 5 11 Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 8 / 14

Overview Proof Sketch Hierarchy Rectangle Partitions x ( mod 2 ) Alice Bob No / Maybe, y ( mod 3 ) 6 | x + y ? x y 0 6 2 8 4 10 1 7 3 9 5 11 ◮ Each bit splits inputs into 2 rectangles. 0 ◮ After c bits have 2 c rectangles. 6 2 8 4 10 1 7 3 9 5 11 Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 8 / 14

Overview Proof Sketch Hierarchy Rectangle Partitions x ( mod 2 ) Alice Bob No / Maybe, y ( mod 3 ) Yes / No 6 | x + y ? x y 0 6 2 8 4 10 1 7 3 9 5 11 ◮ Each bit splits inputs into 2 rectangles. 0 ◮ After c bits have 2 c rectangles. 6 2 8 4 10 1 7 3 9 5 11 Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 8 / 14

Overview Proof Sketch Hierarchy Rectangle Partitions Alice Bob x y 0 1 2 3 4 5 6 7 8 9 10 11 ◮ Each bit splits inputs into 2 rectangles. 0 ◮ After c bits have 2 c rectangles. 1 2 ◮ Can show EQ requires 2 n rectangles. 3 4 5 6 7 8 9 10 11 Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 8 / 14

Overview Proof Sketch Hierarchy Triangle Partitions f ( x ) g ( y ) Alice Oracle Bob � f ( x ) ≥ g ( y ) � � f ( x ) ≥ g ( y ) � ≥ x y g ( y ) ◮ Each call splits inputs into 2 triangles. f ( x ) Marc Vinyals (TIFR) Equality Alone Does not Simulate Randomness 9 / 14