Algorithms and lower bounds for de-Morgan formulas of low- communication leaf gates Sajin Koroth (Simon Fraser University) Joint with Valentine Zhenjian Dimitrios Igor Carboni Kabanets Lu Myrisiotis Oliveira
Outline • Background • Circuit model : Formula [ s ] ∘ • Prior work • Results • Lower bounds • PRG’s • SAT algorithm’s • Learning algorithms • Overview of the lower bound technique
Parallel vs Sequential computation • Most of linear algebra can be done in parallel • Gaussian elimination is an outlier • Intuitively its an inherently sequential procedure • There are theoretical reasons to believe so • There is an e ffi cient sequential algorithm
P vs NC 1 Class P of poly-time solvable problems Are there problems with e ffi cient sequential algorithms which do not have e ffi cient parallel algorithms ? Modeled as circuits
Circuit complexity Internal gates • Complexity parameters : • Size : # of gates • Depth : length of the longest path from root to leaf • Fan in : 2, Fan out • Formulas : Leaf gates • Underlying DAG is a tree • No reuse of computation • Depth = log ( Size )
Circuit complexity Class = Poly-Size Formulas NC 1 F • E ffi cient parallel computation (formally CREW PRAM): size ( F ) = n O (1) • Polynomially many processors depth ( F ) = O (log n ) • Logarithmic computation time x 1 x 2 x n x 5 In formula, depth ( F ) = O (log size ( F ))
Circuit complexity P vs rephrased NC 1 • A Boolean function (candidates: Perfect matching, Gaussian elimination etc) f • That can be computed in poly-time ( ) f ∈ P • Any de-Morgan formula computing it has super-poly size ( f ∉ NC 1 )
P vs NC 1 State of the art • Andreev’87 : Ω ( n 2.5 − o (1) ) for a function in called the Andreev function P • Also, Andreev’87 : Ω ( n 1+ Γ− o (1) ) , where is the shrinkage exponent Γ • Paterson and Zwick’93 : Γ ≥ 1.63 • Hastad’98 (breakthrough) : Γ ≥ 2 − o (1) • Tal’14 : Γ = 2 Ω ( log 2 n log log n ) n 3 • Best l.b. for Andreev’s function (Tal’14) : Ω ( log n (log log n ) 2 ) n 3 • Best l.b. for a function in (Tal’16) : P
Cubic formula lower bounds Andreev’s function f ( , x 1 x 2 x 3 x n ⋯⋯ 2 log n = n Truth Table of a bit function ( ) log n h ) = y 1 y 2 y 3 y n ⋯⋯ h z log n z 2 z 1 ⊕ ⊕ ⊕ n log n y n − y n y y 2 n y 1 y 2 y 3 y ⋯⋯ ⋯⋯ ⋯⋯ n n n log n log n log n log n
Cubic formula lower bounds Hastad’s result Ω ( log 2 n log log n ) n 3 • (Tal’14) : • Doesn’t work if there are parity gates at bottom
Our Model Augmenting de-Morgan formulas • de-Morgan formulas : leaf gates, input literals • Our model : leaf gates, low communication functions Leaf gates of low cc Leaf gates
Our model Reformulation • Formula [ s ] ∘ • Size s de-Morgan formula • : A family of Boolean functions • Leaf gates are functions g ∈ • Our model : • - low communication complexity Boolean functions s = ˜ • O ( n 2 )
Communication complexity • Yao’s 2-party model m 1 m 2 • Input divided into 2 parts x , y m k y x f ( x , y ) • Goal : compute f ( x , y ) with minimal communication
Our model Complexity of Andreev’s function f ( , x 1 x 2 x 3 x n ⋯⋯ 2 log n = n Truth Table of a bit function ( ) log n h ) = y 1 y 2 y 3 y n ⋯⋯ n de-Morgan formula of size log n h Leaf gates z log n z 2 z 1 Communication complexity Of Parity = 2 bits ⊕ ⊕ ⊕ n log n y n − y n y y 2 n y 1 y 2 y 3 y ⋯⋯ ⋯⋯ ⋯⋯ n n n log n log n log n log n
Our model Prior work - Bipartite Formulas F • Input is divided into two parts, x , y • Every leaf can gate can access any Boolean function of either or but not x y both Communication complexity Of a bipartite function = 1 bit • Models a well known measure - graph complexity g 1 g 2 g 3 g s • Tal’16: Bipartite formula complexity of ˜ Ω ( n 2 ) is IP n x 1 x 2 x 3 x n y 1 y 2 y 3 y n ⋯⋯ ⋯⋯ • Earlier methods could not do super linear
Our model Connection to Hardness Magnification N = 2 n • : Given the truth table of a function on bits ( ) MCSP N [ k ] f n • Yes : if has a circuit of size at most f k • No : otherwise • Meta computational problem with connections to Crypto, learning theory, circuit complexity etc • OPS’19: • If there exists an such that MCSP N [2 o ( n ) ] Formula [ N 1+ ϵ ] ∘ XOR is not in ϵ • then, NP ∉ NC 1
Our model Connection to PRG for polytopes • Polytope : AND of LTF’s • LTF : sign ( w 1 x 1 + … + w n x n − θ ) • w 1 , …, w n , θ ∈ ℝ • Ex : 3 x 1 + 4 x 2 + 5 x 7 ≥ 12 • Nisan’94 : Randomized communication complexity O (log n ) • PRG’s for polytopes : Approximate volume computation
Our model Interesting low communication bottom gates • Bipartite functions • Parities • LTF’s (Linear threshold functions) • PTF’s (Polynomial threshold functions)
Our results Target function - Generalized inner product x 1 1 x 1 2 x 1 x 1 ⋯⋯ • Generalization of binary inner n 3 k product ∧ ∧ ∧ IP n ( x , y ) = ∑ x i y i x 2 2 x 2 x 2 1 x 2 ⋯⋯ • n 3 k i ∈ [ n ] n ( x 1 , x 2 , …, x k ) = ∑ ∧ ∧ ∧ i ∈ [ n / k ] ∏ x j GIP k • i j ∈ [ k ] x k 2 x k x k 1 x k ⋯⋯ n 3 k ∧ ∧ ∧ z 1 z 2 z k ⊕ 0/1
Our results Lower bound • Let GIP k be computed on average by , F ∈ Formula [ s ] ∘ n x [ F ( x ) = GIP k • That is, Pr n ( x )] ≥ 1/2 + ϵ s = Ω ( ϵ /2 n 2 ( ) ⋅ log 2 (1/ ϵ ) ) n 2 • Then, k 2 ⋅ 16 k ⋅ R k ϵ /2 n 2 R k • : Randomized communication of with error in the ϵ /2 n 2 ( ) number on forehead communication complexity model
Our results MCSP lower bounds s = ˜ O ( n 2 ) • If MCSP N [2 cn ] is computed , then Formula [ s ] ∘ XOR • Contrast : OPS’19: MCSP N [2 o ( n ) ] • If there exists an such that is not in ϵ Formula [ N 1+ ϵ ] ∘ XOR • then, NP ∉ NC 1 MCSP N [2 o ( n ) ] • Our techniques cannot handle
Our results PRG • A pseudo random generator is said to fool a function class if G ℱ ϵ z ∈ {0,1} l ( n ) [ f ( G ( z )) = 1 ] − x ∈ {0,1} n [ f ( x ) = 1 ] Pr Pr ≤ ϵ • • is any function from f ℱ G : {0,1} l ( n ) → {0,1} n • • is the seed, z l ( n ) ⋘ n • Smaller the seed length compared to the better n
Our results PRG • Parities at the bottom can make things harder. • AC 0 best known PRG seed length poly (log n ) AC 0 ∘ XOR • best known only (1 − o (1)) n
Our results PRG • There is a PRG that -fools Formula [ s ] ∘ XOR ϵ • Seed length : O ( s ⋅ log s ⋅ log(1/ ϵ ) + log n ) • Seed length is optimal, unless lower bound can be improved
Our results PRG • Natural generalization to Formula [ s ] ∘ • There is a PRG that -fools Formula [ s ] ∘ ϵ s ⋅ ( R k − NIH • Seed length : n / k + O ( ( ) + log s ) ⋅ log(1/ ϵ ) + log k ) ⋅ log k ϵ /6 s • Number in hand
Our results PRG - Corollaries • (Ours + Vio15) : There is a PRG O ( n 1/2 ⋅ m 1/4 ⋅ log n ⋅ log( n / ϵ )) • Seed length : • {0,1} n -fools intersection of halfspaces over m ϵ • Our results beats earlier results when and m = O ( n ) ϵ ≤ 1/ n
Our results PRG - Corollaries • There is a PRG O ( n 1/2 ⋅ s 1/4 ⋅ log n ⋅ log( n / ϵ )) • Seed length : • -fools Formula [ s ] ∘ SYM ϵ • First of its kind • Blackbox counting algorithm (Whitebox due to CW19)
Our results SAT Algorithm • Given circuit class 𝒟 • Circuit SAT : Given , is there an , x C ( x ) = 1 C ∈ 𝒟 • #Circuit SAT : Given , how many , C ∈ 𝒟 x C ( x ) = 1
Our results SAT Algorithm • Randomized #SAT algorithm for Formula [ s ] ∘ • Running time 2 n − t 1/2 n for LTFs log n t = Ω • s ⋅ log 2 s ⋅ R 2 1/3 ( ) • First of its kind #SAT for unbounded depth Boolean circuits with PTF’s at the bottom
Our results Learning algorithm • There is PAC-learning algorithm • Learns Formula [ n 2 − γ ] ∘ XOR • Accuracy : , Confidence : ϵ δ • Time complexity : poly (2 n /log n ,1/ ϵ , log(1/ δ )) • Formula [ n 2 − γ ] 2 o ( n ) can be learned in [Rei11] • Crypto connection: • is assumed to compute PRFs (BIP+18) MOD 3 ∘ XOR • If true, Formula [ n 2.8 ] ∘ XOR 2 o ( n ) can’t be learned in time
Lower bound technique Outline • GIP k cannot even be weakly approximated by low communication n complexity functions • Weakness of : Size formula can be “approximated” by Formula [ s ] ∘ s degree polynomial s • GIP k is weakly approximated by a collection of leaf gates n
Lower bound technique Part I • GIP k cannot even be weakly approximated by low communication n complexity functions • In the number on forehead model • Protocol computes GIP k with error (uniform distribution) ϵ n n /4 k − log(1/(1 − 2 ϵ )) • Then commn.comp >
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