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How to Fool People to Work on Circuit Lower Bounds Ran Raz Weizmann Institute & Microsoft Research

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The Only Barrier for Proving Super-Poly Lower Bounds…

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Why Super-Poly Lower Bounds Were Still not Proved ? Maybe because not enough people are working on it…

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The Secret Plan: Fooling people to work on circuit lower bounds… Coming up with innocent looking clean and simple problems that are seemingly unrelated to proving circuit lower bounds, and whose solution would imply strong circuit lower bounds

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Arithmetic Circuits: Field: F Variables: X 1,...,X n Gates: Every gate in the circuit computes a polynomial in F[X 1,...,X n ] Example: (X 1 ¢ X 1 ) ¢ (X 2 + 1)

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The Holy Grail: Super-polynomial lower bounds for circuit or formula size I will present two innocent looking problems that imply such bounds

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Elusive Functions and Lower Bounds for Arithmetic Circuits

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Polynomial Mappings: f = (f 1,...,f m ) : C n ! C m is a polynomial mapping of degree d if f 1,...,f m are polynomials of (total) degree d f is explicit if given a monomial M and index i, the coefficient of M in f i can be computed in poly time [Val]

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The Moments Curve: f: C ! C m f(x) = (x,x 2,x 3,...,x m ) Fact: 8 affine subspace A ( C m 8 :C m-1 ! C m of (total) degree 1,

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The Exercise that Was Never Given: Give an explicit f: C ! C m s.t.: 8 : C m-1 ! C m of degree 2, We require: f of degree · [R08]: Any explicit f ) super-polynomial lower bounds for the permanent

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Elusive Functions: f: C n ! C m is (s,r)-elusive if 8 : C s ! C m of degree r, [R08]: explicit constructions of elusive functions imply lower bounds for the size of arithmetic circuits

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Proof Idea: Consider : C s ! C m of degree r, that maps a circuit to the polynomial computed by it = polynomials that can be computed by small circuits. Proving lower bounds, Finding points outside Since f hits a hard function Add input variables of f as additional input variables

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Lower Bounds for Depth-d Circuits: [SS91], [R08]: Lower bounds of n 1+ (1/d) (using elusive functions)

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Tensor-Rank and Lower Bounds for Arithmetic Formulas

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Tensor-Rank: A: [n] r ! F is of rank 1 if 9 a 1,…,a r : [n] ! F s.t. A = a 1 a 2 … a r, that is A(i 1,…,i r ) = a 1 (i 1 ) ¢¢¢ a r (i r ) Rank(A) = Min k s.t. A=A 1 +…+A k where A 1,…,A k are of rank 1 8 A: [n] r ! F Rank(A) · n r-1 (generalization of matrix rank)

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Tensors and Polynomials: Given A: [n] r ! F and n ¢ r variables x 1,1,…,x r,n define

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Tensor-Rank and Arithmetic Circuits: [Str73]: explicit A:[n] 3 ! F of rank m ) explicit lower bound of (m) for arithmetic circuits (for f A ) (may give lower bounds of up to (n 2 )) (best known bound: (n)) [R09]: 8 r · logn/loglogn explicit A:[n] r ! F of rank n r(1-o(1)) ) explicit super-poly lower bound for arithmetic formulas (for f A )

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Depth-3 vs. General Formulas: Tensor-rank corresponds to depth-3 set-multilinear formulas (for f A ) Corollary : strong enough lower bounds for depth-3 formulas ) super-poly lower bounds for general formulas Folklore: strong enough bounds for depth-4 circuits ) exp bounds for general circuits [AV08]: any exp bound for depth-4 circuits ) exp bound for general circuits

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The Tensor-Product Approach [Str]: Given A 1 :[n 1 ] r ! F, A 2 :[n 2 ] r ! F Define A = A 1 A 2 : [n 1 ¢ n 2 ] r ! F by A((i 1,j 1 ),…,(i r,j r )) = A 1 (i 1,…,i r ) ¢ A 2 (j 1,…j r ) For r=2, Rank(A) = Rank(A 1 ) ¢ Rank(A 2 ) Is Rank(A) > Rank(A 1 ) ¢ Rank(A 2 )/n o(1) ( 8 r) ? YES ) super-poly lower bounds for arithmetic formulas

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The Tensor-Product Approach [Str]: Given A 1 :[n 1 ] r ! F, A 2 :[n 2 ] r ! F Define A = A 1 A 2 : [n 1 ¢ n 2 ] r ! F by A((i 1,j 1 ),…,(i r,j r )) = A 1 (i 1,…,i r ) ¢ A 2 (j 1,…j r ) For r=2, Rank(A) = Rank(A 1 ) ¢ Rank(A 2 ) Is Rank(A) > Rank(A 1 ) ¢ Rank(A 2 )/n o(1) ( 8 r) ? YES ) super-poly lower bounds for arithmetic formulas Proof: Let m=n 1/r Take A 1,…,A r :[m] r ! F of high rank Let A = A 1 A 2 … A r : [n] r ! F How do we find A 1,…,A r of high rank ? We fix their r ¢ n entries as inputs !

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Main Steps of the Proof: 1) New homogenization and multilinearization techniques 2) Defining syntactic-rank of a formula (bounds the tensor-rank) 3) 8 s we find the formula of size s with the largest syntactic-rank 4) Compute the largest syntactic- rank of a poly-size formula

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Conclusions (of Step 1): For r · logn/loglogn 1) super-poly lower bounds for homogenous formulas ) super-poly lower bounds for general formulas 2) super-poly lower bounds for set-mult formulas ) super-poly lower bounds for general formulas

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Homogenization: Given a formula C of size s for a homogenous polynomial f of deg r give a homogenous formula D for f [Str73]: D of size s O(log r) (optimality conjectured in [NW95]) [R09]: D of size (where d = product depth of C) If s=poly(n), and r · logn/loglogn Size(D)=poly(n)

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