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

How to Fool People to Work

on Circuit Lower Bounds

Ran RazWeizmann Institute &

Microsoft Research

The Only Barrier for Proving Super-Poly Lower Bounds…

Why Super-Poly Lower Bounds Were Still not Proved ?

Maybe because not enough people are working on it…

The Secret Plan:

Fooling people to work on circuit lower bounds…

Coming up with innocent lookingclean and simple problems thatare seemingly unrelated to

provingcircuit lower bounds, and whose solution would imply strong

circuitlower bounds

Arithmetic Circuits:Field: FVariables: X1,...,Xn

Gates:

Every gate in the circuit computes

a polynomial in F[X1,...,Xn]

Example: (X1 ¢ X1) ¢ (X2 + 1)

The Holy Grail:

Super-polynomial lower bounds

for circuit or formula size

I will present two innocent looking

problems that imply such bounds

Elusive Functions and Lower

Bounds for Arithmetic

Circuits

Polynomial Mappings:

f = (f1,...,fm) : Cn ! Cm is a polynomial mapping of degree d iff1,...,fm are polynomials of (total)degree d

f is explicit if given a monomial M and index i, the coefficient of M infi can be computed in poly time [Val]

The Moments Curve:

f: C ! Cm

f(x) = (x,x2,x3,...,xm)

Fact: 8 affine subspace A ( Cm

8 :Cm-1 ! Cm of (total) degree 1,

The Exercise that Was Never Given:

Give an explicit f: C ! Cm s.t.:8 : Cm-1 ! Cm of degree 2,

We require: f of degree ·

[R08]: Any explicit f ) super-polynomial lower bounds for the permanent

Elusive Functions:

f: Cn ! Cm is (s,r)-elusive if8 : Cs ! Cm of degree r,

[R08]: explicit constructions ofelusive functions imply lower

boundsfor the size of arithmetic

circuits

Proof Idea:Consider : Cs ! Cm of degree r, that mapsa 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 functionAdd input variables of f as additionalinput variables

Lower Bounds for Depth-d Circuits:

[SS91], [R08]:

Lower bounds of n1+(1/d)

(using elusive functions)

Tensor-Rank and Lower

Bounds for Arithmetic

Formulas

Tensor-Rank:A: [n]r ! F is of rank 1 if9 a1,…,ar : [n] ! F s.t.A = a1 a2 … ar , that isA(i1,…,ir) = a1(i1) ¢¢¢ ar(ir)

Rank(A) = Min k s.t. A=A1+…+Ak

where A1,…,Ak are of rank 1

8 A: [n]r ! F Rank(A) · nr-1

(generalization of matrix rank)

Tensors and Polynomials:

Given A: [n]r ! F and n¢r variables

x1,1,…,xr,n define

Tensor-Rank and Arithmetic Circuits:

[Str73]: explicit A:[n]3!F of rank m ) explicit lower bound of (m) for arithmetic circuits (for fA)(may give lower bounds of up to (n2))(best known bound: (n))

[R09]: 8 r · logn/loglognexplicit A:[n]r!F of rank nr(1-o(1)) ) explicit super-poly lower bound for arithmetic formulas (for fA)

Depth-3 vs. General Formulas:

Tensor-rank corresponds to depth-3

set-multilinear formulas (for fA)Corollary: strong enough lower

boundsfor depth-3 formulas ) super-polylower bounds for general formulasFolklore: strong enough bounds for depth-4circuits ) exp bounds for general circuits[AV08]: any exp bound for depth-4circuits ) exp bound for general circuits

The Tensor-Product Approach [Str]:Given A1:[n1]r!F, A2:[n2]r!F

Define A = A1 A2 : [n1¢n2]r ! F by

A((i1,j1),…,(ir,jr)) = A1(i1,…,ir) ¢ A2(j1,…jr)

For r=2, Rank(A) = Rank(A1)¢Rank(A2)

Is Rank(A) > Rank(A1)¢Rank(A2)/no(1) (8r) ?

YES ) super-poly lower bounds for arithmetic formulas

The Tensor-Product Approach [Str]:Given A1:[n1]r!F, A2:[n2]r!F

Define A = A1 A2 : [n1¢n2]r ! F by

A((i1,j1),…,(ir,jr)) = A1(i1,…,ir) ¢ A2(j1,…jr)

For r=2, Rank(A) = Rank(A1)¢Rank(A2)

Is Rank(A) > Rank(A1)¢Rank(A2)/no(1) (8r) ?

YES ) super-poly lower bounds for arithmetic formulas

Proof: Let m=n1/r Take A1,…,Ar:[m]r!F of high rank

Let A = A1 A2 … Ar : [n]r ! F

How do we find A1,…,Ar of high rank ?

We fix their r¢n entries as inputs !

Main Steps of the Proof:1) New homogenization and multilinearization techniques2) Defining syntactic-rank of a formula (bounds the tensor-

rank)3) 8s we find the formula of size s with the largest syntactic-rank4) Compute the largest syntactic-

rank of a poly-size formula

Conclusions (of Step 1):For r · logn/loglogn1) super-poly lower bounds for homogenous formulas ) super-

polylower bounds for general

formulas 2) super-poly lower bounds for set-mult formulas ) super-polylower bounds for general

formulas

Homogenization:Given a formula C of size s for ahomogenous polynomial f of deg r give a homogenous formula D for

f[Str73]: D of size sO(log r)

(optimality conjectured in [NW95])

[R09]: D of size

(where d = product depth of C) If s=poly(n), and r · logn/loglognSize(D)=poly(n)

Thanks!