Elusive Functions, and Lower Bounds for Arithmetic Circuits

Elusive Functions, and Lower Bounds for Arithmetic Circuits Ran Raz Weizmann Institute

Arithmetic Circuits: • Field: C • Variables: X1,...,Xn • Gates: • Every gate in the circuit computes • a polynomial in C[X1,...,Xn] • Example:(X1¢ X1) ¢ (X2+ 1)

Size Lower Bounds: • [Strassen,Baur-Strassen]: • A lower bound of (n log n) for the • size of arithmetic circuits • Open Problem: • Better lower bounds • The Holy Grail: • Super-polynomial lower bounds • (say, for the permanent)

Our Main Results: • 1) A family of (seemingly unrelated) problems that imply lower bounds for arithmetic circuits • 2) Polynomial lower bounds for constant depth arithmetic circuits (for polynomials of constant degree)

Polynomial Mappings: • f = (f1,...,fm): Cn! Cmis a • polynomial mapping of degreedif • f1,...,fmare polynomials of (total) • degreed • f is explicit if given a monomial M • and index i, the coefficient of M in • fi 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 ! Cms.t.: • 8: Cm-1! Cm of degree2, • We require: f of degree · • Our result:Any explicitf • )super-polynomial lower bounds • for the permanent

Elusive Functions: • f: Cn! Cmis (s,r)-elusive if • 8: Cs! Cm of degreer, • Our Result: explicit constructions of • elusive functions imply lower bounds for • the size of arithmetic circuits

The Degree of f: • An (s,r)-elusive f:Cn!Cm of deg 2d • ) (s,r)-elusive g:Cnd!Cm of deg n¢d • Hence: • Enough to consider f of deg ·n

f:Cn!Cmis (s,r)-elusiveif 8:Cs!Cm of degreer, • (m=m(n),s=s(n),r=r(n)) • Result 1: • Explicit(s,r)-elusive f : Cn! Cm • with s ¸ m0.9,r=2,n · mo(1) • )super-polynomial lower bounds • for the permanent • (f is explicit if given a monomial M and index i, the • coefficient of M in fi is computed in time poly(n))

f:Cn!Cmis (s,r)-elusiveif 8:Cs!Cm of degreer, • (m=m(n),s=s(n),r=r(n)) • Result 2: • Explicit(s,r)-elusive f : Cn! Cm • with m=nr¸ s > poly(n), • )super-polynomial lower bounds • for the permanent • (f is explicit if given a monomial M and index i, the • coefficient of M in fi is computed in time poly(n))

f:Cn!Cmis (s,r)-elusiveif 8:Cs!Cm of degreer, • (m=m(n),s=s(n),r=r(n)) • Result 3: • Explicit(s,2r-1)-elusive f: Cn! Cm • with m=nr+1, • )lower bounds of • (f is explicit if given a monomial M and index i, the • coefficient of M in fi is computed in time poly(n))

Results for Known  : (example) • r=3, m=n3, s=n2.5, • Given : Cs! Cm of degree3, • Give an explicit f: Cn! Cms.t.: • Explicit f )A Lower bound of • (n1.25) • A win-win result

Sketch of Proof: Notation: • Fix r, (say, r= (1)) • m = number of monomials of degree r, over x1,...,xn • Cm = Cr[x1,...,xn] = homogenous polynomials of degree r • For a polynomial f 2 Cm, • Comp(f) = complexity of f

Sketch of Proof: Lemma: • 8 s, 9: Cs’! Cm of degree2r-1, • (with s’ ¼ s2) , s.t.: • 1)Comp(f) · s)f 2 Image() • 2)f 2 Image()) Comp(f) · s’ • Image()= polynomials that can be computed by small circuits. • Proving lower bounds = Finding points outside Image()

Sketch of Proof: The lower bound: • Assume f: Cn! Cm, s.t. • 8 z1,..,zn2 C, f(z1,..,zn) 2 Cr[x1,..,xn] • Let • h(z1,..,zn,x1,..,xn) = f(z1,..,zn)(x1,..,xn) • Comp(h) · s ) • 8 z1,..,znComp(f(z1,..,zn)) · s ) • 8 z1,..,znf(z1,..,zn) 2 Image() ) • Thus: Comp(h) > s!!

Lower Bounds for the Permanent: • If h is explicit and the lower bound • is super-polynomial then • lower bounds for h ) • lower bounds for the permanent

Lower Bounds for Depth-d Circuits: • 8 d, we give g: Cn! C of degree O(d) • (with coefficients in {0,1}), s.t., • Any depth d circuit for g is of size ¸ • n1+(1/d) • If d=O(1) then deg(g)=O(1),and size¸ • n1+(1) • Previously: (for g of degree O(1)), only • bounds of n¢ld(n) (slightly superlinear) • [Pud,RS]

The End

Elusive Functions, and Lower Bounds for Arithmetic Circuits