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

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

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,

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 · Our result: Any explicit f ) super-polynomial lower bounds for the permanent

Elusive Functions: f: C n ! C m is (s,r)-elusive if 8  : C s ! C m of degree r, 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:C n ! C m of deg 2 d ) (s,r)-elusive g:C nd ! C m of deg n ¢ d Hence: Enough to consider f of deg · n

f:C n ! C m is (s,r)-elusive if 8  :C s ! C m of degree r, (m=m(n),s=s(n),r=r(n)) Result 1: Explicit (s,r)-elusive f : C n ! C m with s ¸ m 0.9, r=2, n · m o(1) ) super-polynomial lower bounds for the permanent (f is explicit if given a monomial M and index i, the coefficient of M in f i is computed in time poly(n))

f:C n ! C m is (s,r)-elusive if 8  :C s ! C m of degree r, (m=m(n),s=s(n),r=r(n)) Result 2: Explicit (s,r)-elusive f : C n ! C m with m=n r ¸ 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 f i is computed in time poly(n))

f:C n ! C m is (s,r)-elusive if 8  :C s ! C m of degree r, (m=m(n),s=s(n),r=r(n)) Result 3: Explicit (s,2r-1)-elusive f: C n ! C m with m=n r+1, ) lower bounds of (f is explicit if given a monomial M and index i, the coefficient of M in f i is computed in time poly(n))

Results for Known  : (example) r=3, m=n 3, s=n 2.5, Given  : C s ! C m of degree 3, Give an explicit f: C n ! C m s.t.: Explicit f ) A Lower bound of  (n 1.25 ) A win-win result

Sketch of Proof: Notation: Fix r, (say, r=  (1)) m = number of monomials of degree r, over x 1,...,x n C m = C r [x 1,...,x n ] = homogenous polynomials of degree r For a polynomial f 2 C m, Comp(f) = complexity of f

Sketch of Proof: Lemma: 8 s, 9  : C s’ ! C m of degree 2r-1, (with s’ ¼ s 2 ), 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: C n ! C m, s.t. 8 z 1,..,z n 2 C, f(z 1,..,z n ) 2 C r [x 1,..,x n ] Let h(z 1,..,z n,x 1,..,x n ) = f(z 1,..,z n )(x 1,..,x n ) Comp(h) · s ) 8 z 1,..,z n Comp(f(z 1,..,z n )) · s ) 8 z 1,..,z n f(z 1,..,z n ) 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: C n ! C of degree O(d) (with coefficients in {0,1}), s.t., Any depth d circuit for g is of size ¸ n 1+  (1/d) If d=O(1) then deg(g)=O(1), and size ¸ n 1+  (1) Previously: (for g of degree O(1)), only bounds of n ¢ d (n) (slightly superlinear) [Pud,RS]

The End