Multilinear NC 1  Multilinear NC 2

1. Multilinear NC1 Multilinear NC2 Ran Raz Weizmann Institute

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

3. Classes of Arithmetic Circuits: • NC1:Size: poly(n)Degree: poly(n) • Depth: O(log n) • (poly-size formulas) • NC2:Size: poly(n)Degree: poly(n) • Depth: O(log2n) • P:Size: poly(n) Degree: poly(n)

4. Valiant Skyum Berkowitz Rackoff: • Arithmetic NC2 =Arithmetic P • [H]: poly-size arithmetic circuit ! • quasipoly-size arithmetic formula • Outstanding open problem: • Arithmetic NC1Arithmetic NC2 • Are arithmetic formulas weaker • than arithmetic circuits?

5. Multilinear Circuits: • [NW]: • Every gate in the circuit computes • a multilinear polynomial • Example:(X1¢ X2) + (X2¢ X3) • (no high powers of variables)

6. Motivation: • 1) For many functions, non-multilinear circuits are very counter-intuitive • 2) For many functions, most (or all) known circuits are multilinear • 3) Multilinear polynomials: interesting subclass of polynomials • 4)Multilinear circuits: strong subclass of circuits (contains other classes) • 5) Relations to quantum circuits[Aaronson]

7. Previous Work : • [NW 95]:Lower bounds for a subclass of constant depth multilinear circuits • [Nis, NW, RS]: Lower bounds for other subclasses of multilinear circuits • [R 04]: Multilinear formulas for Determinant and Permanent are of size • [Aar 04]: Lower bounds for multilinear formulas for other functions

8. Our Result: • Explicitf(X1,...,Xn), with coeff. • in{0,1},s.t., over any field: • 1) 9poly-size NC2 multilinear circuit forf • 2) Any multilinear formula forfis of size multilinear NC1 multilinear NC2

9. Partial Derivatives Matrix [Nis]: • f=a multilinear polynomial over • {y1,...,ym} [ {z1,...,zm} • P=set of multilinear monomials in • {y1,...,ym}. |P|=2m • Q=set of multilinear monomials in • {z1,...,zm}. |Q|=2m

10. Partial Derivatives Matrix [Nis]: • f=a multilinear polynomial over • {y1,...,ym} [ {z1,...,zm} • P=set of multilinear monomials in • {y1,...,ym}. |P|=2m • Q=set of multilinear monomials in • {z1,...,zm}. |Q|=2m • M = Mf = 2m dimensional matrix: • For every p 2 P, q 2 Q, • Mf(p,q)= coefficient ofpqinf

11. Example: • f(y1,y2,z1,z2)=1+y1y2-y1z1z2 • Mf=

12. Partial Derivatives Method [N,NW] • [Nis]: Iffis computed by a noncommutative formula of size s then Rank(Mf)= poly(s) • [NW,RS]:The same for other classes of formulas • Is the same true for multilinear formulas ?

13. Counter Example: • Mfis a permutation matrix • Rank(Mf) =2m

14. We Prove: • Partition(at random){X1,...,X2m} • ! {y1,...,ym} [ {z1,...,zm} • If f has poly-size multilinear • formula, then (w.h.p.): If for every partitionRank(Mf)=2m then any multilinear formula for f is of super-poly-size ( )

15. High-Rank Polynomials: • Define:f(X1,..,X2m)isHigh-Rank • iffor every partitionRank(Mf)=2m f is High-Rank !any multilinear formula for f is of super-poly-size

16. Our Result: Step 1 • Explicit f(X1,..,X2m)overC,s.t.: • 1)9poly-sizeNC2multilinearcircuit forf • 2)fis High-Rank • (coefficients different than 0,1) • (We use algebraicly independent • constants from C)

17. Our Result: Step 2 • Explicit f(X1,..,X2m,X’1,..,X’r), with • coeff. in{0,1},and r=poly(m),s.t. • (over any field) • 1)9poly-sizeNC2multilinearcircuit forf • 2)a1,..,aralgeb. independent !f(X1,..,X2m,a1,..,ar)is High-Rank

18. Our Result: Step 3 • If F is a finite field take F ½ G • of infinite transcendental dimension • (G contains an infinite number of • algeb. independent elements) • Step 2!lower bound overG • !lower bound overF

19. The End