Multilinear NC1  Multilinear NC2
Ran Raz
Weizmann Institute
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)
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)
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 ?
Multilinear Circuits:
[NW]:
Every gate in the circuit computes
a multilinear polynomial
Example: (X1 ¢ X2) + (X2 ¢ X3)
(no high powers of variables)
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]
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
Our Result:
Explicit f(X1,...,Xn), with coeff.
in {0,1}, s.t., over any field:
1) 9 poly-size NC2 multilinear
circuit for f
2) Any multilinear formula for f is
of size
multilinear NC1  multilinear NC2
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
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 of pq in f
Example:
f(y1,y2,z1,z2) = 1 + y1y2 - y1z1z2
Mf
=
1 0
0
0
0
0 -1
y1
0
0
0
0
y2
1 0
0
0
y1y2
1 z1 z2
0
z1 z2
1
Partial Derivatives Method [N,NW]
[Nis]: If f is 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 ?
Counter Example:
Mf is a permutation matrix
Rank(Mf) = 2m
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 partition Rank(Mf)=2m
then any multilinear formula for f
is of super-poly-size (
)
High-Rank Polynomials:
Define: f(X1,..,X2m) is High-Rank
if for every partition Rank(Mf)=2m
f is High-Rank ! any multilinear
formula for f is of super-poly-size
Our Result: Step 1
Explicit f(X1,..,X2m) over C, s.t.:
1) 9 poly-size NC2 multilinear
circuit for f
2) f is High-Rank
(coefficients different than 0,1)
(We use algebraicly independent
constants from C)
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) 9 poly-size NC2 multilinear
circuit for f
2) a1,..,ar algeb. independent !
f(X1,..,X2m,a1,..,ar) is High-Rank
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 over G
lower bound over F
The End