Multilinear NC1 ? Multilinear NC2

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Multilinear NC1 ? Multilinear NC2

• Ran Raz
• Weizmann Institute

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• Arithmetic Circuits (and Formulas)
• Field F
• Variables X1,...,Xn
• Gates
• Every gate in the circuit computes
• a polynomial in FX1,...,Xn
• Example (X1 X1) (X2 1)

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• 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)

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• 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 ?

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• Multilinear Circuits
• NW
• Every gate in the circuit computes
• a multilinear polynomial
• Example (X1 X2) (X2 X3)
• (no high powers of variables)

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• 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

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• 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

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• 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
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• 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

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• 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

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• Example
• f(y1,y2,z1,z2) 1 y1y2 - y1z1z2
• Mf

1 0 0 0
0 0 0 -1
0 0 0 0
1 0 0 0
1
y1
y2
y1y2
1 z1 z2 z1z2
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• 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 ?

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• Counter Example
• Mf is a permutation matrix
• Rank(Mf) 2m

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• 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 ( )
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• 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
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• 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)

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• Our Result Step 2
• Explicit f(X1,..,X2m,X1,..,Xr), with
• coeff. in 0,1, and rpoly(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

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• 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

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The End