Elusive Functions, and Lower Bounds for Arithmetic Circuits

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Elusive Functions, and Lower Bounds for
Arithmetic Circuits

• Ran Raz
• Weizmann Institute

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• Arithmetic Circuits
• Field C
• Variables X1,...,Xn
• Gates
• Every gate in the circuit computes
• a polynomial in CX1,...,Xn
• Example (X1 X1) (X2 1)

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• 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)

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• 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)

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• Polynomial Mappings
• f (f1,...,fm) Cn ! Cm is a
• polynomial mapping of degree d if
• f1,...,fm are polynomials of (total)
• degree d
• f is explicit if given a monomial M
• and index i, the coefficient of M in
• fi can be computed in poly time Val

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• 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,

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• The Exercise that Was Never Given
• Give an explicit f C ! Cm s.t.
• 8 ? Cm-1 ! Cm of degree 2,
• We require f of degree
• Our result Any explicit f
• ) super-polynomial lower bounds
• for the permanent

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

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• The Degree of f
• An (s,r)-elusive fCn!Cm of deg 2d
• ) (s,r)-elusive gCnd!Cm of deg nd
• Hence
• Enough to consider f of deg n

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• fCn!Cm is (s,r)-elusive if 8 ?Cs!Cm of degree
  r,
• (mm(n),ss(n),rr(n))
• Result 1
• Explicit (s,r)-elusive f Cn ! Cm
• with s m0.9, r2, 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))

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• fCn!Cm is (s,r)-elusive if 8 ?Cs!Cm of degree
  r,
• (mm(n),ss(n),rr(n))
• Result 2
• Explicit (s,r)-elusive f Cn ! Cm
• with mnr s gt 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))

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• fCn!Cm is (s,r)-elusive if 8 ?Cs!Cm of degree
  r,
• (mm(n),ss(n),rr(n))
• Result 3
• Explicit (s,2r-1)-elusive f Cn ! Cm
• with mnr1,
• ) 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))

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• Results for Known ? (example)
• r3, mn3, sn2.5,
• Given ? Cs ! Cm of degree 3,
• Give an explicit f Cn ! Cm s.t.
• Explicit f ) A Lower bound of
• ?(n1.25)
• A win-win result

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• Sketch of Proof Notation
• Fix r, (say, r ?(1))
• m number of monomials of degree r, over
  x1,...,xn
• Cm Crx1,...,xn homogenous polynomials of
  degree r
• For a polynomial f 2 Cm,
• Comp(f) complexity of f

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• Sketch of Proof Lemma
• 8 s, 9 ? Cs ! Cm of degree 2r-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(?)

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• Sketch of Proof The lower bound
• Assume f Cn ! Cm, s.t.
• 8 z1,..,zn 2 C, f(z1,..,zn) 2 Crx1,..,xn
• Let
• h(z1,..,zn,x1,..,xn) f(z1,..,zn)(x1,..,xn)
• Comp(h) s )
• 8 z1,..,zn Comp(f(z1,..,zn)) s )
• 8 z1,..,zn f(z1,..,zn) 2 Image(?) )
• Thus Comp(h) gt s !!

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• 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

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• 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 dO(1) then deg(g)O(1), and size
• n1?(1)
• Previously (for g of degree O(1)), only
• bounds of nld(n) (slightly superlinear)
• Pud,RS

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The End