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Volume 2 (2006) Article 6 pp. 121-135
Separation of Multilinear Circuit and Formula Size
by Ran Raz
Received: September 30, 2005
Revised: March 4, 2006
Published: May 2, 2006
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Keywords: algebraic complexity, arithmetic circuits, log-depth, multilinear polynomials, lower bounds, separation of complexity classes
ACM Classification: F.2.2, F.1.3, F.1.2, G.2.0
AMS Classification: 68Q15, 68Q17, 68Q25, 94C05

Abstract: [Plain Text Version]

An arithmetic circuit or formula is multilinear if the polynomial computed at each of its wires is multilinear. We give an explicit polynomial $f(x_1,\dots,x_n)$ with coefficients in $\{0,1\}$ such that over any field:

  1. $f$ can be computed by a polynomial-size multilinear circuit of depth $O(\log^2 n)$.
  2. Any multilinear formula for $f$ is of size $n^{\Omega(\log n)}$.
This gives a superpolynomial gap between multilinear circuit and formula size, and separates multilinear $NC_1$ circuits from multilinear $NC_2$ circuits.