Volume 2 (2006)
Article 6 pp. 121-135

Separation of Multilinear Circuit and Formula Size

Received: September 30, 2005

Revised: March 4, 2006

Published: May 2, 2006

Revised: March 4, 2006

Published: May 2, 2006

**Keywords:**algebraic complexity, arithmetic circuits, log-depth, multilinear polynomials, lower bounds, separation of complexity classes

**Categories:**complexity theory, lower bounds, circuits, arithmetic circuits, formulas, arithmetic formulas, complexity classes, separation of complexity classes, polynomials - multivariate

**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:

- $f$ can be computed by a polynomial-size multilinear circuit of depth $O(\log^2 n)$.
- Any multilinear formula for $f$ is of size $n^{\Omega(\log n)}$.