Ran Raz

An arithmetic formula is multi-linear if the polynomial computed

by each of its sub-formulas is multi-linear. We prove that any

multi-linear arithmetic formula for the permanent or the

determinant of an $n \times n$ matrix is of size super-polynomial

in $n$.

Ran Raz

An arithmetic circuit or formula is multilinear if the polynomial

computed at each of its wires is multilinear.

We give an explicit example for a polynomial $f(x_1,...,x_n)$,

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 ...
more >>>

Yael Tauman Kalai, Ran Raz

An interactive-PCP (say, for the membership $x \in L$) is a

proof that can be verified by reading only one of its bits, with the

help of a very short interactive-proof.

We show that for membership in some languages $L$, there are

interactive-PCPs that are significantly shorter than the known

more >>>

Ankit Gupta, Neeraj Kayal, Youming Qiao

Informally stated, we present here a randomized algorithm that given blackbox access to the polynomial $f$ computed by an unknown/hidden arithmetic formula $\phi$ reconstructs, on the average, an equivalent or smaller formula $\hat{\phi}$ in time polynomial in the size of its output $\hat{\phi}$.

Specifically, we consider arithmetic formulas wherein the ... more >>>

Neeraj Kayal, Chandan Saha, Ramprasad Saptharishi

We consider arithmetic formulas consisting of alternating layers of addition $(+)$ and multiplication $(\times)$ gates such that the fanin of all the gates in any fixed layer is the same. Such a formula $\Phi$ which additionally has the property that its formal/syntactic degree is at most twice the (total) degree ... more >>>

Neeraj Kayal, Nutan Limaye, Chandan Saha, Srikanth Srinivasan

We show here a $2^{\Omega(\sqrt{d} \cdot \log N)}$ size lower bound for homogeneous depth four arithmetic formulas. That is, we give

an explicit family of polynomials of degree $d$ on $N$ variables (with $N = d^3$ in our case) with $0, 1$-coefficients such that

for any representation of ...
more >>>