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 $n$.
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 >>>
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 >>>
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 >>>
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 >>>
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 >>>
An arithmetic formula is an arithmetic circuit where each gate has fan-out one. An \emph{arithmetic read-once formula} (ROF in short) is an arithmetic formula where each input variable labels at most one leaf. In this paper we present several efficient blackbox \emph{polynomial identity testing} (PIT) algorithms for some classes of ... more >>>
In this work, we initiate the study of the space complexity of the Polynomial Identity Testing problem (PIT).
First, we observe that the majority of the existing (time-)efficient ``blackbox'' PIT algorithms already give rise to space-efficient ``whitebox'' algorithms for the respective classes of arithmetic formulas via a space-efficient ...
more >>>
