In this paper we explore the noncommutative analogues, $\mathrm{VP}_{nc}$ and
$\mathrm{VNP}_{nc}$, of Valiant's algebraic complexity classes and show some
striking connections to classical formal language theory. Our main
results are the following:
(1) We show that Dyck polynomials (defined from the Dyck languages of formal language theory) are complete for ... more >>>
Let $U_{k,N}$ denote the Boolean function which takes as input $k$ strings of $N$ bits each, representing $k$ numbers $a^{(1)},\dots,a^{(k)}$ in $\{0,1,\dots,2^{N}-1\}$, and outputs 1 if and only if $a^{(1)} + \cdots + a^{(k)} \geq 2^N.$ Let THR$_{t,n}$ denote a monotone unweighted threshold gate, i.e., the Boolean function which takes ... more >>>
We show that for any coprime $m,r$ there is a circuit of the form $\text{MOD}_m\circ \text{AND}_{d(n)}$ whose correlation with $\text{MOD}_r$ is at least $2^{-O\left( \frac{n}{d(n)} \right) }$. This is the first correlation lower bound for arbitrary $m,r$, whereas previously lower bounds were known for prime $m$. Our motivation is the ... more >>>