A polynomial threshold function (PTF) is defined as the sign of a polynomial $p\colon\bool^n\to\mathbb{R}$. A PTF circuit is a Boolean circuit whose gates are PTFs. We study the problems of exact and (promise) approximate counting for PTF circuits of constant depth.

Satisfiability (#SAT). We give the first zero-error randomized algorithm ... more >>>

Kane and Williams proved average-case wire lower bounds at the $n^{5/2}/\mathrm{polylog}, n$ scale for an explicit function against depth-two linear-threshold circuits. We prove an almost-everywhere near-cubic wire lower bound for a language in $\mathrm{E}^{\mathrm{NP}}$. For every fixed $c>0$, there is one language $F_c$ and positive constants $b_{S,c}$ and $b_{T,c}$ such ... more >>>