A family of Boolean circuits $\{C_n\}_{n\geq 0}$ is called \emph{$\gamma(n)$-weakly uniform} if
there is a polynomial-time algorithm for deciding the direct-connection language of every $C_n$,
given \emph{advice} of size $\gamma(n)$. This is a relaxation of the usual notion of uniformity, which allows one
to interpolate between complete uniformity (when $\gamma(n)=0$) ... more >>>

We propose a new family of circuit-based sampling tasks, such that non-trivial algorithmic solutions to certain tasks from this family imply frontier uniform lower bounds such as ``NP is not in uniform ACC^0" and ``NP does not have uniform polynomial-size depth-two threshold circuits". Indeed, the most general versions of our ... more >>>

Many results in fine-grained complexity reveal intriguing consequences from solving various SAT problems even slightly faster than exhaustive search. We prove a ``self-improving'' (or ``bootstrapping'') theorem for Circuit-SAT, $\#$Circuit-SAT, and its fully-quantified version: solving one of these problems faster for ``large'' circuit sizes implies a significant speed-up for ``smaller'' circuit ... more >>>