We show that for any integer $d>0$, depth $d$ Frege systems are NP-hard to automate. Namely, given a set $S$ of depth $d$ formulas, it is NP-hard to find a depth $d$ Frege refutation of $S$ in time polynomial in the size of the shortest such a refutation. This extends ... more >>>

A $d$-dimensional simplicial complex $X$ is said to support a direct product tester if any locally consistent function defined on its $k$-faces (where $k\ll d$) necessarily come from a function over its vertices. More precisely, a direct product tester has a distribution $\mu$ over pairs of $k$-faces $(A,A')$, and given ... more >>>

Given a family X of subsets of [n] and an ensemble of local functions $\{f_s:s\to\Sigma \;|\; s\in X\}$, an agreement test is a randomized property tester that is supposed to test whether there is some global function $G:[n]\to\Sigma$ such that $f_s=G|_s$ for many sets $s$. For example, the V-test chooses ... more >>>