We consider the problem of identifying a planted assignment given a random $k$-SAT formula consistent with the assignment. This problem exhibits a large algorithmic gap: while the planted solution can always be identified given a formula with $O(n\log n)$ clauses, there are distributions over clauses for which the best known ... more >>>

We develop a new method to prove communication lower bounds for composed functions of the form $f\circ g^n$ where $f$ is any boolean function on $n$ inputs and $g$ is a sufficiently ``hard'' two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of $f \circ ... more >>>

We investigate the space complexity of refuting $3$-CNFs in Resolution and algebraic systems. No lower bound for refuting any family of $3$-CNFs was previously known for the total space in resolution or for the monomial space in algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a ... more >>>