Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are necessary to prove a given theorem. In this work, we systematically explore the reverse mathematics of complexity lower bounds. We explore reversals in the setting of bounded arithmetic, with Cook's theory $\mathbf{PV}_1$ as the base ... more >>>

A search-to-decision reduction is a procedure that allows one to find a solution to a problem from the mere ability to decide when a solution exists. The existence of a search-to-decision reduction for time-bounded Kolmogorov complexity, i.e., the problem of checking if a string $x$ can be generated by a ... more >>>

The planted clique conjecture states that no polynomial-time algorithm can find a hidden clique of size $k \ll \sqrt{n}$ in an $n$-vertex Erd\H{o}s--R\'enyi random graph with a $k$-clique planted. In this paper, we prove the equivalence among many (in fact, \emph{most}) variants of planted clique conjectures, such as search ... more >>>