There are various notions of balancing set families that appear in combinatorics and computer science. For example, a family of proper non-empty subsets $S_1,\ldots,S_k \subset [n]$ is balancing if for every subset $X \subset \{1,2,\ldots,n\}$ of size $n/2$, there is an $i \in [k]$ so that $|S_i \cap X| = ... more >>>

We investigate the computational power of an arbitrary distinguisher for (not necessarily computable) hitting set generators as well as the set of Kolmogorov-random strings. This work contributes to (at least) two lines of research. One line of research is the study of the limits of black-box reductions to some distributional ... more >>>

A major open problem in proof complexity is to prove super-polynomial lower bounds for AC^0[p]-Frege proofs. This system is the analog of AC^0[p], the class of bounded depth circuits with prime modular counting gates. Despite strong lower bounds for this class dating back thirty years (Razborov, '86 and Smolensky, '87), ... more >>>