In this paper, we study the problem of computing the majority function by low-depth monotone circuits and a related problem of constructing low-depth sorting networks. We consider both the classical setting with elementary operations of arity $2$ and the generalized setting with operations of arity $k$, where $k$ is a ... more >>>

Consider the recently introduced notion of \emph{probabilistic
time-bounded Kolmogorov Complexity}, pK^t (Goldberg et al,
CCC'22), and let MpK^tP denote the language of pairs (x,k) such that pK^t(x) \leq k.
We show the equivalence of the following:
- MpK^{poly}P is (mildly) hard-on-average w.r.t. \emph{any} samplable
distribution $\D$;
- ... more >>>

One of the major open problems in complexity theory is to demonstrate an explicit function which requires super logarithmic depth, to tackle this problem Karchmer, Raz and Wigderson proposed the KRW conjecture about composition of two functions. While this conjecture seems out of our current reach, some relaxed conjectures are ... more >>>