We show that computing the majority of $n$ copies of a boolean function $g$ has randomised query complexity $\mathrm{R}(\mathrm{Maj} \circ g^n) = \Theta(n\cdot \bar{\mathrm{R}}_{1/n}(g))$. In fact, we show that to obtain a similar result for any composed function $f\circ g^n$, it suffices to prove a sufficiently strong form of the ... more >>>

Proving circuit lower bounds has been an important but extremely hard problem for decades. Although one may show that almost every function $f:\mathbb{F}_2^n\to\mathbb{F}_2$ requires circuit of size $\Omega(2^n/n)$ by a simple counting argument, it remains unknown whether there is an explicit function (for example, a function in $NP$) not computable ... more >>>

We prove logarithmic depth lower bounds in Stabbing Planes for the classes of combinatorial principles known as the Pigeonhole principle and the Tseitin contradictions. The depth lower bounds are new, obtained by giving almost linear length lower bounds which do not depend on the bit-size of the inequalities and in ... more >>>