We present a top-down lower-bound method for depth-$4$ boolean circuits. In particular, we give a new proof of the well-known result that the parity function requires depth-$4$ circuits of size exponential in $n^{1/3}$. Our proof is an application of robust sunflowers and block unpredictability.

Random $\Delta$-CNF formulas are one of the few candidates that are expected to be hard to refute in any proof system. One of the frontiers in the direction of proving lower bounds on these formulas is the $k$-DNF Resolution proof system (aka $\mathrm{Res}(k)$). Assume we sample $m$ clauses over $n$ ... more >>>

We prove that a modification of Andreev's function is not computable by $(3 + \alpha - \varepsilon) \log{n}$ depth De Morgan formula with $(2\alpha - \varepsilon)\log{n}$ layers of AND gates at the top for any $1/5 > \alpha > 0$ and any constant $\varepsilon > 0$. In order to do ... more >>>

Restricted branching programs capture various complexity measures like space in Turing machines or length of proofs in proof systems. In this paper, we focus on the application in the proof complexity that was discovered by Lovasz et al. '95 who showed the equivalence between regular Resolution and read-once branching programs ... more >>>

We prove that there is a constant $K$ such that \emph{Tseitin} formulas for an undirected graph $G$ requires proofs of
size $2^{\mathrm{tw}(G)^{\Omega(1/d)}}$ in depth-$d$ Frege systems for $d<\frac{K \log n}{\log \log n}$, where $\tw(G)$ is the treewidth of $G$. This extends H{\aa}stad recent lower bound for the grid graph ... more >>>