Some operations over Boolean functions are considered. It is shown that
the operation of the geometrical projection and the operation of the
monotone extension can increase the complexity of Boolean functions for
formulas in each finite basis, for switching networks, for branching
programs, and read-$k$-times ... more >>>

We prove that any monotone switching network solving directed connectivity on $N$ vertices must have size $N^{\Omega(\log N)}$

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We prove tight size bounds on monotone switching networks for the NP-complete problem of
$k$-clique, and for an explicit monotone problem by analyzing a pyramid structure of height $h$ for
the P-complete problem of generation. This gives alternative proofs of the separations of m-NC
from m-P and of m-NC$^i$ from ... more >>>

The two-player pebble game of Dymond–Tompa is identified as a barrier for existing techniques to save space or to speed up parallel algorithms for evaluation problems.

Many combinatorial lower bounds to study L versus NL and NC versus P under different restricted settings scale in the same way as the ... more >>>

An approximate computation of a Boolean function by a circuit or switching network is a computation which computes the function correctly on the majority of the inputs (rather than on all inputs). Besides being interesting in their own right, lower bounds for approximate computation have proved useful in many subareas ... more >>>

Monotone span programs are a linear-algebraic model of computation which were introduced by Karchmer and Wigderson in 1993. They are known to be equivalent to linear secret sharing schemes, and have various applications in complexity theory and cryptography. Lower bounds for monotone span programs have been difficult to obtain because ... more >>>

For a universal constant $\alpha > 0$, we prove size lower bounds of $2^{\alpha N}$ for computing an explicit monotone function in NP in the following models of computation: monotone formulas, monotone switching networks, monotone span programs, and monotone comparator circuits, where $N$ is the number of variables of the ... more >>>

A circuit $\mathcal{C}$ samples a distribution $\mathbf{X}$ with an error $\epsilon$ if the statistical distance between the output of $\mathcal{C}$ on the uniform input and $\mathbf{X}$ is $\epsilon$. We study the hardness of sampling a uniform distribution over the set of $n$-bit strings of Hamming weight $k$ denoted by $\mathbf{U}^n_k$ ... more >>>