  The best known size lower bounds against unrestricted circuits have remained
around $3n$ for several decades. Moreover, the only known technique for proving
lower bounds in this model, gate elimination, is inherently limited to proving
lower bounds of less than $5n$. In this work, we propose a non-gate-elimination
approach for obtaining circuit lower bounds, via certain depth-three lower
bounds. We prove that every (unbounded-depth) circuit of size $s$ can be
expressed as an OR of $2^{s/3.9}$ $16$-CNFs. For DeMorgan formulas, the best
known size lower bounds have been stuck at around $n^{3-o(1)}$ for decades.
Under a plausible hypothesis about probabilistic polynomials, we show that
$n^{4-\varepsilon}$-size DeMorgan formulas have
$2^{n^{1-\Omega(\varepsilon)}}$-size depth-3 circuits which are approximate
sums of $n^{1-\Omega(\varepsilon)}$-degree polynomials over ${\mathbb F}_2$.
While these structural results do not immediately lead to new lower bounds,
they do suggest new avenues of attack on these longstanding lower bound
problems.
  Our results complement the classical depth-$3$ reduction results of Valiant,
which show that logarithmic-depth circuits of linear size can be computed by an
OR of $2^{\varepsilon n}$ $n^{\delta}$-CNFs, and slightly stronger results for
series-parallel circuits. It is known that no purely graph-theoretic reduction
could yield interesting depth-3 circuits from circuits of super-logarithmic
depth. We overcome this limitation (for small-size circuits) by taking into
account both the graph-theoretic and functional properties of circuits and
formulas.
  We show that improvements of the following pseudorandom constructions imply
new circuit lower bounds: dispersers for varieties, correlation with constant
degree polynomials, matrix rigidity, and hardness for depth-$3$ circuits with
constant bottom fan-in.
