Circuit Depth Reductions (1811.04828v4)

Abstract: 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.