On the depth of randomly generated circuits

Abstract

This research is motivated by the Circuit Value Problem; this problem is well known to be inherently sequential. We consider Boolean circuits with descriptions length d that consist of gates with a fixed fanin f and a constant number of inputs. Assuming uniform distribution of descriptions, we show that such a circuit has expected depth O(log d). This improves on the best known result. More precisely, we prove for circuits of size n their depth is asymptotically eflnn with extremely high probability. Our proof uses the coupling technique to bound circuit depth from above and below by those of two alternative discrete time processes. We are able to establish the result by embedding the processes in suitable continuous time branching processes. As a simple consequence of our result we obtain that monotone CVP is in the class average NC.