On the Depth of Randomly Generated Circuits

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 xed fan-in 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 ef ln n 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. 1 The Problem and Motivation A circuit is a directed acyclic graph whose nodes are labeled from a given set: the nodes with in-degree 0 are called inputs and all the remaining nodes are gates. Circuits are a widely used model for both sequential and parallel computations by giving appropriate labels to nodes. For parallel computations, the size of a circuit { the number of its gates { corresponds to the cost of the structure, while its depth { the length of a longest path { corresponds to its computation time. Furthermore, circuits consist an excellent model for