Circuits with Medium Fan-In

Abstract:

We consider boolean circuits in which every gate may compute an arbitrary boolean function of k other gates, for a parameter k. We give an explicit function f:{0,1}^n to {0,1} that requires at least Omega(log^2 n) non-input gates when k = 2n/3. When the circuit is restricted to being layered and depth 2, we prove a lower bound of n^{Omega(1)} on the number of non-input gates. When the circuit is a formula with gates of fan-in k, we give a lower bound  Omega(n^2/k \log n) on the total number of gates. 

Our model is connected to some well known approaches to proving lower bounds in complexity theory. Optimal lower bounds for the Number-On-Forehead model in communication complexity, or for bounded depth circuits in AC_0, or extractors for varieties over small fields would imply strong lower bounds in our model. On the other hand, new lower bounds for our model would prove new time-space tradeoffs for branching programs and impossibility results for (fan-in 2) circuits with linear size and logarithmic depth. In particular, our lower bound gives a different proof for a known time-space tradeoff for oblivious branching programs.

Joint work with Pavel Hrubes.