6.1 Existence of Monotone Circuits for the Majority Function

Proof: We will think of the random circuits we are going to construct as built exclusively of Maj3 gates, that is, gates with fan-in three and unbounded fan-out, which compute the majority function on their three inputs. Because each of these gates can be constructed from four binary AND, OR gates (exercise!), this simplification does not affect the asymptotic size and depth of the circuit. The circuit we construct consists of a full ternary tree of depth D = k log n, the nodes of which are all Maj3 gates, for some constant k that will be specified later. Each leaf of the tree is connected to three inputs. The randomness appears in the way inputs and leaves are connected: each leaf gate independently picks three inputs independently and u.a.r. (with replacement) from among all n inputs to the circuit. This completes the description of our random circuit C (see Figure 6.1). Notice that C always has size poly(n) and depth O(log n).