2 Monotone Functions and Monotone Circuits

In the last lecture we looked at lower bounds for constant-depth circuits, proving that PARITY cannot be computed by constant-depth circuits, i.e. PARITY / ∈ AC0. General circuit lower bounds for explicit functions are quite weak: the best we can prove after years of effort is that there is a function, which requires circuits of size 5n − o(n). In this lecture we will examine what happens if we place natural restrictions on a circuit. Namely, we will prove that detecting a clique in a graph requires superpolynomial circuits.