On the Constant-Depth Complexity of k-Clique

We prove a lower bound of omega(n(k/4)) on the size of constant-depth circuits solving the k-clique problem on n-vertex graphs (for every constant k). This improves a lower bound of omega(n(k/89d2)) due to Beame where d is the circuit depth. Our lower bound has the advantage that it does not depend on the constant d in the exponent of n, thus breaking the mold of the traditional size-depth tradeoff. Our k-clique lower bound derives front a stronger result of independent interest. Suppose f(n) : {0, 1}((n/2)) --> {0, 1} is a sequence of functions computed by constant-depth circuits of size O(n(t)). Let G be art Erdos-Renyi random graph with vertex set {1, ... , n} and independent edge probabilities n(-alpha) where alpha <= 1/2t-1. Let A be a uniform random k-element subset of {1,..., n} (where k is any constant independent of n) and let K-A denote the clique supported on A. We prove that f(n) (G) = f(n) (G boolean OR K-A) asymptotically almost sit rely. These results resolve a long-standing open question in finite model theory (going back at least to Immerman in 1982). The m-variable fragment of first-order logic, denoted by FOm, consists of the first-order sentences which involve at most m variables. Our results imply that the bounded variable hierarchy FO1 subset of FO2 subset of ... subset of FOm subset of ... is strict in terms of expressive power on finite ordered graphs. It was previously unknown that FO3 is less expressive than full first-order logic on finite ordered graphs.