On the constant-depth complexity of k-clique

We prove a lower bound of ω(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 ω(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 from 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 an Erdos-Renyi random graph with vertex set {1,...,n} and independent edge probabilities n^-α where α ≤ 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 ∪ K_A) asymptotically almost surely. These results resolve a long-standing open question in finite model theory (going back at least to Immerman in 1982). The <i>m-variable fragment of first-order logic</i>, denoted by FO^m, consists of the first-order sentences which involve at most m variables. Our results imply that the <i>bounded variable hierarchy</i> FO¹ ⊂ FO² ⊂ ... ⊂ FO^m ⊂ ... 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.