Abstract
We investigate monotone circuits with local oracles [K., 2016], i.e., circuits containing additional inputs y i =y i (x ⃗ ) that can perform unstructured computations on the input string x ⃗. Let μELEMENT OF[0,1] be the locality of the circuit, a parameter that bounds the combined strength of the oracle functions y i (x ⃗ ) , and U n,k ,V n,k SUBSET OF OR EQUAL TO {0,1} m be the set of k -cliques and the set of complete (k-1) -partite graphs, respectively (similarly to [Razborov, 1985]).
Our results can be informally stated as follows. 1. For an appropriate extension of depth-2 monotone circuits with local oracles, we show that the size of the smallest circuits separating U n,3 (triangles) and V n,3 (complete bipartite graphs) undergoes two phase transitions according to μ . 2.
For 5<=k(n)<=n 1/4 , arbitrary depth, and μ<=1/50 , we prove that the monotone circuit size complexity of separating the sets U n,k and V n,k is n Θ(k SQUARE ROOT ) , under a certain restrictive assumption on the local oracle gates. The second result, which concerns monotone circuits with restricted oracles, extends and provides a matching upper bound for the exponential lower bounds on the monotone circuit size complexity of k -clique obtained by Alon and Boppana (1987).