We investigate monotone circuits with local oracles [K., 2016], i.e.,
circuits containing additional inputs yi=yi(x) that can perform
unstructured computations on the input string x. Let μ∈[0,1] be
the locality of the circuit, a parameter that bounds the combined strength of
the oracle functions yi(x), and Un,k,Vn,k⊆{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 Un,3
(triangles) and Vn,3 (complete bipartite graphs) undergoes two phase
transitions according to μ.
2. For 5≤k(n)≤n1/4, arbitrary depth, and μ≤1/50, we
prove that the monotone circuit size complexity of separating the sets
Un,k and Vn,k is nΘ(k), 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).Comment: Updated acknowledgements and funding informatio