  The gossip problem, in which information (known as secrets) must be shared
among a certain number of agents using the minimum number of calls, is of
interest in the conception of communication networks and protocols. We extend
the gossip problem to arbitrary epistemic depths. For example, we may require
not only that all agents know all secrets but also that all agents know that
all agents know all secrets. We give optimal protocols for various versions of
the generalised gossip problem, depending on the graph of communication links,
in the case of two-way communications, one-way communications and parallel
communication. We also study different variants which allow us to impose
negative goals such as that certain agents must not know certain secrets. We
show that in the presence of negative goals testing the existence of a
successful protocol is NP-complete whereas this is always polynomial-time in
the case of purely positive goals.
