  Recent research in machine teaching has explored the instruction of any
concept expressed in a universal language. In this compositional context, new
experimental results have shown that there exist data teaching sets
surprisingly shorter than the concept description itself. However, there exists
a bound for those remarkable experimental findings through teaching size and
concept complexity that we further explore here. As concepts are rarely taught
in isolation we investigate the best configuration of concepts to teach a given
set of concepts, where those that have been acquired first can be reused for
the description of new ones. This new notion of conditional teaching size
uncovers new insights, such as the interposition phenomenon: certain prior
knowledge generates simpler compatible concepts that increase the teaching size
of the concept that we want to teach. This does not happen for conditional
Kolmogorov complexity. Furthermore, we provide an algorithm that constructs
optimal curricula based on interposition avoidance. This paper presents a
series of theoretical results, including their proofs, and some directions for
future work. New research possibilities in curriculum teaching in compositional
scenarios are now wide open to exploration.
