be constructed, we do not further discuss their definitions in this paper. In [@holzl2011towards] a corresponding definition was already introduced for $\sqsubset$.

The formulation of the adaptation sequence used to prove self-adaptivity naturally implies some kind of temporal structure. So basing said structure around $\mathbb{N}$ implies a very simple, linear and discrete model of time. More complex temporal evolution of systems is also already touched upon in [@holzl2011towards]. As noted, there may be several ways to define such a temporal structure on systems. We refer to related and future work for a more intricate discussion on this matter.

So, non-trivial self-adaptation does imply some structure for any self-adaptive system $S$ of type $\mathcal{V} = (V_i)_{i \in I}$: Mainly, there needs to be a subset of the type $\mathcal{V}' \subseteq \mathcal{V}$ that is used to encode the whole relation behind $S$ so that the already improved instances can sufficiently be passed on to the general adaptation mechanism.

For a general adaptation mechanism (as we previously assumed to be part of a system) to be able to improve a system’s adaptivity, it needs to be able to access some representation of its goals and its fitness function. This provides a grey-box view 