all $(E, \gamma, \phi) \in \mathcal{A}$ it holds that $S \otimes E \models \gamma$.

Note that in Definition \[def:adaptation-domain-opt\] we only augmented the data structure for adaptation domains but did not actually alter the condition to check for the fulfillment of an adaptation domain. This means that for an adaptation domain $\mathcal{A}$, a system needs to fulfill all goals in $\mathcal{A}$ but is not actually tested on the fitness defined by $\phi$. We could define a fitness threshold $f$ we require a system $S$ to surpass in order to adapt to $\mathcal{A}$ in the formalism. But such a check, written $f \preceq \phi(S \otimes E)$, could already be included in the Boolean goals if we use a logic that is expressive enough.

Instead, we want to use the fitness function as soft constraints: We expect the system to perform as well as possible on this metric, but we do not (always) require a minimum level of performance. However, we can use fitness to define a fitness preorder on systems:

\[def:optimization\] Given two systems $S$ and $S'$ as well as an adaptation space $\mathcal{A}$, $S'$ is at least as optimal as $S$, written $S \preceq_\mathcal{A} S'$, iff for all $(E, \gamma, 