$S'$, $S'$ is at least as adaptive as $S$, written $S \sqsubseteq S'$ iff for all adaptation spaces $\mathcal{A} \in \mathfrak{A}$ it holds that $S \Vdash \mathcal{A} \Longrightarrow S' \Vdash \mathcal{A}$.

Both Definitions \[def:adaptation-domain\] and \[def:adaptation-space\] can be augmented to include soft constraints or optimization goals. This means that in addition to checking against boolean goal satisfaction, we can also assign each system $S$ interacting with an environment $E$ a *fitness* $\phi(S \otimes E) \in F$, where $F$ is the type of fitness values. We assume that there exists a preorder $\preceq$ on $F$, which we can use to compare two fitness values. We can then generalize Definition \[def:adaptation-domain\] and \[def:adaptation-space\] to respect these optimization goals.

\[def:adaptation-domain-opt\] Let $S$ be a system. Let $\mathcal{E}$ be a set of environments that can be combined with $S$ using a combination operator $\otimes$. Let $\Gamma$ be a set of Boolean goals. Let $F$ be a set of fitness values and $\preceq$ be a preorder on $F$. Let $\Phi$ be a a set of fitness functions with codomain $F$. An *adaptation domain* $\mathcal{A}$ is a set $\mathcal{A} \subseteq \mathcal{E} \times \Gamma \times \Phi$. $S$ can adapt to $\mathcal{A}$, written $S \Vdash \mathcal{A}$ iff for 