Ideal and real Success Sets

The similarity-function $simil()$ presupposes a relationship between the actual working set $G'$ as well as the ideal success set $G^{+}$. But the question is, how such a relationship should look like. We have the additional postulate that there must exist some ordering between the elements of the image set $ran()$ of the evaluation function $ran(eval())$ as $\forall x,y \in ran(eval())( x\neq y \Rightarrow (x < y) \vee (y < x))$, otherwise the fitness values can give no 'hint' for an optimization.

For an evaluation function $eval()$ I have to assume elements $g'$ of a working set $G'$. These elements must additionally be associated with some 'observable' = 'real' values $v \in VAL$ in the world $W$ which can be mapped onto a set of a 'success criterion' assumed as $g^{+} \in G^{+}$. If we assume these success values as points in an n-dimensional space and the observable values $VAL$ as well, then we can compute some distance between some $v$ and some $g^{+}$ with $dist(v,g^{+})$. In this case it would be possible to enable a minimal ordering of the distance values if the set of success elements would not be completely randomly distributed. Thus there should exist some property of the set $G^{+}$ which can be 'computed' in a finite time.

In the case of a computable success set $G^{+}$ we could then compute for every observable value $v_{g'}$ a minimal distance $mdist(v_{g'},G^{+})$ such that the set of all distance sets as $ran(eval())$ could be ordered.

If the success set is not known in advance an observer can only count the number of the offspring. The number is then an 'indirect hint', a 'hidden index' for the success set. Thus the 'derived numbers' $g^{d}$ from $v_{g'}$ would be an empirical estimate for the ideal success set. The derived numbers should also obey an ordering. Thus $G^{+}_{v}$ is the set of all derived success values as an empirical estimate of $G^{+}$.

'Not knowing the success set $G^{+}$ in advance' characterizes the case of the 'real' world; otherwise we are in an 'ideal' world where the success set is a purely theoretical entity in a 'simulated (= virtual)' world.