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.

Gerd Doeben-Henisch 2013-01-14