## Ideal and real Success Sets

The similarity-function presupposes a relationship between the actual working set as well as the ideal success set . 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 of the evaluation function as , otherwise the fitness values can give no 'hint' for an optimization.

For an evaluation function I have to assume elements of a working set . These elements must additionally be associated with some 'observable' = 'real' values in the world which can be mapped onto a set of a 'success criterion' assumed as . If we assume these success values as points in an n-dimensional space and the observable values as well, then we can compute some distance between some and some with . 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 which can be 'computed' in a finite time.

In the case of a computable success set we could then compute for every observable value a minimal distance such that the set of all distance sets as 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' from would be an empirical estimate for the ideal success set. The derived numbers should also obey an ordering. Thus is the set of all derived success values as an empirical estimate of .

'Not knowing the success set 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