Only Genotype

I begin with the set $G \subseteq \Sigma^{*}$ of all possible information strings (genes) over an alphabet $\Sigma$, the set $G' \subseteq G$ of a real set of information strings doing some 'work' in some 'world' $W \in WORLD$, and the set $G^{+} \subseteq G$ of the actual successful information strings. The set of genes is also called the Genotype.

Successful has to be understood in this context in relation to the presupposed world $W$; a behaving system $PG'$ based on information strings from $G$ will be 'successful' if it uses those information strings $G^{+}$ from $G$ which are labeled as 'successful'. In the case of a changing world $W$ does this set of 'successful' information strings continuously change! This means that one has to include time $T$ as a parameter in the description of these processes. Ideally we have a similarity $simil: G' \times G^{+} \longmapsto \{0,1\}$. The limits of $simil()$ are given by $simil(G',G^{+}) = 0$ if $G'\cap G^{+} = \emptyset$ and $simil(G',G^{+}) = 1$ if $G'\cap G^{+} = G'$.

While we can define the set $G^{+}$ easily in theory and use it for the computation of some fitness values, this is not possible in 'reality'. A real system working with a 'real' set of information strings $G'$ has no direct knowledge about $G^{+}$. It does it's 'job' for some time and then it eventually will 'die'. Only an external observer could 'observe', that the 'offspring' of some of the real information strings are more numerous than others. This means that the important information is only available on a meta-level which is beyond the possibilities of simple populations of information strings.

The 'behavior' of these sets presupposes as a minimal working framework operations like 'evaluation' $eval()$, 'reproduction' $recomb()$, 'crossover' $crossov()$ as well as 'mutation' $mutate()$ repeatedly applied to the set of real information strings $G'$ as well as some world $W$ as source of feedbacks $FIT$. This means we have to assume that there is an evaluation $eval()$ as a meta-function operating on the world $W$ and the population $G'$ with $FIT_{t+1}= eval(W_{t}, G'(t))$ as well as $G'_{t+1}= gacycle(G'_{t},FIT_{t})$ with $gacycle= recomb \otimes crossv \otimes mutate$.

This leads to a first network of concepts as follows:

$$G \subseteq \Sigma^{*}$$ (5.14)

$$G^{+} \subseteq G$$ (5.15)

$$G' \subseteq G$$ (5.16)

$$simil(G',G^{+}) = 0 if G'\cap G^{+} = \emptyset$$ (5.17)

$$simil(G',G^{+}) = 1 if G'\cap G^{+} = G'$$ (5.18)

$$FIT_{t+1} = eval(W_{t}, G'(t))$$ (5.19)

$$G'_{t+1} = gacycle(G'_{t},FIT_{t})$$ (5.20)

$$gacycle = recomb \otimes crossv \otimes mutate$$ (5.21)

This leaves still in many respects a 'conceptual gap'. One open question is to the relationship between the similarity-function $simil()$ and the evaluation-function $eval()$.