Summing up: GAs without Phenotype

After these first examples of biological evolution and modeling GAs a more general structure can be presented (cf. 4.15).

Figure 4.15: General GA-Structure

Thus a GA Structure $ GAS$ reads as follows:

$\displaystyle GAS$ $\displaystyle =$ $\displaystyle \langle \cal{E},\cal{P}, \varSigma,\varXi, \cal{G}, \gamma, o, \alpha, \nu\rangle$ (4.20)
$\displaystyle \gamma$ $\displaystyle :$ $\displaystyle \varSigma^{*} \times \cal{G} \times \cal{G} \longmapsto
\cal{G}$ (4.21)
$\displaystyle o$ $\displaystyle :$ $\displaystyle \cal{G} \longmapsto \cal{P}$ (4.22)
$\displaystyle \alpha$ $\displaystyle :$ $\displaystyle \varSigma \times \cal{P} \longmapsto \varXi$ (4.23)
$\displaystyle \nu$ $\displaystyle :$ $\displaystyle \varXi \times \cal{E} \longmapsto \varSigma$ (4.24)

A $ GAS$ structure includes the set of environments $ \cal{E}$, the set of phenotypes $ \cal{P}$, feedback-strings $ \varSigma^{*}$ as well as action-strings of the phenotypes, $ \varXi$, the set of genomes $ \cal{G}$,a genetic function $ \gamma$ mapping pairs of genomes with feedback $ \varSigma^{*}$ into modified genomes, an interpretation function $ o$ mapping genomes into phenotypes, a systems function $ \alpha$ of a phenotype mapping input strings into output strings, and finally a feedback function $ \nu$ of the environment communicating back the effects of the behavior of a phenotype in an environment.

The possible information space which can be represented by a binary encoded genome, is given by $ 2^{l}$ possible genomes of length $ l$. This genome information space $ G^{l}$ is finite and will be translated a real process called ontogenesis $ o$ (or simply 'growth') into the space of phenotypes which either are constants or functions. As 'constants' do the phenotypes represent the genome information space in a 1-to-1 manner. As (finite) 'functions' the phenotypes can potentially represent an infinite space.

If we take as given that the feedback $ \sigma \in \Sigma$ of a certain environment $ E$ shall help to find those working subsets $ G' \subseteq G^{l}$ of the genome information space, which represent the 'best' solution for a phenotype $ P$ in the environment $ E$, then we have to assume that a certain working subset $ G'$ has to be large enough to be able to cover all those states of the environment $ E$ which are important for a certain task $ \tau \in \varGamma$.

In the case of biological systems we know, that those genomes which have been in existence or are still in existence are not complete super systems in th sense that they are optimal for all tasks but they are finite systems which have some local optimum for a certain type of environment4.5. Because the terrestrial environment is constantly changing this changes have all the times caused a severe pressure onto the genomes. While all the times parts of the genomes have been extincted because they could not quickly enough adapt to a certain environment one has to state that the genetic principle of information encoding as such has meanwhile survived more than 3 billion yeras. This means that it is not a certain genome which survives but the genetic principle how to encode information as plan which can be modified.

From this follows that in nature the genetic algorithm has in a certain time interval $ (t,t')$ not to solve the 'complete' maximum but only a subset of of the global maximum, which will be called here an 'intermediate' maximum. For a concrete system this is a 'maximum', but with regard to the whole environment and a longer period of time this is only an intermediate goal which has to be surpassed in the future. Nevertheless simultaneously there is the general ability to continue the production of new genomes for new environments.

At every point of time the universe as a whole unit represents an information space which is far beyond everything what a single genome can handle. The only way to overcome the preprogrammed genetic limits of n individual system is the coupling of more than one genome in a population of genomes in cooperation. Such a cooperation requires communication and allows for additional artificial (technical) information systems supporting the genomic information space (printed information, electronic information, communication networks, etc.). Thus, it can be concluded, that the concept of GAs can - and probably will - be extended to the case of cooperating genomes4.6.

Gerd Doeben-Henisch 2013-01-14