GAS Example: Numbers and Function

Repeating the introductory example in the more general framework we can write


$\displaystyle GAS1$ $\displaystyle =$ $\displaystyle \langle E1,P1, \varSigma_{1},\varXi_{1}, G1, o_{1}, \alpha_{1},
\nu_{1}\rangle$ (3.25)
$\displaystyle G1$ $\displaystyle =$ $\displaystyle \{1,0\}^{5}$ (3.26)
$\displaystyle P1$ $\displaystyle =$ $\displaystyle \{x\vert x = \sum_{i=0}^{l}{q_{i} \times 2^{i}}\} \& q_{i} \in g_{i} \in G1$ (3.27)
$\displaystyle l$ $\displaystyle =$ $\displaystyle 5$ (3.28)
$\displaystyle \gamma_{1}$ $\displaystyle =$ $\displaystyle mutation \oplus crossover \oplus selection$ (3.29)
$\displaystyle o_{1}$ $\displaystyle =$ $\displaystyle \sum_{i=0}^{l}{q_{i} \times 2^{i}} \& q_{i} \in g_{i} \in G1$ (3.30)
$\displaystyle \alpha_{1}$ $\displaystyle =$ $\displaystyle \emptyset$ (3.31)
$\displaystyle \nu_{1}$ $\displaystyle :$ $\displaystyle P1 \longmapsto P1^2$ (3.32)

The $ GAS_{1}$ structure includes the environment set $ \cal{E}$$ _{1}$, which consist only of a feedback (fitness) function $ \nu_{1}$ of the environment feeding back the effects of the behavior of a phenotype in an environment. In case of $ GAS_{1}$ there is no real action string (see below). The fitness is computed directly on the basis of the phenotype without action (see below). The fiteness-function is given as $ y=x^2$ (which could easily replaced by any other kind of function $ y=f(x)$). This environmental fiteness function generates 'fitness' as a fitness-string $ \sigma \in \varSigma_{1}$ as a decimal number (which can be encoded as binary string).

The set of phenotypes $ \cal{P}$$ _{1}$ is a set of decimal numbers which whill be computed out of the genomes by the interpretation (growth-)function $ o_{1}$, mapping genomes into phenotypes by translating binary strings into decimal numbers (which could be done in different ways).

A system function $ \alpha_{1}$ as part of the phenotype mapping input strings into output strings is not available and therefore also no action-strings of the phenotypes, $ \varXi_{1}$.

The genetic function $ \gamma_{1}$ is realized by a concatenation of the functions $ selection$, $ crossover$, and $ mutation$. The fitness feedbacks encoded in the input strings $ \Sigma$ is implicitly given as input of these partial functions.

Gerd Doeben-Henisch 2012-03-31