GAS Example: Numbers and Function

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

$$GAS1 = \langle E1,P1, \varSigma_{1},\varXi_{1}, G1, o_{1}, \alpha_{1}, \nu_{1}\rangle$$ (3.25)

$$G1 = \{1,0\}^{5}$$ (3.26)

$$P1 = \{x\vert x = \sum_{i=0}^{l}{q_{i} \times 2^{i}}\} \& q_{i} \in g_{i} \in G1$$ (3.27)

$$l = 5$$ (3.28)

$$\gamma_{1} = mutation \oplus crossover \oplus selection$$ (3.29)

$$o_{1} = \sum_{i=0}^{l}{q_{i} \times 2^{i}} \& q_{i} \in g_{i} \in G1$$ (3.30)

$$\alpha_{1} = \emptyset$$ (3.31)

$$\nu_{1} : P1 \longmapsto P1^2$$ (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.