A First Formal Model

From these general considerations we can try to abstract a formal structure which represents - hopefully - all the key properties which are necessary to describe the behavior of self-organizing systems as generally as possible in a way that all the interesting details of concrete adaptive biological systems can be included if needed.

Here a list of proposed assumptions - or postulates, axioms - which we think are important (cf. figure 2.3):

1. Environment: All systems $S_{i} \in Z$ are presupposed to be located in an environment $E$ which can change. These changes can be mapped onto a time-line $T$.

2. Systems: Every system $S_{i} \in Z$ is an input-output system with a system function $\gamma_{i} \in \Gamma$ mapping input $\sigma \in \Sigma^{*}$ into output $\xi \in \Xi^{*}$ as $\gamma_{i}: \Sigma^{*} \longmapsto \Xi^{*}$. Every system $S_{i}$ also possesses at least one plan $p_{i} \in \Pi$ of it's structure, which can be used for the formation of new plans.

3. Life-Time: A living system $S_{i} \in Z$ has a finite duration characterized by a start time $t_{i.0}$ ('birth') and an end time $t_{i.n}$ ('death'). The life-time of a living system is therefore given by the pair $(t_{i.0}, t_{i.n})$.

4. Mating-Function: From a certain point of time during their lifetime $t_{i.r}, t_{j.k}$ two living systems $S_{i}, S_{j}$ can produce a new plan $p_{ij} \in \Pi$ of a new system using their built-in plans $p_{i}, p_{j}$ by applying a mating function $\eta: \{p_{i},p_{j}\}\longmapsto \{p_{ij}\}$. Every new plan $p_{ij}$ is assumed to be in a storage unit $o_{ij} \in \Omega$ from which it can be delivered to built up a new system.

5. Growth-Function: The growth-function $\rho$ can take a new plan $p_{ij}$ from a storage unit $o_{ij}$ and can built up a new system $\rho: \{p_{ij}\} \longmapsto \{S_{ij}\}$. The time to grow is an initial part of the life-time of a system represented as $growth-time = \{(t_{ij.0}, t_{ij.g})\} \subseteq \{(t_{ij.0}, t_{ij.n})\}$ with $(0 < g \leq n$). During the growth-time of a system the system $S_{ij}$ is assumed to start functioning as far as the system-function $\gamma_{ij}$ can do anything.^2.2

6. Learning-Capacity: The system-function $\gamma_{i}$ of a living system $S_{i}$ is assumed to be adaptive, i.e. it is assumed that this function can learn. Learning is defined with regard to the observable behavior of the system: If the behavior of a system $S$ after some point of time $t_{y}$ is better than before some point of time $t_{x}$ (with $x \leq y)$ then the system $S$ is classified as having improved and this is assumed to be the result of some learning based on internal changes of the system-function $\gamma$. The usage of the term 'better' presupposes the availability of an operational criterion which can be applied to the observable behavior and which can be measured in an objective way. Example: the measurable time to find food in a certain environment is becoming 'shorter' after time $t_{y}$ than 'before' $t_{x}$.

Figure 2.3: Genetic Loop of Evolutionary Framework of Engineered Intelligent Systems