GA as Model of Genetic Inheritance

A simple formalization of the above idea of genetic inheritance is described below. It assumes a population $P$ as a set of individual elements called agents $a \in P$ and as a minimal property of an agent it is assumed that he possesses genetic informations $g \in G$ where $G$ is a subset of binary strings of a fixed size $G \subseteq \{1,0\}^{n}$^3.2.

With these assumptions one can define a simple algorithm to simulate genetic inheritance.

$$\vert P\vert > 1$$ (3.1)

$$Fitness : P \longmapsto F$$ (3.2)

$$Selection : P \times F \longmapsto P \times F$$ (3.3)

$$Crossover : P \times P \longmapsto P$$ (3.4)

$$Mutation : P \longmapsto P$$ (3.5)

$$Repeat at Fitness$$ (3.6)

Formula 3.1 assumes that there is an initial population $P$ with at least two members which can produce offspring. If there is a population $P$ then can the $Fitness$-function compute some fitness value $f \in F$ for each agent of the population. The fitness function as such is no part of the GA; it has to be 'provided' by the user of the GA. If there is a population with fitness values for each agent then it is possible to select a subset out of $P \times F$. Usually this subset takes these members of $P$ which have the highest fitness values. Different to real populations are virtual populations understood as staying constant in number. If there are no more members for such a selection then the population $P$ is extincted. Otherwise the GA algorithm continues with 'playing' with the genetic information. There are two main options: (i) keeping the given structures but arrange them in a new way (one example is $Crossover$ mixing the genetic informations of two different agents by keeping the different parts 'as they are'); (ii) changing the given structures in a 'new' way (one example is called $Mutation$ by changing some part of the given information). After these Modifications the original population $P$ has changed to a new format $P'$, which is the starting point for a new cycle of the GA algorithm.

There arise many deep questions regarding mathematical properties of such a GA algorithm. Before these will be discussed some examples will be presented^3.3 an example.