Success in the Real World

In the known real world the success of a system is dependent on his ability to intake enough energy $ e$ to compensate for the consumption of energy in a 'sufficient' manner. Additionally there should be some offspring to compensate for the finite duration of a system $ g' \in G'$. Thus we have the condition that $ consume(e,t,t') \leq intake(e,t,t')$ and $ \vert OFFSPRING(g',t,t')\vert > 0$5.3. Thus there are some 'real' parameters which constrain the success of a real system.

In a simulated world one has to provide such constraining parameters in some analogous way.

Therefore the world $ W$ as the world of the working population $ G'$ as well as every element (= system) $ g' \in G'$ provides a basic mechanism for energy intake, energy consumption, offspring generation as well as process termination.

Based on this minimal framework of operations one can observe the success or failure of a working population $ G'$ only by 'counting' the number of offspring as a 'hint' for the 'quality' of the working system.

Here it is helpful to distinguish between the 'genotype' and the 'phenotype' of a system. Real systems occur in two different modes. While the genotype $ g' \in G'$ gives only the 'instructions' (the 'blueprint') for the phenotype $ pg' \in PG'$ of a system, the phenotype does the observable work in the real world. Thus a phenotype $ pg'$ represents a system which has some input from the world $ W$ as well as it can have some output. The transition from the genotype to the phenotype is realized by some growth-function $ growth: G' \longmapsto PG'$. Thus we have:

$\displaystyle ENV$ $\displaystyle \subseteq$ $\displaystyle POS \times PROP$ (5.22)
$\displaystyle POS$ $\displaystyle \in$ $\displaystyle 2^{X \times Y \times Z}$ (5.23)
$\displaystyle PROP$ $\displaystyle =$ $\displaystyle O \cup G' \cup PG'$ (5.24)
$\displaystyle change$ $\displaystyle :$ $\displaystyle ENV \longmapsto ENV$ (5.25)
$\displaystyle intake$ $\displaystyle :$ $\displaystyle O \times PG' \times DUR \longmapsto E^{PG'}$ (5.26)
$\displaystyle consume$ $\displaystyle :$ $\displaystyle PG' \times E_{PG'} \times DUR \longmapsto E_{PG'}$ (5.27)
$\displaystyle move$ $\displaystyle :$ $\displaystyle PG' \times ENV \longmapsto ENV$ (5.28)
$\displaystyle DUR$ $\displaystyle =$ $\displaystyle TIME \times TIME$ (5.29)
$\displaystyle create$ $\displaystyle :$ $\displaystyle PG' \times PG' \longmapsto PG' \times PG' \times G'$ (5.30)
$\displaystyle growth$ $\displaystyle :$ $\displaystyle G' \longmapsto PG'$ (5.31)
$\displaystyle LIFESPAN$ $\displaystyle =$ $\displaystyle DUR$ (5.32)
$\displaystyle death1$ $\displaystyle :$ $\displaystyle PG' \times LIFESPAN \times CLCK\longmapsto \emptyset$ (5.33)
$\displaystyle death2$ $\displaystyle :$ $\displaystyle PG' \times E \longmapsto \emptyset$ (5.34)
$\displaystyle PG(x)$ $\displaystyle iff$ $\displaystyle \langle E,G', consume, IntakeArea, MoveArea>\rangle$ (5.35)
$\displaystyle IntakteArea$ $\displaystyle \in$ $\displaystyle NAT$ (5.36)
$\displaystyle MoveArea$ $\displaystyle \in$ $\displaystyle NAT$ (5.37)
$\displaystyle W(x)$ $\displaystyle iff$ $\displaystyle \langle ENV, O, G', PG', growth, intake, create, move, eval1, death1, death2, change \rangle$ (5.38)
$\displaystyle eval1$ $\displaystyle =$ $\displaystyle offspring$ (5.39)
$\displaystyle offspring$ $\displaystyle :$ $\displaystyle 2^{PG' \times PG' \times G'} \times DUR \longmapsto NAT$ (5.40)

Thus a world $ W$ consists of an environment $ ENV$ with properties $ PROP$ located at positions $ POS$. A position can be a set of vectors $ (x,y,z)$. The distribution of properties at positions can change $ change()$. Properties can be objects $ O$, working genotypes $ G'$ as well as phenotypes $ PG'$. With the function $ growth()$ one can generate phenotypes $ PG'$ out of genotypes. The function $ intake()$ can enable a working phenotype $ PG'$ to transfer objects, which have been 'picked up' from the 'intake area', into energy $ E$. A phenotype can be moved $ move()$ within the distance of MoveArea. Two phenotypes can create a new working genotype by $ create()$ if they are 'close enough'. There is also a death-function $ death1()$ which deletes a working system if the lifespan has been ended. If within the 'normal' lifespan the energy drops below zero a death will also happen with $ death2()$. The function $ eval()$ computes for every set of indexed genes and phenotypes for some duration a number $ n \in NAT$ as 'fitness value'. The values of the eval()-function obey an ordering.

A basic 'world cycle' could then be characterized as follows:

  1. Set the world clock to zero $ CLCK=0$.
  2. Assume a world $ W$ with some properties as objects $ O$ and certain genotypes $ G'$ at certain positions.
  3. [START] Transfer the genotypes into phenotypes while keeping the positions by applying the function $ growth(g')=pg'$ for every available genotype. Increase the world clock by some duration $ CLCK=CLCK+DUR$.
  4. Realize for every phenotype $ pg'$ an intake of objects $ intake()$ if the object is within the 'intake area' of a phenotype. Increase the world clock by some duration $ CLCK=CLCK+DUR$.
  5. Realize for every phenotype $ pg'$ a consumption of energy $ consume()$. Increase the world clock by some duration $ CLCK=CLCK+DUR$.
  6. Realize for every phenotype $ pg'$ a movement $ move()$. Increase the world clock by some duration $ CLCK=CLCK+DUR$.
  7. Realize for every phenotype $ pg'$ a creation of offspring $ offspring()$. Increase the world clock by some duration $ CLCK=CLCK+DUR$.
  8. Realize for every phenotype $ pg'$ a death-function $ death2()$ or $ death1()$. Increase the world clock by some duration $ CLCK=CLCK+DUR$.
  9. Apply some change to the world with $ change()$.
  10. Calculate the offspring for every Phenotype.
  11. Repeat all actions beginning with START

Gerd Doeben-Henisch 2013-01-14