Reactive Systems (RS)

As described before Reactive systems have some internal states $IS \neq \emptyset$ which determine the behavior but are fixed/ static. As long as the internal states are 'fitting' to the tasks at hand these internal states can help the system to solve these tasks more efficiently than by pure chance. If the tasks or the environment are changing than a fixed system can 'loose its ground'.

$\displaystyle RS$	$\displaystyle \subseteq$	$\displaystyle S$	(8.1)
$\displaystyle RS(s)$	$\displaystyle iff$	$\displaystyle s = \langle I, O, IS, \phi\rangle$	(8.2)
$\displaystyle I$	$\displaystyle :=$	$\displaystyle Inputstrings of the system$	(8.3)
$\displaystyle O$	$\displaystyle :=$	$\displaystyle Outputstrings of the system$	(8.4)
$\displaystyle IS$	$\displaystyle :=$	$\displaystyle Internal States of the system$	(8.5)
$\displaystyle \varphi$	$\displaystyle :$	$\displaystyle I \times IS \longmapsto O$	(8.6)

While in the case of technical systems the engineer or the human user has the responsibility to support a good 'fitting' of the technical system, in the case of nature the nature has not such a 'privileged' direct monitoring. Nature has to find the right 'fitting' by producing 'more' (as offspring) and 'more often' (through generations) by 'chance-driven' constructions of different kinds of blueprints.

It could be an interesting measure of structural complexity to compare the structure of a certain system S with an evolutionary process P which is necessary to 'produce' this system S given some initial 'blueprint' $g \in G$ . This blueprint

has to be 'translated' into a working structure

resembling the system structure S. This has to be repeated as long as the process P needs to find a structure S' which 'performs' as good enough as the structure representing system S. Thus we have

$\displaystyle g \in G$	$\displaystyle :=$	$\displaystyle Initial blueprint$	(8.7)
$\displaystyle F$	$\displaystyle :=$	$\displaystyle Fitness$	(8.8)
$\displaystyle P$	$\displaystyle :$	$\displaystyle G \times F \longmapsto G$	(8.9)
$\displaystyle S'$	$\displaystyle :$	$\displaystyle G \longmapsto S$	(8.10)

Such an evolutionary process is a search process organized as a computation. The necessary amount of the computation is then a direct measure for the structural complexity.

The evolutionary process supporting a biological system can be understood as an equivalent of the engineering process realized by human effort. The extremely acceleration of the human driven technological process can be a hint of the cumulated intelligence within a single human person additionally 'enforced' by manifold cooperations as well as generation bridging. To replace the highly complex human cognition by a 'simple' evolutionary process is therefore rather a step back. It can only be a 'progress' if the evolutionary process includes an equivalent of human cognition as part of the process.