Systems Hierarchy

Figure 6.1: A Framework for Systems
\includegraphics[width=5.0in]{framework_for_systems.eps}

The main topic of the following chapters are learning systems (LS). But learning systems are a subset of the more general concept of systems (S). In this booklet we assume usually that systems are input -output systems. Thus if nothing else is stated we assume tacitly this.

As the diagram 6.1 shows we assume further a basic distinction between biological (BS) and technological (TS) systems. As basic distinctions we assume the following:

General Input-Output Systems (IOS or S)


$\displaystyle S(s)$ $\displaystyle iff$ $\displaystyle s = \langle I, O, \varphi\rangle$ (6.1)
$\displaystyle I$ $\displaystyle :=$ $\displaystyle Inputstrings of the system$ (6.2)
$\displaystyle O$ $\displaystyle :=$ $\displaystyle Outputstrings of the system$ (6.3)
$\displaystyle \varphi$ $\displaystyle :$ $\displaystyle I \longmapsto O$ (6.4)

Thus a general input-output system has a system function $ \varphi$ mapping inputs $ I$ into outputs $ O$ without an restrictions how to do this. We will distinguish two important classes of input-output systems: reactive systems as well as learning systems.

Random Systems (RandS)

A random system behaves completely by chance. It can be used as a benchmarking system to compare this with other more advanced systems.

Reactive Systems (RS)


$\displaystyle RS$ $\displaystyle \subseteq$ $\displaystyle S$ (6.5)
$\displaystyle RS(s)$ $\displaystyle iff$ $\displaystyle s = \langle I, O, IS, \varphi\rangle$ (6.6)
$\displaystyle I$ $\displaystyle :=$ $\displaystyle Inputstrings of the system$ (6.7)
$\displaystyle O$ $\displaystyle :=$ $\displaystyle Outputstrings of the system$ (6.8)
$\displaystyle IS$ $\displaystyle :=$ $\displaystyle Internal States of the system$ (6.9)
$\displaystyle \varphi$ $\displaystyle :$ $\displaystyle I \times IS \longmapsto O$ (6.10)

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 efficient than by pure chance. If the tasks or the environment are changing than a fixed system can 'loose its ground'.

Learning Systems (LS)


$\displaystyle LS$ $\displaystyle \subseteq$ $\displaystyle S$ (6.11)
$\displaystyle LS \cap RS$ $\displaystyle =$ $\displaystyle \emptyset$ (6.12)
$\displaystyle LS(s)$ $\displaystyle iff$ $\displaystyle s = \langle I, O, IS, \varphi\rangle$ (6.13)
$\displaystyle I$ $\displaystyle :=$ $\displaystyle Inputstrings of the system$ (6.14)
$\displaystyle O$ $\displaystyle :=$ $\displaystyle Outputstrings of the system$ (6.15)
$\displaystyle IS$ $\displaystyle :=$ $\displaystyle Internal States of the system$ (6.16)
$\displaystyle \varphi$ $\displaystyle :$ $\displaystyle I \times IS \longmapsto IS \times O$ (6.17)

The important difference between reactive and learning systems is the property, that learning systems an change their internal states $ IS$ depending from their input $ I$ and the past internal states $ IS$. Doing these changes in the 'right' way learning systems can 'learn'. But this is the critical point: 'in the right way'! To 'know' about the right way the learning systems can have two kinds of sources: (i) internal states representing 'good/ bad' as well as (ii) external sources. The internal sources are special internal states $ v \in IS$ which are 'used' by the system to organize 'preferences' in its states. The external sources are embedded in the 'relationship' between the system itself and the environment. Within this relationship with the environment one can distinguish two basic cases: (ii.1) success by 'existence' or (ii.2) success by 'support'. Success by existence means that the system is able to live in a certain environment either itself or by itself and by its 'offspring'. Success by support means that the system is able to find anything which it needs to support its ongoing existence ('energy', 'food', 'water',...).

Populations

In case of populations of systems these systems can either help each other to optimize their success or they can hinder each other. Mutual support is usually the more efficient way to solve complex tasks because most tasks cannot be solved with single systems.

An additional option of populations is the bridging of generations in those cases where the systems have a limited life cycle. Populations which are able to 'bridge' between generations are even more powerful than others because they can solve tasks which require a certain duration.

Biological Systems (BioS)

Biological systems represent in a certain sense hybrid systems organized in populations: There is a blueprint version of the system, the genotype, which can be changed, and their is a structured version of the system, the phenotype, which will be generated by a growth process out of the blueprint version. The structured version is a system somewhere between being reactive and somewhere being a learning system ( $ BioS \subseteq RS \cup LS$). Because the blueprint-version is heavily exchanged within a population there is some chance to improve the system structure (but it can also become more worse). Structural improvements will therefore only occur after several generations of exchange.

Technical Systems (TechS)

Traditionally technical systems are mostly reactive and rarely learning systems. Only recently technical systems become slightly learning systems ( $ TechS \subseteq RS \cup LS$). Furthermore technical systems occur usually only in a structured version fully fitted to do the designed task. An automatic coupling of a blueprint version to a structured version is only very slowly appearing, and these couplings are usually not automatically driven.

Gerd Doeben-Henisch 2013-01-14