The Structure of a Virtual World

To elaborate our theory of learning systems further we have to provide a formal framework which includes all the main components of a virtual world $WORLD$ needed. We assume the following components of the virtual world $WORLD$:

1. Environment [ENV]: An environment with some internal states.
2. Systems [SYS]: Some systems, which can interact with the environment.
3. Interface [$\iota$]: The mapping from environmental states into the systems as well as a mapping from system states to the environment.
4. Measurement [$\mu$]: Defined procedures how one can measure all important parameters of the acting systems.

This can be written as follows:

$$WORLD(w) iff w = \langle ENV,I_{e},O_{e}, SYS^{n},I_{s}^{n}, O_{s}^{n} \iota, \mu\rangle$$ (3.1)

$$ENV := Environments$$ (3.2)

$$I_{e} := Inputstrings for Environment$$ (3.3)

$$O_{e} := Outputstrings of Environment$$ (3.4)

$$SYS^{n} := n-many Systems$$ (3.5)

$$I_{s}^{n} := n-many Input Strings for Systems$$ (3.6)

$$O_{s}^{n} := n-many Output Strings of Systems$$ (3.7)

$$\iota = iin \otimes iout$$ (3.8)

$$iin : O_{s}^{n} \longmapsto I_{e}$$ (3.9)

$$iout : O_{e} \longmapsto I_{s}^{n}$$ (3.10)

$$\mu : ENV \times I_{e} \times O_{e} \longmapsto LOG$$ (3.11)

$$LOG := Data from Measurements$$ (3.12)

The Input $I_{e}$ and the output $O_{e}$ of the environment is the same for all systems.

The measurements $\mu$ are only needed for the validation of the systems against the world and to compare the different kinds of systems with regard to the same tasks.

We will start with a most simple environment $ENV$ including some internal states.

$$ENV(e) iff e = \langle I_{e}, O_{e},IS, \psi\rangle$$ (3.13)

$$I_{e} := Input strings for ENV$$ (3.14)

$$O_{e} := Output strings of ENV$$ (3.15)

$$IS_{e} := Internal states of ENV$$ (3.16)

$$\psi : I_{e} \times IS \longmapsto IS \times O_{e}$$ (3.17)

The structure of this environment resembles the structure of a system (see below).

A minimal system $SYS$ is defined as follows:

$$SYS(s) iff s = \langle I_{s}, O_{s}, IS, \phi\rangle$$ (3.18)

$$I_{s} := Inputstrings of the system$$ (3.19)

$$O_{s} := Outputstrings of the system$$ (3.20)

$$IS := Internal States of the system$$ (3.21)

$$phi : I_{s} \times IS \longmapsto IS \times O_{s}$$ (3.22)

Thus a minimal system is assumed to be an open system which can interact. In the case of $IS=\emptyset$ there is a direct fixed mapping between input and output controlled by the system function $\phi$.

In the case of $IS \neq \emptyset$ there exists internal states $IS$ which can change depending from time. In that case the output strings can be 'mediated' by changing states (e.g. by a changing energy level $E \subseteq IS$ or by the change of 'memory contents' from a memory $M \subseteq IS$).

Figure 3.2: Simple virtual world with open systems