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:

$\displaystyle WORLD(w)$ $\displaystyle iff$ $\displaystyle w = \langle ENV,I_{e},O_{e}, SYS^{n},I_{s}^{n}, O_{s}^{n} \iota, \mu\rangle$ (3.1)
$\displaystyle ENV$ $\displaystyle :=$ $\displaystyle Environments$ (3.2)
$\displaystyle I_{e}$ $\displaystyle :=$ $\displaystyle Inputstrings for Environment$ (3.3)
$\displaystyle O_{e}$ $\displaystyle :=$ $\displaystyle Outputstrings of Environment$ (3.4)
$\displaystyle SYS^{n}$ $\displaystyle :=$ $\displaystyle n-many Systems$ (3.5)
$\displaystyle I_{s}^{n}$ $\displaystyle :=$ $\displaystyle n-many Input Strings for Systems$ (3.6)
$\displaystyle O_{s}^{n}$ $\displaystyle :=$ $\displaystyle n-many Output Strings of Systems$ (3.7)
$\displaystyle \iota$ $\displaystyle =$ $\displaystyle iin \otimes iout$ (3.8)
$\displaystyle iin$ $\displaystyle :$ $\displaystyle O_{s}^{n} \longmapsto I_{e}$ (3.9)
$\displaystyle iout$ $\displaystyle :$ $\displaystyle O_{e} \longmapsto I_{s}^{n}$ (3.10)
$\displaystyle \mu$ $\displaystyle :$ $\displaystyle ENV \times I_{e} \times O_{e} \longmapsto LOG$ (3.11)
$\displaystyle LOG$ $\displaystyle :=$ $\displaystyle Data from Measurements$ (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.

$\displaystyle ENV(e)$ $\displaystyle iff$ $\displaystyle e = \langle I_{e}, O_{e},IS, \psi\rangle$ (3.13)
$\displaystyle I_{e}$ $\displaystyle :=$ $\displaystyle Input strings for ENV$ (3.14)
$\displaystyle O_{e}$ $\displaystyle :=$ $\displaystyle Output strings of ENV$ (3.15)
$\displaystyle IS_{e}$ $\displaystyle :=$ $\displaystyle Internal states of ENV$ (3.16)
$\displaystyle \psi$ $\displaystyle :$ $\displaystyle I_{e} \times IS \longmapsto IS \times O_{e}$ (3.17)

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

A minimal system $ SYS$ is defined as follows:

$\displaystyle SYS(s)$ $\displaystyle iff$ $\displaystyle s = \langle I_{s}, O_{s}, IS, \phi\rangle$ (3.18)
$\displaystyle I_{s}$ $\displaystyle :=$ $\displaystyle Inputstrings of the system$ (3.19)
$\displaystyle O_{s}$ $\displaystyle :=$ $\displaystyle Outputstrings of the system$ (3.20)
$\displaystyle IS$ $\displaystyle :=$ $\displaystyle Internal States of the system$ (3.21)
$\displaystyle phi$ $\displaystyle :$ $\displaystyle I_{s} \times IS \longmapsto IS \times O_{s}$ (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

Gerd Doeben-Henisch 2013-01-14