Basic Dimensions

This induces the following first proposal for a formal account (cf. figure 4.7):

A world consists of environments (ENV) and agents (AGENT). A mapping from the environment into the agent (ainp) can map some locations (LOC) of the environment into an input string ( $\sigma \in \Sigma^{n}$ ), which is placed into the input set (INP) of the agent. Vice versa can the output string ( $\xi \in \Xi^{n}$ ) from the output set (OUT) of an agent be mapped into a set of locations of that environment (ENV) where the agent is actually located. Every location has a position (POS, X,Y,Z) as well as a finite set of properties (PROP).

$\displaystyle WORLD$	$\displaystyle =$	$\displaystyle \langle ENV, AGENT, ainp, aout\rangle$	(4.103)
$\displaystyle ENV$	$\displaystyle \subseteq$	$\displaystyle LOC$	(4.104)
$\displaystyle LOC$	$\displaystyle \subseteq$	$\displaystyle POS \times 2^{PROP}$	(4.105)
$\displaystyle POS$	$\displaystyle \subseteq$	$\displaystyle X \times Y \times Z$	(4.106)
$\displaystyle ainp$	$\displaystyle :$	$\displaystyle 2^{LOC} \times ENV \longmapsto \Sigma^{n} \times AGENT$	(4.107)
$\displaystyle aout$	$\displaystyle :$	$\displaystyle \Xi^{n} \times AGENT \longmapsto 2^{LOC} \times ENV$	(4.108)

**Figure 4.7:** Environment - Agent - Interface
$\includegraphics[width=4.0in]{environment_agent_interface_16nov2010_4.0in.eps}$