Within a framework of a world consisting of an environment and some learning systems we need at least one task which has to be solved.

Intuitively a task is a collection of different 'states' $ S$ which can be classified as 'start states' $ S_{start} \subseteq S$, as 'goal states' $ S_{goal} \subseteq S$ or as 'intermediate states' $ S_{inter} \subseteq S$ with

$\displaystyle \emptyset$ $\displaystyle =$ $\displaystyle S_{start} \cap S_{goal} \cap S_{inter}$ (10.1)

A 'complete task' has at least one start state with a corresponding goal state and at least one connecting 'path' between the start and the goal state. Generally we are looking for minimally necessary states.

A state is a finite set of 'properties', each a category-value pair. Two states $ S_{i}, S_{j}$ are different, if there exists at least one property $ p$ with regard to which the two states are different.

$\displaystyle DIFFERENT(S_{i}, S_{j})$ $\displaystyle iff$ $\displaystyle \exists p( p \in (S_{i} \cup S_{j}) \& p \notin (S_{i} \cap S_{j}))$ (10.2)

Every property of a state must be measurable, otherwise it's not existing for the theory.

States belong to an environment.

Gerd Doeben-Henisch 2013-01-14