Deterministic or Probabilistic Property Sets

As one can see in figure 2.4 the transitions $\delta$ between different P-states (sets of properties) $\delta \in P-states \times 2^{P-states}$ can either be deterministic or non-deterministic. In the deterministic case with $\vert\delta(s)\vert = 1$ follows a successor state $s' = \delta(s)$ always after the predecessor state $s$ has occured. In the non-deterministic case it holds that $\vert\delta(s)\vert > 1$ and the occurence of the different successor P-states follows some probability distribution $P$. Thus if $P$ is a probability distribution

$$P : P-states \times P-states \longmapsto [0,1]$$ (2.14)

$$\sum_{s' \in P-states} P(s,s') = 1$$ (2.15)

then the possible sequence of occuring P-states will depend only on P. There are two interesting alternate cases: (1) The probability function $P$ has some memory and (2) the probability function $P$ shows some dynamics, i.e. it is changing during time. Furthermore there could be a combination of (1) and (2) mixing memory as well as dynamics.