To facilitate the notation, suppose in this section that
for all . In Section 10.1.3 this will be lifted.
Suppose that the initial state,
, is known. If the action
is applied, then the set of possible next states is
such that
(10.10)
which is just a special version of (10.5). Now
suppose that an action will be applied. The forward
projection must determine which states could be reached from by
applying followed by . This can be expressed as
(10.11)
This idea can be repeated for any number of iterations but becomes
quite cumbersome in the current notation. It is helpful to formulate
the forward projection recursively. Suppose that an action history
is fixed. Let
denote the forward
projection at stage , given that is the forward projection
at stage . This can be computed as
(10.12)
This may be applied any number of times to compute from an
initial condition
.
Example 10..3 (Nondeterministic Forward Projections)
Recall the first model given in Example 10.1, in which
and
. Suppose that , and is applied. The one-stage forward projection is
. If is applied again, the two-stage
forward projection is
. Repeating this
process, the -stage forward projection is
.