#### Nondeterministic forward projections

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 .

Steven M LaValle 2012-04-20