Backward and bidirectional versions

There is a perfect symmetry to the concepts presented so far in this section. A reachability lattice similar to the one in Figure 14.13 can be obtained by integrating backward in time. This indicates action sequences and associated initial states from which a fixed state can be reached. Note that applying the ideas in the reverse direction does not require the system to be symmetric. Given that the graphs exist in both directions, bidirectional search can be performed. By using the forward and backward time-limited reachability cones, the initial and goal states can be connected to a common lattice, which is started, for example, at the origin.