whether the preconditions of op3 are achievable in the sense that they are added by another action. This is the case since the only precondition C is added by op 2 . Thus, D is removed from F*, which becomes empty now. The action op4 is put back into the set O*,
which now becomes identical with the action set OA . This set, in turn, is identical with the original action set O as no action deletes A . The fixpoint process terminates and B will not be ordered before A as it can be aieved using the action op4 . This correctly reflects the fact that there exists a plan from the state s(A,-B) = Result(I,(op2)) = {C, A} to a state that satisfies B without destroying A.

![0_image_0.png](0_image_0.png)

As already pointed out, the intention behind the fixpoint procedure is the following:
Starting from a state s ( A,-B ); we want to know which facts can become TRUE without destroying A, and consequently, which actions can become applicable. In the first step, only actions that do not use any of the facts in FA are applicable, as all those facts are deleted from the state description when A is added. However, such actions may make facts in FAA TRUE, so we want to remove those facts from FAA. If we manage to find all the facts that can be made true without destroying A, then the final set F* will contain only those facts that do not hold in a state reachable from s{A,¬B} without destroying A. In this case, the final action set O* will contain all the actions that can be applied after s(A,-B); and we can safely use this action set to determine whether another goal B can still be achieved or nt.

However, as we only use the approximative test pA(f, O*) with f ∈ F* to find out if a fact in the current F* set is achievable, there may be facts which are achievable without destroying A, but which remain in the set F*. This could exclude actions from the set O* whi can be safely applied after s(A,-B). Under certain restrictions, however, we can prove that this will not happen. In order to do so, we need to impose a restriction on the particular state s{A,-B}, in which we achieved the goal A: If none of the preconditions of actions, which add facts contained in FA, occur in the state s(A,-B), then the fixpoint procedure will remove all facts from F A that are achievable without destroying A . We will use this property of the fixpoint procedure later to show that our heuristic ordering relation approximates reasonable orderings.