These are exactly the facts that are always deleted by o when achieving A, no matter which effect we use.

The intersection of the sets D ( o ) for all actions o yields the desired set FAA . Let us consider the following small example to clarify the computation.

φ ( o ) = { U } {W}{¬X}; φι ( ο ) = { V, W } φ 2 ( 0 ) = { W }
We obtain D 1 ( o ) = {¬X} U {¬Y} = {¬X,¬Y}, because the precondition of φ 2 ( o ) is implied by the first conditional effect φ 1 ( o ). As φ 1 ( o ) is the only effect that can achieve A ,
we get D(o) = D 1 (o) = {¬X, ¬Y}.

We obtain a smaller set D ( o ), if we add A as an unconditional positive effect of the ation.

φ ( o ) = { U } W, A + {¬X } : φι (ο) = { V, W φ 2 ( 0 ) = { W }
In this case, we need to intersect the sets D 0 ( 0 ) = {¬X} and D 1 ( 0 ) = {¬X,¬Y},
yielding D ( o ) = {¬X}. This reflects the fact that, when achieving A via the unconditional effect of o t only X gets removed from the state.

The fixpoint computation requires to adapt the computation of O*. First, we repeat the same steps as in the case of simple STRIPS actions and consider the unconditional negative effects and the intersection of the preconditions with the False set:
0* := 0 \ {o | A ∈ effo (o) V F B A ∩ pr eo (o) ≠ 0 }
Then, we additionally remove from each action the conditional effects that either imply the deletion of A or have an impossible effect condition.

$$r_{\mathrm{{\tiny~\Delta\!~}}}:-\,r\,e u_{\mathrm{{\tiny~\Delta\!~}}}(\,C\,\,)\,=\,\updownarrow r$$
$$\mathbb{F}_{D A}^{A}\ \neq\emptyset\}$$

0* := r ed(0*) = {r ed(0)|0 E 0*}
Here, red is a function red(o) : o → o' such that Φ(o') = Φ(o) \ {φ k (o) | A ∈ D k (o) V prek (o) ∩ F B A ≠ ∅}
Finally, we need to redefine Definition 12, which expresses the conditions under which a fact is believed to be possibly achievable given a certain set of operators O . Definition 18 (Possibly Achievable Atoms for ADL) An atom p is possibly achievable given an action set O (written pA(p, O)) if and only if
∃ o ∈ O, φ i ∈ Φ(o) : p ∈ eff, + (o) ∧

$$\begin{array}{l}{{p\in e H_{i}^{+}(o)\,\land}}\\ {{\forall\,p^{\prime}\in\left(p r\,e_{i}(o)\cup p r\,e_{0}(o)\right):\ \exists\,o^{\prime}\in{\mathcal{O}},\ \phi_{i}\in\Phi(o^{\prime}):p^{\prime}\in e H_{i}^{+}(o^{\prime})}}\end{array}$$

holds, i.e., there is a positive effect for p and all of its conditions and preconditions can be made true by other effects in the reduced action set.