Each action a
′ ∈ A′is made from one action a ∈
A such that **P re**(a
′) = ∅, and E(a
′) consists of 2 |P re g(a)|+|Add g(a)|+|Del g(a)|conditional effects e. For each conditional effect e: - **cons**(e) is the union of the following sets:
- the certain preconditions **P re**(a),
- the set of possible preconditions of a that are realized, and hidden propositions representing their realization:
P re(a) ∪ {p pre a|p ∈ P re(a)} ∪ {nppre a|p ∈ **P re** g(a) \
P re(a)},
- the set of hidden propositions corresponding to the realization of possible add (delete) effects of a: {q add a|q ∈
Add(a)} ∪ {nqadd a|q ∈ Add g(a) \ Add(a)} ({r del a|r ∈
Del(a)}∪{nrdel a|r ∈ Del g(a)\Del(a)}, respectively);
- the *single* outcome ε of e is defined as add(ε) = Add(a)∪
Add(a), del(ε) = Del(a) ∪ Del(a), and P r(ε) = 1,
where P re(a) ⊆ P re g(a), Add(a) ⊆ Add g(a) and Del(a) ⊆
Del g(a) represent the sets of realized preconditions and effects of the action. In other words, we create a conditional effect for each subset of the union of the possible precondition and effect sets of the action a. Note that the inclusion of new propositions derived from P re(a), Add(a), Del(a) and their "complement" sets P re g(a)\P re(a), Add g(a)\Add(a),
Del g(a) \ Del(a) makes all condition sets of the action a
′
mutually exclusive. As for other cases (including those in which some precondition in **P re**(a) is excluded), the action has no effect on the resulting state, they can be ignored. The condition sets, therefore, are also exhaustive.

The initial belief state bI consists of 2 |Fnew|states s
′ ⊆ F
′
such that p ∈ s
′iff p ∈ I (∀p ∈ F), each represents a complete domain model Di **∈ hhDii** e and with the probability Pr(Di). The goal is G′ = G, and the acceptable goal satisfaction probability is ρ
′ = ρ.

Theorem 1. *Given a plan* π = (a1**, ..., a**n) *for the problem* Pe*, and* π
′ = (a
′ 1
, ..., a′n) *where* a
′
k is the compiled version of ak (1 ≤ k ≤ n*) in* P
′. Then R(π,Pe) ≥ ρ iff π
′ *achieves* all goals with at least ρ *probability in* P
′.

Proof (sketch). According to the compilation, there is oneto-one mapping between each complete model Di **∈ hhDii** e in Pe and a (complete) state s
′
i0 ∈ bI in P
′. Moreover, if Di has a probability of Pr(Di) to be the real model, then s
′ i0 also has a probability of Pr(Di) in the belief state bI of P
′.

Given our projection over complete model Di, executing π from the state I with respect to Di results in a sequence of complete state (si1, ..., si(n+1)). On the other hand, executing π
′from {s
′ i0
} in P
′results in a sequence of belief states ({s
′
i1
}**, ...,** {s
′
i(n+1)}). With the note that p ∈ s
′
i0 iff p ∈ I (∀p ∈ F), by induction it can be shown that p ∈ s
′
ij iff p ∈ sij (∀j ∈ {1**, ..., n** + 1}, p ∈ F). Therefore, si(n+1) |= G iff s
′
i(n+1) |= G = G′.

Since all actions a
′
i are deterministic and s
′
i0 has a probability of Pr(Di) in the belief state bI of P
′, the probability Figure 2: An example of compiling the action *pick-up* in an

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

![0_image_1.png](0_image_1.png) incomplete domain model (top) into CPP domain (bottom).

The hidden propositions p pre pick−up, q add pick−up and their negations can be interpreted as whether the action requires light balls and makes balls dirty. Newly introduced and relevant propositions are marked in bold.

that π
′achieves G′is Ps
′

i(n+1)|=G Pr(Di), which is equal to R(π,Pe) as defined in Equation 2. This proves the theorem.

Example: Consider the action *pick-up(?b - ball,?r - room)* in the Gripper domain as described above. In addition to the possible precondition *(light ?b)* on the weight of the ball ?b, we also assume that since the modeler is unsure if the gripper has been cleaned or not, she models it with a possible add effect *(dirty ?b)* indicating that the action might make the ball dirty. Figure 2 shows both the original and the compiled specification of the action.

## 6. Experimental Results

We tested the compilation with Probabilistic-FF (PFF), a state-of-the-art planner, on a range of domains in the International Planning Competition.We first discuss the results on the variants of the Logistics and Satellite domains, where domain incompleteness is deliberately modeled on the preconditions and effects of actions (respectively). Our purpose here is to observe how generated plans are robustified to satisfy a given robustness threshold, and how the amount of incompleteness in the domains affects the plan generation phase. We then describe the second experimental setting