6.3 The Graphplan Planner 123
In the rest of the chapter, we will denote the set of mutex pairs in A
i
as µA
i
, and
the set of mutex pairs in P
i
as µP
i
. Let us remark that:
•
dependency between actions as well as mutex between actions or propositions
are symmetrical relations, and
•
for ∀i : P
i−1
⊆ P
i
, and A
i−1
⊆ A
i
.
Proposition 6.3 If two propositions p and q are in P
i−1
and (p, q)/∈ µP
i−1
, then (p, q)/∈
µP
i
. If two actions a and b are in A
i−1
and (a, b)/∈ µA
i−1
, then (a, b)/∈ µA
i
.
Proof Every proposition p in a level P
i
is supported by at least its no-op action α
p
.
Two no-op actions are necessarily independent. If p and q in P
i−1
are such
that (p, q)/∈ µP
i−1
, then (α
p
, α
q
)/∈ µA
i
. Hence, a nonmutex pair of propositions
remains nonmutex in the following level. Similarly, ...