In the Asp heuristic, for each action a and for each fact p Î Add(a), a rule C®p is formed,
where C=Pre(a). Assuming a set of
rules, it is said that a fact p is reachable from a state S if p
Î S or
there is a rule C ® p
such that each fact q Î C is
reachable from S.
So,
a function g(p,S) is defined, which
inductively assigns a number i to
each fact p, where i is an
estimate of the number of steps needed to achieve p from S, i.e. the distance of p
from S. More specifically, g(p,S)
is set to 0 for every fact p Î S,
while g(p,S) is set to i+1, i
³ 0, for each fact p for which a rule C ® p
exists, such that . Thus:
|
|
0, if
p Î S |
(4) |
i+1, if for some C®p, |
|||
¥, if
p is not reachable from S |
In the
case where there are more than one rules C®p for a fact p, the rule with
the minimum cost is chosen. Note that a fact p that was initially
achieved by a rule C1®p, may be re-achieved, later, by another rule C2®p with smaller
cost. That is because not all the preconditions of the second rule had been
achieved at the time when the first rule was applied. The task of applying
rules continues until no rule that can achieve a fact with smaller cost exists.
The distances computed in this way are unique.
For a set of facts P, their distance from S is defined as:
|
(5) |
The Asp planner uses g(P,S) to estimate
the distances between each intermediate state S and the Goals. So, the Asp
heuristic function is defined as:
|
(6) |
The Asp heuristic does not take into account the delete lists of the actions. The simplified problem that is created by ignoring the delete lists is referred to as the relaxed problem and the corresponding actions are referred to as the relaxed actions. The complexity for constructing HASP(S) is linear, with respect to the number of ground actions and the number of ground facts.
Ioannis
Refanidis
14-8-2001