We tested the efficiency of the
XOR-constraints based decomposition in two domains: A simplified mystery
domain, where resources have been removed, and the grid domain of the Aips-98 competition. We did not use the
logistics domain for these experiments, since logistics problems
are not difficult for the original Grt
and the small profit from solving the easier sub-problems is compensated by the
extra pre-processing cost of each sub-problem.
We removed resources
from the original mystery domain because otherwise it would be probable
to obtain unsolvable subproblems. As it has been noted in Section 5,
decomposing a problem may lead to loss of completeness, thus the technique is
unsuitable for domains where deadlocks may arise, as the original mystery
one. Note that by removing resources, all mystery problems become
solvable.
The XOR-constraints
that have been defined for the simplified mystery domain were the
following:
( ( xor ( at ?Truck
* ) ) (truck ?Truck ))
( ( xor ( at ?Package * ) (in ?Package
* ) ) ( package ?Package ) )
while for the grid domain
were the following ones:
( ( xor ( at-robot * )
) )
( ( xor ( locked
?Place ) ( open ?Place ) ) ( place ?Place )
)
( ( xor ( at ?Key * ) ( holding
?Key ) ) ( key ?Key ) )
Note that in the grid
domain an XOR-constraint denoting that the arm is either empty, or the robot
holds a key has not been defined, since this would lead to pointless
decompositions.
In both domains, we
ran Grt with and without the
problem decomposition technique. Additionally, in order to demonstrate the
contribution of the irrelevant objects elimination technique when solving the
sub-problems, we conducted experiments for this case in the simplified mystery
domain. We did not consider this case in the grid domain, because no
irrelevant objects can be detected there. Figure 11 presents the results.
(a) Simplified Mystery
(b) Grid Domain
Figure 11: Solution time (in msecs)
and length with and without
the XOR-constraints based problem decomposition technique.
As for the simplified
mystery domain, Grt without the
problem decomposition technique generally produced shorter plans, as expected.
On the other hand, the use of the XOR-constraints accelerated the problem
decomposition process, especially in case of difficult problems. Actually, if
we only consider the seven most difficult problems, the improvement achieved by
the decomposition is 60% on average. Note however that, when the irrelevant
objects elimination technique was not used, there was no improvement. In not
difficult problems there is no acceleration, since, as in the case of the logistics
problems, the small profit from the faster solution of the easier sub-problems
is compensated by the cost of repeating the pre-processing phase for each one
of them.
The grid domain
was the most difficult one of the Aips-98
competition. The contestants managed to solve only the first problem. Grt without XOR-constraints could only
solve the first problem, too. On the other hand, with the XOR-constraints based
decomposition, Grt was able to
solve the first four problems in the time limit of 5 minutes, while in the
fifth problem it ran out of memory. It is worth noting that this domain
produces multiple levels of decompositions. Figure 12 presents these levels for
the strips-grid-y-2 problem.
As far as we know, the
only planner that can cope with the grid problems effectively is Ff. We ran Ff in the five grid problems and it solved the first
four, within the time limit of 5 minutes, with the following results
(length/time): 14/230, 39/840, 40/7810 and 45/3280, which are considerably
better compared to the performance of Grt.
Figure 12: Decomposition for the
strips-grid-y-2 problem using XOR-constraints.
Ioannis
Refanidis
14-8-2001