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9.      Conclusion and Future Work

In this paper we presented the Grt planning system, a heuristic state-space planner, which constructs its heuristic in a domain-independent way. The fundamental difference between Grt and other heuristic state-space planners is that Grt constructs its heuristic once, in a pre-processing phase and in a backward direction, using regression from the goals. Grt attempts to track the positive and negative interactions that occur between the problem facts when trying to achieve them, in order to produce better estimates.

Grt employs several new techniques that improve its efficiency. These are the automated identification of incomplete goal states, the identification and enrichment of inadequate domain representations, the elimination of irrelevant objects and the adoption of a numerical representation of resources. Finally, a knowledge-based method that uses domain axioms in the form of XOR-constraints, in order to decompose difficult problems into easier sub-problems that have to be solved sequentially, has adopted.

The paper presented extensive comparative results in a large number of domains. In the comparisons, besides Grt, four of the most powerful domain independent planners took part. The results showed that no planner clearly outperforms all the others.

Concerning solution time, in most of the domains Grt and Ff were the fastest planners. The explanation behind this observation lies in that these planners construct their heuristic either once (in the case of Grt), or a few times only (in the case of Ff). For example, in the elevator domain, where delete effects do not exist and Ff constructs a relaxed planning graph only once, it is extremely fast. On the contrary, in the gripper and the puzzle domains, where Ff needs to reconstruct the relaxed planning graphs, its efficiency decreases drastically with respect to the Grt's one.

Hsp-2 was not faster than the other planners in any domain, being always outperformed by Ff. This was expected, since the two planners use the forward direction both for the construction of their heuristics and for traversing the state-space, however Ff constructs its heuristic less times than Hsp-2. Our impression is that the Ff heuristic is also more informative and more accurate than the one of Hsp-2. Concerning Altalt, although it constructs its heuristic once, it did not manage to be faster than the others in any domain and this is (we believe) due to the problems that arise from the backward direction in which it traverses the state-space. So, this is an indication that in the case where opposite directions are used for the heuristic construction and the search phase, as Grt, Altalt and Hspr do, it is preferable to use the backward direction for the heuristic construction and the forward direction for the search phase. This is why the problems that arise when constructing the heuristic backwards may be confronted more easily than the problems that arise when traversing the state-space backwards.

Domain analysis techniques, which occur in pre-processing phase, also play an important role. Stan, which is primarily based on these techniques, had many variations in its performance. In transportation domains, like the logistics and the elevator ones, where Stan exploits specialized heuristics, it was among the fastest planners. In the gripper domain, where Stan exploits its symmetry analysis, its performance was also excellent. In other domains, as for example the freecell or the blocks, it was not competitive due to its Graphplan basic architecture, which is not considered a fast technology any more.

Ff also employed a domain analysis technique concerning goal ordering, which played an important role in the blocks problems. It would be very interesting to see the adaptation and the impact of this technique to other planners as well. As far as we know, Hsp-2 and Altalt are not using any domain analysis technique. Grt exploited only the domain enrichment technique in the elevator domain, however this technique is an integral part of its heuristic mechanism, in order to overcome some problems that arise from the backward heuristic construction.

An interesting observation concerns the performance of Grt in the bigger problems of the logistics, freecell, gripper and puzzle domains, where Grt exhibited better performance than in the smaller problems of the same domains, compared to the other planners. We believe that this is due to the fact that Grt constructs its heuristic once, while the repeated construction of the heuristics for the other planners is an inhibitory factor in the bigger problems.

The conclusions drawn above ignore a significant factor, which is the specific implementation, i.e. the approaches adopted by the various planners for "trivial" tasks, such as the computation of all the ground facts and actions of a problem or the computation of the applicable actions to a given state, the optimization of the code and of course potential "bugs". For example, in order to find the applicable actions to a state, Grt uses constraint satisfaction techniques to progressively instantiate the action schemas for each state, whereas most of the other planners exploit connectivity graphs between the facts of a problem and the pre-instantiated actions. Our experiments with Grt have shown that a significant portion of the processing time is spent in the determination of the applicable actions to a state. This is the reason why we have developed a parallel version of Grt, named PGRT (Vrakas et. al., 1999; 2000), which makes use of this observation and has been proved very efficient in all domains. However, it is in our future plans to develop a connectivity graph also in Grt and to compare it to the existent approach.

Differences that are due to the code optimization or potential "bugs" cannot be easily detected, but we believe that all the planners, both the oldest and the newest ones are well-optimized programs. In the future we would like to see theoretical comparisons between the computational complexities of the various techniques and algorithms, apart from their experimental evaluation that is usually adopted.

Concerning plan length, Grt produced better plans than the other planners in the freecell domain, in the gripper domain (along with other planners), in many blocks problems when a 3-action schemas representation was used and in some logistics problems. Stan exhibited the best behavior in most of the domains and we believe this is due to its Graphplan basic architecture, which always produces optimal parallel plans and, in many cases, sequential plans also. Ff behaved well in the logistics and the blocks problems, with the 4-action schemas representation (in the latter case probably due to the goal ordering technique), however it produced lengthy plans in other domains, as the freecell, the gripper and the puzzle ones.

Hsp-2 produced longer plans than Grt in many domains, as for example the logistics, the freecell and the gripper domains and the blocks one, when a 3-actions representation was used. This observation means that in these domains the related facts employed by the Grt heuristic proved more valuable than the forward and repeated reconstruction of the Hsp-2 heuristic. Finally, Altalt has not been distinguished for the quality of its plans in any domain.

Our general impression from the experiments is that there are specific domains that favor specific planners. So, what is important is to investigate the reasons for that. We are currently working in exploring the internal characteristics of each domain, classifying them into more general categories that share common features, and associate these features with specific heuristic search techniques. A first attempt for a domain classification can also be found in (Hoffmann, 2001).

An alternative view of the above problem concerns the way a domain is encoded. The same planner in the same domain may alter its performance when a different representation is adopted. We faced this problem with the blocks-world, with the 4- and 3-actions schemas domain representations, where the performance of Grt varies significantly, while the performance of other planners is also altered. We also faced this problem with the elevator and movie domains, which were the motivation for the development of the domain enrichment technique. Our conviction is that domain-independent planning is strongly domain-representation dependent.

Concerning Grt, we plan to extend it so as to handle more expressive domains, supporting most of the features of the PDDL language (types, quantifications, negations, disjunctions, etc). At this time we are working with an extension of Grt, which has the ability to take into account multiple criteria (i.e. solution time, resources, safety, profit etc.). We are also interested in incorporating domain analysis techniques, as they have been developed in Stan and Discoplan, in order to take advantage of specialized methods for handling specific types of problems or sub-problems. Finally, we will investigate the possibility and the utility of combining domain independent planning techniques with domain dependent ones, without loosing the generality of the planning system.

 

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Next: Acknowledgments Up: The GRT Planning System Previous: Puzzle

Ioannis Refanidis

14-8-2001