Driven by the previous discussion on types of merge strategies, we devise another hybrid merge strategy, which is again based on the causal graph. As a precomputation step, it computes the strongly connected components (SCCs) of the causal graph and decides on an order in which these SCCs should be considered. During the merge-and-shrink computation, the strategy then repeatedly merges all atomic factors for variables within an SCC, considering the SCCs in the specified order, which results in one product system for each SCC. From there on, the strategy merges these resulting products to form the final transition system.
This merge strategy, or rather this framework to enhance existing merge strategies which we call SCC framework, has several parameters: first, it requires to specify the already mentioned order in which SCCs should be considered for computing the product factors for each SCC (order of SCCs). Secondly, it needs a merge strategy that is able to decide on a merge order of the atomic factors for variables within SCCs (secondary merge strategy). Thirdly, it requires either another merge strategy that dictates the merge order of the product systems of SCCs, or alternatively, it again needs an order in which the SCCs are considered, which then can be used for linearly merging all product systems of SCCs (third parameter).
For the order of SCCs, we consider the following four variants: a topological and reverse topological sort of the SCCs, based on the derived directed graph where each SCC is a single “supervertex”, and two orders in which SCCs are sorted by size, either decreasing or increasing, breaking ties by the topological order just described. For the merge strategy that decides on merging the atomic factors for variables within SCCs (secondary merge strategy), we can use any merge strategy that is able to select a next pair of transition systems out of a subset of the atomic factors of a planning task. This is of course easy for score-based merge strategies like DFP or sbMIASM, but also possible for linear merge strategies, since they define a variable order we can use directly. For other precomputed merge strategies like MIASM, it is in general not clear how to extract a subtree of the precomputed merge tree that contains only the necessary (atomic) factors, since they are not necessarily part of the same subtree.
For the third parameter, i.e. the order in which the product systems should be merged, we can again use the same different orders of SCCs to define a linear merge order on the product systems of SCCs, or alternatively, any merge strategy capable of deciding on a merge order given the product systems of SCCs. This is again trivially possible for all score-based merge strategies, but more complicated for precomputed merge strategies. We can use linear merge strategies using the technique described above that maps each variable of the underlying variable order to the factor “representing” that variable, but for the same reasons as above, we do not know how to use MIASM for that purpose.
In our original work introducing the merge strategy SCC-DFP (Sievers et al.,2016), we only reported results for the variant that uses the topological sort for the order of SCCS and DFP for the secondary merge strategy and the third parameter, hence not interpreting the strategy as a framework where any simple merge strategy could be plugged in. In the experimental study of this thesis, we also evaluate the alternative orders of SCCs, and we do not only use DFP as the secondary merge strategy and for the third parameter, but also consider sbMIASM and the linear merge strategies in the way described above.6 For simplicity, we always use merge strategies
Abbreviation transformation strategy
CGGL linear merge strategy: causal graph goal level (Helmert, Haslum, & Hoffmann,2007) F shrink strategy: f-preserving (Helmert, Haslum, & Hoffmann,2007)
B shrink strategy: based on (approximating) bisimulation (Nissim, Hoffmann, & Helmert,2011) G shrink strategy: based on greedy bisimulation (Nissim, Hoffmann, & Helmert,2011) GCGL linear merge strategy: goal causal graph level (Nissim, Hoffmann, & Helmert,2011)
L linear merge strategy: (Fast Downward’s variable order) level (Nissim, Hoffmann, & Helmert,2011) RL linear merge strategy: (Fast Downward’s variable order) reverse level (Nissim, Hoffmann, & Helmert,2011) DFP non-linear merge strategy: due to Dräger, Finkbeiner, and Podelski (2006), adapted to planning by
Sievers, Wehrle, and Helmert (2014)
MIASM non-linear merge strategy: maximum intermediate abstraction size merge strategy (Fan, Müller, & Holte,2014) symm-X non-linear merge strategy: merge strategy X enhanced with factored symmetries, called symm by
Sievers, Wehrle, Helmert, Shleyfman, and Katz (2015)
sbMIASM non-linear merge strategy: score-based MIASM, called DYN-MIASM by Sievers, Wehrle, and Helmert (2016) SCC-X non-linear merge strategy: individually merging SCCs of the CG, using X as secondary merge strategy
(Sievers, Wehrle, & Helmert,2016)
Table 5.1.: Abbreviations of transformation strategies with a brief summary and their source in the literature, in chronological order.
for the third parameter (and not any order of SCCs that induces a linear merge order), and we always use the same merge strategy as for the second parameter. We call the resulting merge strategy SCC-X if using merge strategy X as secondary merge strategy.
5.5. Conclusions
To summarize this chapter, we provide an overview of the transformation strategies we discussed in this chapter and which we will use in our experimental study. Table5.1lists the transformation strategies in chronological order, for each showing the abbreviation we use, a brief summary, and the source in the literature.
We discussed several new merge strategies and defined different types of merge strategies that have different strengths and weaknesses:
• Precomputed merge strategies fix a merge tree before the computation of the merge-and- shrink algorithm. Their advantage is being able to capture maximal causal dependencies of planning tasks, but they fail to take into account the impact of other transformation strategies which comes apparent only during the merge-and-shrink computation. Further- more, they tend to be more complex to understand and to cause difficulties if combining them with other merge strategies. All linear merge strategies (CGGL, GCGL, L, RL), which can be understood as a variable order, and MIASM fall into this category.
• Stateless or score-based merge strategies are in some sense the opposite of precomputed merge strategies: they select the next pair of transition systems exclusively based on the current factored transition system and do not memorize any information across several merge decisions. They are myopic in the sense that the cannot plan ahead the current
strategies both as the secondary merge strategy and for the third parameter, because in that case all information is available from the beginning on.
merge decision, but they excel in taking into account the impact of other transformation strategies by greedily maximizing scores defined solely on the current factored transition system. Furthermore, they can usually be described mathematically and are simple to understand. They are also particularly flexible because they can easily be combined with each other. DFP and sbMIASM are representatives of this type of merge strategies. • Hybrid merge strategies are those that are neither entirely precomputed nor solely score-
based merge strategies. They can pick the best of both precomputed and score-based merge strategies, but they also have an disadvantage: they are usually similarly complex as precomputed merge strategies, and they also cannot easily be combined with other merge strategies. For example, it is unclear if there exists a meaningful way of combining a merge strategy with both factored symmetries and the SCC framework, which are the two ways of producing hybrid merge strategies we have seen.
In future work, besides investigating a possible combination of the frameworks for using fac- tored symmetries and SCCs, we would also like to come up with a better integration of MIASM and factored symmetries. Similarly, we want to combine MIASM with the SCC framework. Given that score-based merge strategies are simple and flexible, yet powerful, we also think that there are many more criteria that could serve as a basis for even more informative score-based merge strategies, such as, e.g., looking at the number of transitions or states of factors, their g- and h-values, or the amount of combinable labels to maximize label reductions.
Furthermore, the approach that both the SCC framework and MIASM follow can be viewed as an even wider framework: in a precomputation phase, it partitions the variables of the planning task according to some criteria, in the spirit of how the SCC framework partitions variables based on SCCs and how MIASM partitions variables based on the expected ratio of dead states. Then it first merges all atomic factors for variables within a partition before merging the resulting products to form the final transition system, using any secondary merge strategy for these two subtasks.