How to Relax a Bisimulation?

Merge-and-shrink abstraction (M&S) is an approach for constructing admissible heuristic functions for cost-optimal planning. It enables the targeted design of abstractions, by allowing to choose individual pairs of (abstract) states to aggregate into one. A key question is how to actually make these choices, so as to obtain an informed heuristic at reasonable computational cost. Recent work has addressed this via the well-known notion of bisimulation. When aggregating only bisimilar states – essentially, states whose behavior is identical under every planning operator – M&S yields a perfect heuristic. However, bisimulations are typically exponentially large. Thus we must relax the bisimulation criterion, so that it applies to more state pairs, and yields smaller abstractions. We herein devise a fine-grained method for doing so. We restrict the bisimulation criterion to consider only a subset K of the planning operators. We show that, if K is chosen appropriately, then M&S still yields a perfect heuristic, while abstraction size may decrease exponentially. Designing practical approximations for K, we obtain M&S heuristics that are competitive with the state of the art. Introduction Heuristic forward state-space search with A and admissible heuristics is a state of the art approach to cost-optimal domain-independent planning. The main research question in this area is how to derive the heuristic automatically. That is what we contribute to herein. We design new variants of the merge-and-shrink heuristic, short M&S, whose previous variant (Nissim, Hoffmann, and Helmert 2011b) won a 2nd price in the optimal planning track of the 2011 International Planning Competition (IPC), and was part of the 1st-prize winning portfolio (Helmert et al. 2011). M&S uses solution cost in a smaller, abstract state space to yield an admissible heuristic. The abstract state space is built incrementally, starting with a set of atomic abstractions corresponding to individual variables, then iteratively merging two abstractions (replacing them with their synchronized product) and shrinking them (aggregating pairs of states into one). In this way, M&S allows to select individual pairs of (abstract) states to aggregate. A key question, that governs both the computational effort taken and the quality of the resulting heuristic, is how to actually select these state pairs. Copyright c © 2012, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. M&S was first introduced for planning by Helmert et al. (2007), with only a rather naı̈ve method for selecting the state pairs to aggregate. Nissim et al. (2011a) more recently addressed this via the notion of bisimulation, adopted from the verification literature (e. g., (Milner 1990)). Two states s, t are bisimilar, roughly speaking, if every transition label (every planning operator) leads into equivalent abstract states from s and t. If one aggregates only bisimilar states, then the behavior of the transition system (the possible paths) remains unchanged. This property is invariant over both the merging and shrinking steps in M&S, and thus the resulting heuristic is guaranteed to be perfect. Unfortunately, bisimulations are exponentially big even in trivial examples, including benchmarks like, for example, Gripper. A key observation made by Nissim et al. is that bisimulation is unnecessarily strict for our purposes. In verification, paths must be preserved because the to-be-verified property shall be checked within the abstracted system. However, here we only want to compute solution costs. Thus it suffices to preserve not the actual paths, but only their cost. Nissim et al. design a label reduction technique, that changes the path inscriptions (the associated planning operators) but not their costs. This leads to polynomial behavior in Gripper and some other cases, but the resulting abstractions are still much too large in most planning benchmarks. Nissim et al. address this by (informally) introducing what they call greedy bisimulation, which “catches” only a subset of the transitions: s, t are considered bisimilar already if every transition decreasing remaining cost leads into equivalent abstract states from s and t. That is, “bad transitions” – those increasing remaining cost – are ignored. This is a lossy relaxation to bisimulation, i. e., a simplification that results in smaller abstractions but may (and usually does) yield an imperfect heuristic in M&S: “bad transition” is defined locally, relative to the current abstraction, which does not imply that the transition is globally bad. For example, driving a truck away from its own goal may be beneficial for transporting a package. Under such (very common) behavior, greedy bisimulation is not invariant across the M&S merging step, because the relevant transitions are not caught. We herein adopt the same approach for relaxing bisimulation – we catch a subset of the transitions – but we take a different stance for determining that subset. We first select a subset of labels (operators). Then, throughout M&S, ha l-0 07 65 02 7, v er si on 1 14 D ec 2 01 2 Author manuscript, published in "22nd International Conference on Automated Planning and Scheduling (ICAPS) (2012)"