How to Relax a Bisimulation?

Abstract

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.

Non-Linear Merging Strategies for Merge-and-Shrink Based on Variable Interactions

“…Unlike the shrinking strategies (Helmert, Haslum, and Hoffmann 2007;Nissim, Hoffmann, and Helmert 2011;Katz, Hoffmann, and Helmert 2012), all the existing merging strategies make decisions independent of the initial state, i.e., given planning tasks that differ only in their initial states those methods produce the same merging order. CGL and RL depend only on causal graphs and variable level, which do not take initial states into consideration.…”

“…Previous studies of M&S have focused more on shrinking strategies (Nissim, Hoffmann, and Helmert 2011;Katz, Hoffmann, and Helmert 2012) than on merging strategies. The existing merging strategies used in planning include:…”

“…An abstraction A of a transition graph Θ = S, L, T, s 0 , S * is a pair Θ α , α , where Θ α = S α , L, T α , s α 0 , S α * is an abstract transition graph and α is an abstraction mapping, i.e., a function mapping S to S α such that (Katz, Hoffmann, and Helmert 2012). Let s ∈ S α be an abstract state in A.…”

“…The goal of shrinking is to reduce the number of abstract states of the two abstractions so that the size of their synchronized product is less than the size bound N . Several shrinking strategies have been studied (Helmert, Haslum, and Hoffmann 2007;Nissim, Hoffmann, and Helmert 2011;Katz, Hoffmann, and Helmert 2012).…”

MS-Lite: A Lightweight, Complementary Merge-and-Shrink Method

“…Non-greedy bisimulation shrinking does not always end up with the coarsest bisimulation abstractions but are guaranteed to produce h-preserving abstractions. In contrast, greedy bisimulation shrinking (Katz, Hoffmann, and Helmert 2012) refines the abstractions until they are bisimilar or runs out of resources, and is thus an active shrinking strategy. Label reduction is used mainly for reducing the bisimulation size (Sievers, Wehrle, and Helmert 2014).…”