hm(P) = h1(Pm): Alternative Characterisations of the Generalisation From hmax To hm

The hm (m = 1, . . .) family of admissible heuristics for STRIPS planning with additive costs generalise the h heuristic, which results when m = 1. We show that the step from h to hm can be made by changing the planning problem instead of the heuristic function. This furthers our understanding of the hm heuristic, and may inspire application of the same generalisation to admissible heuristics stronger than h. As an example, we show how it applies to the additive variant of hm obtained via cost splitting. Introduction The h heuristic, introduced by Bonet and Geffner in 1999, is the first of the “new generation” of admissible heuristics for (propositional STRIPS) planning with additive action costs, and it is certainly the most widely known and understood. The relaxation underlying the h heuristic is taking the cost of achieving a conjunction of atoms to equal the cost of achieving the most costly single atom in the conjunction. This assumption makes the heuristic simple, conceptually and computationally, often allowing more advanced admissible heuristics to be related to h (e.g. Helmert and Domshlak 2009). However, it also limits the power of the heuristic. In particular, the h heuristic is invariant under delete relaxation and therefore bounded above by h, the optimal delete-relaxation heuristic. This implies, for instance, that the h estimate can not include the cost of any action more than once, even though an action may be needed an exponential number of times in a plan. The h (m = 1, . . .) family of admissible heuristics is based on the same relaxation as h, but parameterise it by the maximum size m of conjunctions considered. Thus h = h. This makes the h heuristics more powerful. For instance, form > 1, h is not bounded by h, and even equals the real optimal cost function h for sufficiently large m. However, the complexity of computing the h heuristic rises exponentially with m, and for practical purposes it is typically limited to m = 2 or m = 3. Thus, a case can be made for seeking ways to generalise admissible heuristics more powerful than h in a maner analogous to the way h generalises h. Copyright c © 2009, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. This paper presents a new characterisation of the h heuristics, by showing that h can be obtained as h (i.e., h) over a modified planning problem. That is, for a planning problem P we construct a new problem P and show that h(P ) = h(P). The size of P is polynomial in the size of P (though exponential inm). However, P does not preserve real plan costs, i.e., h(P) may be greater than h(P ). This has the undesirable implication that applying an admissible heuristic to P does not necessarily yield an estimate that is admissible for P . Alternative constructions are also sketched. The new characterisation does not directly lead to a practical way of generalising an arbitrary admissible heuristic from 1 to m. Nor is it a more efficient way to compute h: computing h(P) typically requires more time and memory than computing h(P ). What it does offer is some new insight into the mechanics of the h heuristic, which, hopefully, suggests how the step from 1 to m can be carried out. As an example, we apply this insight to generalise Helmert’s and Domshlak’s (2009) cost partitioning scheme for additive h to additive h.