Additive Counterexample-Guided Cartesian Abstraction Refinement

We recently showed how counterexample-guided abstraction refinement can be used to derive informative heuristics for optimal classical planning. In this work we introduce an algorithm for building additive abstractions and demonstrate that they outperform other state-of-the-art abstraction heuristics on many benchmark domains. Introduction Recently, we presented a new algorithm for deriving admissible heuristics for classical planning (Seipp and Helmert 2013) based on the counterexample-guided abstraction refinement (CEGAR) methodology (Clarke et al. 2000). Starting from a coarse abstraction of a planning task, we iteratively compute an optimal abstract solution, check if and why it fails for the concrete planning task and refine it so that the same failure cannot occur in future iterations. After a given time or memory limit is hit, we use the abstraction as an admissible search heuristic. As the number of CEGAR iterations grows, one can observe diminishing returns: it takes more and more iterations to obtain further improvements in heuristic value. Therefore, we propose building multiple smaller additive abstractions instead of a single big one and show that this increases the number of solved benchmark tasks. Background We consider optimal planning in the classical setting, using a SAS-like (Bäckström and Nebel 1995) representation. Definition 1. Planning tasks. We define a planning task as a 5-tuple Π = 〈V,O, c, s0, s?〉. • V is a finite set of state variables vi, each with an associated finite domain D(vi). A fact is a pair 〈v, d〉 with v ∈ V and d ∈ D(v). A partial state is a function s defined on a subset of V . This subset is denoted by Vs. For all v ∈ Vs, we must have s(v) ∈ D(v). Partial states defined on all variables are called states, and S(Π) is the set of all states of Π. The update of partial state s with partial state t, s⊕ t, is the partial state defined on Vs∪Vt which agrees with t on all v ∈ Vt and with s on all v ∈ Vs \ Vt. Copyright c © 2013, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. • O is a finite set of operators. Each operator o has a precondition pre(o) and effect eff(o), which are partial states. The cost function c assigns a cost c(o) ∈ N0 to each operator. • s0 is the initial state and s? is a partial state, the goal. A planning task Π = 〈V,O, c, s0, s?〉 induces a transition system with states S(Π), labelsO, initial state s0, goal states {s ∈ S(Π) | s? ⊆ s} and transitions {〈s, o, s⊕ eff(o)〉 | s ∈ S(Π), o ∈ O, pre(o) ⊆ s}. Optimal planning is the problem of finding a shortest path from the initial to a goal state in the transition system induced by a planning task, or proving that no such path exists. For a formal definition of transition systems we refer to Seipp and Helmert (2013).