Planning with Multi-Valued Landmarks

Landmark heuristics are perhaps the most accurate current known admissible heuristics for optimal planning. A disjunctive action landmark can be seen a form of at-least-one constraint on the actions it contains. In many domains, some critical propositions have to be established for a number of times. Propositional landmarks are too weak to express this kind of constraints. In this paper, we propose to generalize landmarks to multi-valued landmarks to represent the more general cardinality constraints. We present a class of local multi-valued landmarks that can be efficiently extracted from propositional landmarks. By encoding multi-valued landmarks into CNF formulas, we can also use SAT solvers to systematically extract multi-valued landmarks. Experiment evaluations show that multivalued landmark based heuristics are more close to h∗ and compete favorably with the state-of-the-art of admissible landmark heuristics on benchmark domains. The most accurate current admissible heuristics for optimal sequential planning are based on landmarks (Helmert and Domshlak 2009). A (disjunctive) action landmark for Π is a set of actions that has non-empty intersection with every solution plan for Π. Action landmarks can be seen as a simple form of cardinality constraints: at least one of its actions must be included in any plan. In many domains, propositions have to be established for many times, for example, in the blocks world domain, if the length of an optimal plan π∗ for Π is n, the proposition HANDEMPTY will be established for at least n/2 times. Since only PUTDOWN and STACK actions have HANDEMPTY as add effects, the size of the action landmark that contains all PUTDOWN and STACK actions has a lower bound of n/2 . Propositional landmarks are too weak for these cases. In this paper, we introduce multi-valued landmarks to model these cardinality constraints. Definition 1. A multi-valued landmark l for planning task Π is a set of actions {a1, · · · , ak} satisfying the constraint ∗Funded by Nanjing Univ. graduate school program(2011CL7). †Funded by 973 program(2011 CB505300), NSFC(61021062, 61105069), Jiangsu Key R&D program(BE2011171, BE2012161). Copyright c © 2013, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. lb ≤ |l ∩ π| ≤ ub for any plan π. The lb and ub is referred to as l’s lower bound and upper bound respectively. One particular example of multi-valued landmarks is the global multi-valued landmark L that consists of all actions in A: L = {a|a ∈ A}. A collection of action landmarks is complete for Π if the cost of a minimum-cost hitting set equals h∗(Π) (Bonet and Castillo 2011). The minimal cost hitting set is an optimal plan forΠ. Similarly, a multi-valued landmark set L can also defined to be complete forΠ if the tight lower bound induced byL is equal to h∗(Π). It can be shown that the global multivalued landmark set {L} for planning taskΠ is complete for its delete relaxation Π. In planning tasks with delete effects, actions might be required more than once. {L} is not complete for these tasks. However, if we assume that the length of satisficing plan π is bounded, the number of each action also has a upper bound |π|. We can construct a complete multi-valued landmark set for Π by repeating each action for |π| times. L′ = {a@t|t ∈ [0, |π|), a ∈ A} It can also be easily shown that {L′} is complete for Π. Given a complete global landmark set {L} for Π, we can exploit L to find an optimal plan for Π by encoding L into CNF formula as an at-most-k constraint. F (L) = CNF (Π) ∧ CNF ( ∑