The computational complexity of avoiding spurious states in state space abstraction

Abstraction is a powerful technique for speeding up planning and search. A problem that can arise in using abstraction is the generation of abstract states, called spurious states, from which the goal state is reachable in the abstract space but for which there is no corresponding state in the original space from which the goal state can be reached. Spurious states can be harmful, in practice, because they can create artificial shortcuts in the abstract space that slow down planning and search, and they can greatly increase the memory needed to store heuristic information derived from the abstract space (e.g., pattern databases). This paper analyzes the computational complexity of creating abstractions that do not contain spurious states. We define a property—the downward path preserving property (DPP)—that formally captures the notion that an abstraction does not result in spurious states. We then analyze the computational complexity of (i) testing the downward path preserving property for a given state space and abstraction and of (ii) determining whether this property is achievable at all for a given state space. The strong hardness results shown carry over to typical description languages for planning problems, including sas and propositional strips. On the positive side, we identify and illustrate formal conditions under which finding downward pathion is a powerful technique for speeding up planning and search. A problem that can arise in using abstraction is the generation of abstract states, called spurious states, from which the goal state is reachable in the abstract space but for which there is no corresponding state in the original space from which the goal state can be reached. Spurious states can be harmful, in practice, because they can create artificial shortcuts in the abstract space that slow down planning and search, and they can greatly increase the memory needed to store heuristic information derived from the abstract space (e.g., pattern databases). This paper analyzes the computational complexity of creating abstractions that do not contain spurious states. We define a property—the downward path preserving property (DPP)—that formally captures the notion that an abstraction does not result in spurious states. We then analyze the computational complexity of (i) testing the downward path preserving property for a given state space and abstraction and of (ii) determining whether this property is achievable at all for a given state space. The strong hardness results shown carry over to typical description languages for planning problems, including sas and propositional strips. On the positive side, we identify and illustrate formal conditions under which finding downward path ∗Corresponding author Email addresses: [email protected] (Sandra Zilles), [email protected] (Robert C. Holte) Most of this work was carried out while Sandra Zilles was at the University of Alberta. Preprint submitted to Elsevier July 2, 2010 preserving abstractions is provably tractable.