Discovering Admissible Heuristics by Abstracting and Optimizing: A Transformational Approach

We present an implemented model for discovering a class of state-space search heuristics. First, abstractions of a state-space problem are generated by dropping information from the problem definition. An optimal solution path for any such abstracted problem gives a lower bound on the true distance to the goal. This bound can be used as an admissible evaluation function for guiding the base-level search. Moreover, if the abstracted goal is unreach-able from an abstracted state, the original state can safely be pruned. However, using exhaustive search to evaluate the abstracted problem is generally too slow. Therefore, optimization is used to speed up the computation of the lower bound (or solvability test), for example by factoring the abstracted problem into independent subproblems. We analyze the conditions under which the resulting heuristic is faster than brute force search. Our implementation , named ABSOLVER, has several general transformations for abstracting and simplifying state-space problems, including a novel method for problem factoring. ABSOLVER appears to be the first mechanical generator of heuristics guaranteed to find optimal solution paths. We have used it to derive known and novel heuris-tics for various state space problems, including Rubik's Cube. Jersey. The opinions expressed in this paper arc those of the authors and do not reflect any policies, either expressed or implied, of any granting agency. We thank 1 Introduction Finding optimal (least cost) solutions to large state-space problems is generally intractable without good admissible heuristics (evaluation functions that return lower-bound estimates of distance to goal). When coupled with search algorithms that ensure optimality, like A* [Nilsson, 1980] or iterative-deepening A* (IDA*) [Korf, 1985a], such heuristics can reorder the search so that solutions are found much earlier. They can also reduce search by pruning states that lie more than a specified distance from the goal. This distance may be the exact solution length for problems where it is known a priori, an upper bound on acceptable solution length, or just infinity, in which case the predicate tests whether the goal is reachable at all from a given state. However, good admissible heuristics can be hard to find. For example, after extensive study, Korf was unable to find a single good heuristic evaluation function for Rubik's Cube [Korf, 1985b]. He concluded that "if there does exist a heuristic, its form is probably quite complex." The long-term goal of our research is to develop a system that can discover good admissible …