A General Theory of Additive State Space Abstractions

TL;DR

This paper introduces a universal framework for additive state space abstractions, enabling their application to a wide range of puzzles and significantly improving search efficiency.

Contribution

It provides a general definition of additive abstractions, proves their correctness, and demonstrates their effectiveness on various complex puzzles.

Findings

Additive abstractions can be applied to puzzles like Rubik's Cube and Pancake puzzle.

Using these abstractions reduces search time by up to three orders of magnitude.

A test for heuristic accuracy can further improve search efficiency.

Abstract

Informally, a set of abstractions of a state space S is additive if the distance between any two states in S is always greater than or equal to the sum of the corresponding distances in the abstract spaces. The first known additive abstractions, called disjoint pattern databases, were experimentally demonstrated to produce state of the art performance on certain state spaces. However, previous applications were restricted to state spaces with special properties, which precludes disjoint pattern databases from being defined for several commonly used testbeds, such as Rubiks Cube, TopSpin and the Pancake puzzle. In this paper we give a general definition of additive abstractions that can be applied to any state space and prove that heuristics based on additive abstractions are consistent as well as admissible. We use this new definition to create additive abstractions for these testbeds and show experimentally that well chosen additive abstractions can reduce search time substantially for the (18,4)-TopSpin puzzle and by three orders of magnitude over state of the art methods for the 17-Pancake puzzle. We also derive a way of testing if the heuristic value returned by additive abstractions is provably too low and show that the use of this test can reduce search time for the 15-puzzle and TopSpin by roughly a factor of two.