We have found the first optimal solutions to random instances of Rubik's Cube. The median optimal solution length appears to be 18 moves. The algorithm used is iterative-deepening-A* (IDA*), with a lowerbound heuristic function based on large memory-based lookup tables, or "pattern databases" (Culberson and Schaeffer 1996). These tables store the exact numberofmoves required to solve various subgoals of the problem, in this case subsets of the individual movable cubies. We characterize the effectiveness of an admissible heuristic function by its expected value, and hypothesize that the overall performance of the program obeys a relation in which the product of the time and space used equals the size of the state space. Thus, the speed of the program increases linearly with the amount of memory available. As computer memories become larger and cheaper, we believe that this approach will become increasingly cost-effective.

Abstraction techniques are important for solving constraint satisfaction problems with global constraints and low solution density. In the presence of global constraints, backtracking search is unable to prune partial solutions. It therefore operates like pure generate-and-test. Abstraction improves on generate-and-test by enabling entire subsets of the solution space to be pruned early in a search process. This paper describes how abstraction spaces can be characterized in terms of approximate symmetries of the original, concrete search space. It defines two special types of approximate symmetry, called "range symmetry" and "domain symmetry", which apply to function finding problems. It also presents algorithms for automatically synthesizing hierarchic problem solvers based on range or domain symmetry. The algorithms operate by analyzing declarative descriptions of classes of constraint satisfaction problems. Both algorithms have been fully implemented. This paper concludes by presenting data from experiments testing the two synthesis algorithms and the resulting problem solvers on NP-hard scheduling and partitioning problems.

Introduction Search is a universal problem-solving mechanism in artificial intelligence (AI). In AI problems, the sequence of steps required for solution of a problem are not known a priori, but often must be determined by a systematic trial-and-error exploration of alternatives. The problems that have been addressed by AI search algorithms fall into three general classes: single-agent pathfinding problems, two-player games, and constraint-satisfaction problems. Classic examples in the AI literature of pathfinding problems are the sliding-tile puzzles, including the 3 \Theta 3 Eight Puzzle (see Fig. 1) and its larger relatives the 4 \Theta 4 Fifteen Puzzle, and 5 \Theta 5 Twenty-Four Puzzle. The Eight Puzzle consists of a 3 \Theta 3 square frame containing eight numbered square tiles, and an empty position called the blank. The legal operators are to slide any tile that is h

In this paper we present a production system which acts on fixed length vectors of labels. Our goal is to automatically generate heuristics to search the state space for shortest paths between states efficiently. The heuristic values which guide search in the state space are obtained by searching for the shortest path in an abstract space derived from the definition of the original space. In PSVN, a state is a fixed length vector of labels and abstractions are generated by simply mapping the set of labels to another smaller set of labels (domain abstraction). A domain abstraction on labels induces a state space abstraction and this abstract space preserves important properties of the original space while usually being significantly smaller in size. It is guaranteed that the shortest path between two states in the original space is at least as long as the shortest path between their images in the abstract space. Hence, such abstractions provide admissible heuristics for search algorith...

A memory-based heuristic is a function, h(s), stored in the form of a lookup table (pattern database): h(s) is computed by mapping s to an index and then retrieving the appropriate entry in the table. (Korf 1997) conjectures for search using memory-based heuristics that m \Delta t is a constant, where m is the size of the heuristic's lookup table and t is search time. In this paper we present a method for automatically generating memorybased heuristics and use this to test Korf's conjecture in a large-scale experiment. Our results confirm that there is a direct relationship between m and t. Introduction A heuristic is a function, h(s), that computes an estimate of the distance from state s to a goal state. In a memory-based heuristic this computation consists of mapping s to an index which is then used to look up h(s) in a table. Even heuristics that have a normal functional definition are often precomputed and stored in a lookup table in order to speed up search ((Priedi...

Weighted deduction with aggregation is a powerful theoretical formalism that encompasses many NLP algorithms. This paper proposes a declarative specification language, Dyna; gives general agenda-based algorithms for computing weights and gradients; briefly discusses Dyna-to-Dyna program transformations; and shows that a first implementation of a Dyna-to-C++ compiler produces code that is efficient enough for real NLP research, though still several times slower than hand-crafted code. 1

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 Rubik’s 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. 1.

This thesis demonstrates how the power of symbolic processing can be exploited in the learning of low level control functions. It proposes a novel hybrid architecture with a tight coupling between a variant of symbolic planning and reinforcement learning. This architecture combines the strengths of the function approximation of subsymbolic learning with the more abstract compositional nature of symbolic learning. The former is able to represent mappings of world states to actions in an accurate way. The latter allows a more rapid solution to problems by exploiting structure within the domain. A control function is learnt over time through interaction with the world. Symbols are attached to features in the functions. The symbolic attachments act as anchor points used to transform the function of a previously learnt task to that of a new task. The solution of more complex tasks is achieved through composing simpler functions, using the symbolic attachments to determine the composition. T...

Evaluation functions are an essential component of practical search algorithms for optimization, planning and control. Examples of such algorithms include hillclimbing, simulated annealing, best-first search, A*, and alpha-beta. In all of these, the evaluation functions are typically built manually by domain experts, and may require considerable tweaking to work well. I will investigate the thesis that statistical machine learning can be used to automatically generate high-quality evaluation functions for practical combinatorial problems. The data for such learning is gathered by running trajectories through the search space. The learned evaluation function may be applied either to guide further exploration of the same space, or to improve performance in new problem spaces which share similar features. Two general families of learning algorithms apply here: reinforcement learning and meta-optimization. The reinforcement learning approach, dating back to Samuel's checkers player [ 1959 ...

The heuristics used for planning and search often take the form of pattern databases generated from abstracted versions of the given state space. Pattern databases are typically stored space, which limits the size of the abstract state space and therefore the quality of the heuristic that can be used with a given amount of memory. In the AIPS-2002 conference Stefan Edelkamp introduced an alternative representation, called symbolic pattern databases, which, for the Blocks World, required two orders of magnitude less memory than a lookup table to store a pattern database. This paper presents experimental evidence that Edelkamp’s result is not restricted to a single domain. Symbolic pattern databases, in the form of Algebraic Decision Diagrams, are one or more orders of magnitude smaller than lookup tables on a wide variety of problem domains and abstractions.