The game of checkers has roughly 500 billion billion possible positions (5 x 10(20)). The task of solving the game, determining the final result in a game with no mistakes made by either player, is daunting. Since 1989, almost continuously, dozens of computers have been working on solving checkers, applying state-of-the-art artificial intelligence techniques to the proving process. This paper announces that checkers is now solved: Perfect play by both sides leads to a draw. This is the most challenging popular game to be solved to date, roughly one million times as complex as Connect Four. Artificial intelligence technology has been used to generate strong heuristic-based game-playing programs, such as Deep Blue for chess. Solving a game takes this to the next level by replacing the heuristics with perfection.
The efficiency of A* searching depends on the quality of the lower bound estimates of the solution cost. Pattern databases enumerate all possible subgoals required by any solution, subject to constraints on the subgoal size. Each subgoal in the database provides a tight lower bound on the cost of achieving it. For a given state in the search space, all possible subgoals are looked up in the pattern database, with the maximum cost over all lookups being the lower bound. For sliding tile puzzles, the database enumerates all possible patterns containing N tiles and, for each one, contains a lower bound on the distance to correctly move all N tiles into their correct final location. For the 15-Puzzle, iterative-deepening A* with pattern databases (N = 8) reduces the total number of nodes searched on a standard problem set of 100 positions by over 1000-fold.
Despite recent progress in AI planning, many benchmarks remain challenging for current planners. In many domains, the performance of a planner can greatly be improved by discovering and exploiting information about the domain structure that is not explicitly encoded in the initial PDDL formulation. In this paper we present and compare two automated methods that learn relevant information from previous experience in a domain and use it to solve new problem instances. Our methods share a common four-step strategy. First, a domain is analyzed and structural information is extracted, then macro-operators are generated based on the previously discovered structure. A filtering and ranking procedure selects the most useful macro-operators. Finally, the selected macros are used to speed up future searches.We have successfully used such an approach in the fourth international planning competition IPC-4. Our system, Macro-FF, extends Hoffmann's state-of-the-art planner FF 2.3 with support for two kinds of macro-operators, and with engineering enhancements. We demonstrate the effectiveness of our ideas on benchmarks from international planning competitions. Our results indicate a large reduction in search effort in those complex domains where structural information can be inferred.
Many enhancements to the alpha-beta algorithm have been proposed to help reduce the size of minimax trees. A recent enhancement, the history heuristic, is described that improves the order in which branches are considered at interior nodes. A comprehensive set of experiments is reported which tries all combinations of enhancements to determine which one yields the best performance. Previous work on assessing their performance has concentrated on the benefits of individual enhancements or a few combinations. However, each enhancement should not be taken in isolation; one would like to find the combination that provides the greatest reduction in tree size. Results indicate that the history heuristic and transposition tables significantly out-perform other alpha-beta enhancements in application generated game trees. For trees up to depth 8, when taken together, they account for over 99% of the possible reductions in tree size, with the other enhancements yielding insignificant gains.
When solving instances of problem domains that feature a large branching factor, A* may generate a large number of nodes whose cost is greater than the cost of the optimal solution. We designate such nodes as surplus. Generating surplus nodes and adding them to the OPEN list may dominate both time and memory of the search. A recently introduced variant of A* called Partial Expansion A* (PEA*) deals with the memory aspect of this problem. When expanding a node n, PEA* generates all of its children and puts into OPEN only the children with f = f (n). n is re-inserted in the OPEN list with the f -cost of the best discarded child. This guarantees that surplus nodes are not inserted into OPEN. In this paper, we present a novel variant of A* called Enhanced Partial Expansion A* (EPEA*) that advances the idea of PEA* to address the time aspect. Given a priori domain- and heuristic- specific knowledge, EPEA* generates only the nodes with f = f(n). Although EPEA* is not always applicable or practical, we study several variants of EPEA*, which make it applicable to a large number of domains and heuristics. In particular, the ideas of EPEA* are applicable to IDA* and to the domains where pattern databases are traditionally used. Experimental studies show significant improvements in run-time and memory performance for several standard benchmark applications. We provide several theoretical studies to facilitate an understanding of the new algorithm.
This paper discusses a new work-scheduling algorithm for parallel search of single-agent state spaces, called Transposition-Table-Driven Work Scheduling, that places the transposition table at the heart of the parallel work scheduling. The scheme results in less synchronization overhead, less processor idle time, and less redundant search effort. Measurements on a 128-processor parallel machine show that the scheme achieves close-to-linear speedups; for large problems the speedups are even superlinear due to better memory usage. On the same machine, the algorithm is 1.6 to 12.9 times faster than traditional work-stealing-based schemes.
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