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In August 2004, we thought the proof was complete, but an inspection showed that some trivial positions with an advantage of 7 or more checkers had been eliminated from the proof

Solving Checkers

IJCAI, pp.292-297, (2005)

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Abstract

AI has had notable success in building high- performance game-playing programs to compete against the best human players. However, the availability of fast and plentiful machines with large memories and disks creates the possibility of a game. This has been done before for simple or relatively small games. In this paper, we present new id...More

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Introduction
  • High-performance game-playing programs have been a major success story for AI. In games such as chess, checkers, Othello, and Scrabble, mankind has been humbled by the machine.
  • The game is ultra-weakly solved and a strategy is known for achieving the game-theoretic value from the opening position, assuming reasonable computing resources.
  • Computations from the end of the game backward have resulted in a database of © ( !¥£ positions ( pieces on the board) for which the game-theoretic value has been computed.
Highlights
  • High-performance game-playing programs have been a major success story for AI
  • A persistent B-tree is used to store information on every position examined in every opening, including both the back-end results and the backed-up values computed by the front end
  • The White Doctor opening was chosen as the first candidate to solve because of its importance in tournament practice to the checkers-playing community
  • The White Doctor proof began in November 2003 and a heuristic proof for a threshold of 125 was completed one month later
  • For the eight months, computer cycles were spent increasing the heuristic threshold to a win and verifying that there were no bugs in the proof.2
  • In August 2004, we thought the proof was complete, but an inspection showed that some trivial positions with an advantage of 7 or more checkers had been eliminated from the proof
Results
  • Subsets of a game with a small number of pieces on the board can be exhaustively enumerated to compute which positions are wins, losses or draws.
  • The front-end manager maintains a large persistent proof tree, and selects, stores, and organizes the search results.
  • The proof manager saves all back-end search results so that they can be re-used for searches with a different heuristic threshold.
  • A persistent B-tree is used to store information on every position examined in every opening, including both the back-end results and the backed-up values computed by the front end.
  • The heuristic search value is used to determine whether the current position is relevant to the current proof threshold.
  • Values outside the threshold range are considered as likely wins/losses by the proof tree manager.
  • If the heuristic value is more than twice the size of the threshold, no further processing is done; the formal proof search is postponed until later in the proof when the authors will have more confidence that the authors really need this result.
  • It is possible that the value of a search is influenced by a draw-byrepetition score, and this result is saved in the proof tree or in a transposition table.
  • In the front-end manager, GHI is avoided by not saving any value in the proof tree that is unsafely influenced by a draw-byrepetition score somewhere in its subtree.
  • This holds even for a fairly high initial heuristic threshold, where likely wins/losses are positions that human checkers experts agree are “game over”.
Conclusion
  • The leaf node in the tree is the result of a 100 second proof number search, terminating in an endgame database position.
  • The analysis of this opening evolved over 75 years, had many human analysts contributing, requires analysis of lines that are over 50 ply long, and requires extensive insight into the game to assess positions and identify lines to explore.
  • To solve checkers for the initial starting position, with no moves made, roughly 50 openings need to be computed (¦ ̈§ cutoffs eliminate most of the openings).
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