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Temperature Discovery Search

Temperature Discovery Search (TDS) is a game tree search method using AI techniques to compute or estimate the temperature of combinatorial games through forward minimax search, offering a systematic approach and trade-off control. It significantly outperforms AB searchers in practical play. It evaluates terminal positions, move generation, and mean computation.

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Temperature Discovery Search

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  1. Temperature Discovery Search Martin M¨uller and Markus Enzenberger and Jonathan Schaeffer Temperature Discovery Search (TDS) is a new minimaxbased game tree search method designed to compute or approximate the temperature of a combinatorial game. TDS is based on the concept of an enriched environment.

  2. Introduction • Solve games by minimax search typically increases exponentially with the required search depth. • One example are combinatorial games which can be broken down into a sum of independent subgames. • Go endgames can be analyzed as combinatorial games.

  3. A method for discovering the temperature of a local game by a forward minimax search. The contributions of this paper are: • A novel use of AI search techniques to solve a problem in the domain of combinatorial games. • The first general forward search method to compute the mean and temperature of any finite loop-free combinatorial game.

  4. The first known systematic method to approximate mean and temperature by forward search. • Several techniques to control the trade-off between speed and accuracy, and their empirical evaluation. • A series of experiments in practical play of hot sum games, showing that TDS in combination with the Hotstrat algorithm convincingly outperforms an ab searcher.

  5. Combinatorial Games • A major difference between normal ab minimax search and local analysis of a sum game is that while ab solves a single optimization problem, in a sum game there is a trade-off between a local gain and the right to play first in another game. • Combinatorial game theory provides tools to quantify this trade-off.

  6. Temperature Discovery Search • Searching a Sum Game G + C • Move generation • All moves in G plus one move in C. • Terminal Positions • Two types of positions are leaf nodes • normal terminal position • pseudo-terminal position • Evaluation

  7. Evaluation • 1. Evaluation of G: • Normal terminal positions are evaluated by their exact integer value. • Nonterminal positions in G can be evaluated by a heuristic evaluation function h(G) to improve move ordering or enable heuristic TDS • 2. Coupons taken: • The evaluation is the sum of all coupons taken by Left minus the sum of all coupons taken by Right. • 3. Remaining coupons: • The evaluation is V(C , p) . This assumes that both players take the remaining coupons in turn, beginning with the player to move.

  8. Example • C ( 17/8 ) • C ( 16/8 ) • C ( 15/8 ) • C ( 14/8 ) • C ( 13/8 ) • C ( 12/8 ) • C ( 11/8 ) • A2–A3*B2 • C ( 10/8 ) • C ( 9/8 ) • C ( 8/8 ) • C ( 7/8 ) • C ( 6/8 ) • C ( 5/8 ) • C ( 4/8 ) • C ( 3/8 ) • C ( 2/8 ) • C ( 1/8 ) • C ( 0 ) • C ( -1/8 ) • C ( -2/8 ) • C ( -3/8 ) • C3–B4*C3.

  9. Determining the mean: V( G+C,p)=μ(G)+V(C,p) V(C,p)=┌ 17/2 ┐* (1/8) = 9/8 μ(G)=15/8-9/8=6/8 • Determining the temperature:

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