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Enormous Search Space Enough Speed for Interactive Play

65. 65. 62. 65. 90. 62. 83. 65. 35. 90. 49. 62. 30. 83. 80. 2. n. Alpha-Beta Search Space . min. 2. m. 3. sum. 1. 1. 1. 1. 1. th ∞ ,th ∞. ( 2 , 2 ). th 2 ,th ∞-1. th 4 ,th ∞-1. a. b. ( 3 , 1 ). ( 2 , 1 ). th 3 th 2. th 3 th 3. ( 1 , 2 ). ( 1 , 2 ). c. d.

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Enormous Search Space Enough Speed for Interactive Play

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  1. 65 65 62 65 90 62 83 65 35 90 49 62 30 83 80 2 n Alpha-Beta Search Space min 2 m 3 sum 1 1 1 1 1 th∞,th∞ (2,2) th2,th∞-1 th4,th∞-1 a b (3,1) (2,1) th3 th2 th3 th3 (1,2) (1,2) c d (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) Imai Laboratory Introduction Game Tree Search The Purpose of Game Research Current Strength of Computers Fair Comparison Clear Goal Checkers : Defeated Human Champion in 1994 Beat Human Champion Show the Ability of Computers Strength ! Solved in 2007 Reversi : Defeated Human Champion 6-0 in 1996 Difficulty Chess : Defeated Human Champion in 1997 Enormous Search Space Enough Speed for Interactive Play Shogi (Japanese Chess) : Only some hundreds of people can beat strongest programs Go : Strongest Programs are at weak Ideal Test Bed for AI research amateur player level (about 1dan) Alpha-Beta search and its variants 2 Person Zero-Sum Perfect Information Games can be solved by mini-max search 1 player tries to maximize the score, while the other tries to minimize the score. (Zero-Sum) Max node Min node There is no need to search all nodes, to find out a provably optimal leaf node. Using alpha-cut and beta-cut, we can reduce search space. Original Search Space = N Df-pn (Depth First Proof Number Search) Df-pn[2] is a depth-first version of proof number search PNS (Proof Number Search) [1] is a strong algorithm for AND/OR tree search OR node Attacker’s win if one of the child nodes is a WIN Represents attacker’s turn Proof number : The lower bound of the number of nodes which is needed to be proved in order to prove (attacker’s WIN) Attacker expands from the node with the smallest proof number AND node Attacker’s win if all the child nodes are WIN for the attacker Represents defender’s turn Df-pn uses 2 thresholds each for proof numbers and disproof numbers In other words, players try to minimize the opponent’s options Works fast with small size of memory Df-pn and it’s variants are currently the best algorithms for checkmate search. Df-pn+ [2] : Df-pn with heuristic (dis)proof number generation. Best Tsume-shogi solver. Df-pn(r) [3] : Df-pn+ with exact repetition handling. Best TsumeGo solver. Df-pn(l) [4] : Combination of Df-pn+ and l search. Solves Capturing Problem in Go. References (Imai lab. Members in Red) [1] Victor Allis, “Searching for Solutions in Games and Artificial Intelligence,” Ph.D. Thesis, University of Limburg, Maastricht, 1994. [2] Ayumu Nagai, “Df-pn Algorithm for Searching AND/OR Trees and Its Applications”, Ph.D. Thesis, University of Tokyo, 2002. [3] Akihiro Kishimoto, “Correct and Efficient Search Algorithms in the Presence of Repetitions,” Ph.D. Thesis, University of Alberta, 2005 . [4] Kazuki Yoshizoe, Akihiro Kishimoto and Martin Müller, “Lambda Depth-First Proof Number Search and Its Application to Go,” In Proceedings of 20th International Joint Conference on Artificial Intelligence (IJCAI-07), pages 2404-2409, 2007

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