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Imai Laboratory Introduction Game Tree Search The Purpose of Game Research Clear Goal Beat Human Champion Show the Ability of Computers Current Strength of Computers Fair Comparison Strength ! Difficulty Enormous Search Space Enough Speed for Interactive Play Checkers : Defeated Human Champion in 1994 Solved in 2007 Reversi : Defeated Human Champion 6-0 in 1996 Chess : Defeated Human Champion in 1997 Shogi (Japanese Chess) : Only some hundreds of people can beat strongest programs Go Ideal Test Bed for AI research : Strongest Programs are at weak 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 Max node 1 player tries to maximize the score, while the other tries to minimize the score. (Zero-Sum) 65 Min node 65 62 65 90 62 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. 83 Original Search Space = N 65 35 90 49 62 30 83 Alpha-Beta Search Space 80 Df-pn (Depth First Proof Number Search) PNS (Proof Number Search) [1] is a strong algorithm for AND/OR tree search 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 In other words, players try to minimize the opponent’s options min 3 Represents defender’s turn sum 1 1 1 1 Df-pn [2] is a depth-first version of proof number search th∞,th∞ th2,th∞-1 (2,2) th4,th∞-1 a (3,1) Represents attacker’s turn AND node Attacker’s win if all the child nodes are WIN for the attacker 2 n 2 m OR node Attacker’s win if one of the child nodes is a WIN N th3 th2 b (2,1) th3 c (1,2) th3 d (1,2) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) Df-pn uses 2 thresholds each for proof numbers and disproof numbers 1 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