Solving Tsumego on Computers M2 Hirokazu Ishii Chikayama & Taura Lab

Agenda 1. Introduction 2. Related Work GoTools 3. Tsumego Solver Df-pn (Depth-First Proof-Number Search) Df-pn+ (Depth-First Proof-Number Search+) 4. Conclusion & Future Work

Agenda 1. Introduction 2. Related Work GoTools 3. Tsumego Solver Df-pn (Depth-First Proof-Number Search) Df-pn+ (Depth-First Proof-Number Search+) 4. Conclusion & Future Work

Computer Game Players Have been studied for many years  Easy to evaluate  Suitable to try out various basic technologies Were successful in some games  Othello, backgammon, chess… Great challenges remain in some games  Shogi, Go… I study on Go, especially tsumego.

Game Rule Go is a 2-players game. A move is played on a line intersection. If one or more stones are completely surrounded by other stones, they are captured. The purpose of Go is to conquer a larger part of the board than the opponent.

Game Rule Eye  An eye is an area within a group of stones which is completely surrounded by stones of the group.  If the group gets two eyes, they are absolutely alive.

Definition of Tsumi Variation of Tsumi  Semeai problem  Escape and Disconnect problem  Life and Death problem Goal is to distinguish whether there are two eyes or not.

Method AND/OR tree search  OR NODE It corresponds to a first player’s move. In order to prove tsumi of this node, we must prove that one node of child nodes has tsumi.  AND NODE It corresponds to a second player’s move. In order to disprove this node, we must disprove that all nodes of child nodes are tsumi.

Agenda 1. Introduction 2. Related Work GoTools 3. Tsumego Solver Df-pn (Depth-First Proof-Number Search) Df-pn+ (Depth-First Proof-Number Search+) 4. Conclusion & Future Work

GoTools Has been the best tsume-go solver for 15 years. Uses a depth-first search. Specializes in completely enclosed positions.

Heuristics Static rules  Most of the heuristic rules are static. It is at a particular auspicious point. It completes one or more eyes. ・・・  Static rules rate moves lower or higher. Dynamic rules  The moves refuting opponent moves at subsequent positions also get some credit.

Dynamic Rules Number  The sequence in which the moves are made. Letter  The field where the stone is placed. ○:OR NODE □:AND NODE 1A 2B 3B 4C 5D 6D

Agenda 1. Introduction 2. Related Work GoTools 3. Tsumego Solver Df-pn (Depth-First Proof-Number Search) Df-pn+ (Depth-First Proof-Number Search+) 4. Conclusion & Future Work

Tsumego Solver Important factors are to  Recognize the position  Generate candidate moves  Evaluate these moves and select the next move Evaluation of a tsumego is strictly determined life or death. The only method of finding a strict solution is to search.

Df-pn Search Depth-First Search  Memory and time are used efficiently.  It is necessary to set a threshold. Proof number Disproof number Best-First search Iterative Deepening  Searches are tried iteratively increasing the threshold.

Proof Number Proof number is the minimum number of descendant nodes which must be proven in order to prove the node. We can see proof number of the node as the minimum resource required for search. Proof number is effective for search because we want to search the most promising node.

Disproof Number Disproof number is the minimum number of descendant nodes that must be disproven in order to disprove the node. Disproof number is effective for search as well as the proof number.

Df-pn Search If n is a leaf node  when the value is true  when the value is false When n is an uninspected node  The node might be proven or disproven immediately when inspected

Df-pn Search If n is an internal node  when n is an OR node  when n is an AND node

Df-pn Search Is a depth-first search. Uses two kinds of threshold (proof number and disproof number) １.Assign where r is the root node

Df-pn Search ２.At each node n, the search process continues to search below n until (We call it the ending condition) or

Df-pn Search ３.If n is an OR node  At each node n, select the child n c with the minimum proof number and the child n 2 with the second minimum proof number. Search below n c with assigning

Df-pn Search ４.If n is an AND node  At each node n, select the child n c and the child n 2. Search below n c with assigning ５.If the ending condition holds, the search process returns to the parent node.

A Df-pn Search R D I E H BC F J (∞,∞) ○:OR NODE □:AND NODE G ・・・ [1,1] [1,3] [1,1] [30,1] 30 nodes (2,∞-2) [0,∞] [2,1] [1,3] (2,∞-2) [1,1] [1,3] (3,∞-2) [0,∞][∞,0] [2,2] (31,∞-1) (30,2) [0,∞] [1,1] (31,∞-1) [∞,0][0,∞]

Df-pn+ Intends to distinguish promising moves more accurately and to search them much more deeper. Uses two kinds of additional information during search.

Df-pn+ Two kinds of information  cost (dis)proof (n, n child ) The cost of inspection of n child starting from n.  h (dis)proof (n) Heuristic estimate of the cost to reach any proof solution from position n.

Df-pn+ The formula for calculating proof number and disproof number are modified from df-pn. If node n is an uninspected node

Df-pn+ If n is an internal node At each node n, search below n c with assigning

Bouzy’s 5/21 Algorithm Dilation  If the intersection is not surrounding opponent’s stones, then add to the intersection the number of own interim territory and stones. Erosion  Subtract the number of intersections with opponent’s stones or vacant.

Bouzy’s 5/21 Algorithm

4 Dilation + 2 Erosion

Bouzy’s 5/21 Algorithm In GNU Go ver. 2.6, it was extensively used  5 Dilation and 21 Erosion are used for territory.  5 Dilation and 10 Erosion are used for moyo.  4 Dilation and 0 Erosion are used for area. Higher Bouzy value means higher expectation to form an eye. Because two eyes are required to live, the second maximum Bouzy value may give a good criterion.

Agenda 1. Introduction 2. Related Work GoTools 3. Tsumego Solver Df-pn (Depth-First Proof-Number Search) Df-pn+ (Depth-First Proof-Number Search+) 4. Conclusion & Future Work

Conclusion & Future Work I showed…  Definition of tsumi  Feature of tsumego solver ‘GoTools’  Df-pn+  Bouzy’s 5/21 algorithm and its use in tsumego. Firstly, I have to finish implementing the program. I also plan to apply the combinatorial game theory to the program.