1 Solving Ponnuki-Go on Small Board Paper: Solving Ponnuki-Go on small board Authors: Erik van der Werf, Jos Uiterwijk, Jaap van den Herik Presented by: Niu Xiaozhen

2 Outline n Introduction n Motivation n Method Summary n Results and Analysis n Conclusions

3 Introduction n Ponnuki-Go (also known as Atari-Go), the goal is to be the first to capture one or more of the opponent’s stones n Two rules are different with Go: u Capturing directly ends the game u Passing is not allowed (no tie) n Simpler than Go (no ko-fights)

4 Motivation Why we study Atari-Go? u It contains major concepts of Go such as capturing stones, determining life or death and making territory

5 Motivation (2) Why we study Atari-Go? u A good benchmark for testing the performance of algorithms u Successful algorithms in small board Atari-Go might be useful for computer Go

6 Outline n Introduction n Motivation n Method Summary n Results and Analysis n Conclusions

7 Method Summary n Standard alpha-beta framework with many enhancements: u Iterative deepening Principal Variation Search (PVS) u Transposition table u History heuristic u Enhanced transposition cutoffs u Move ordering

8 Transposition Table n Use the two-deep replacement scheme: n 2 25 (32M) double entries

9 History Heuristic n History Heuristic employs one table for both black and white moves, utilizing the Go proverb “the important move of my opponent is important to me as well”

10 Move Ordering n First the transposition move is tested n Second are the killer moves n Third the rest of the moves are ordered by the history heuristic

11 Evaluation Function n Simple evaluation function is to use a three-valued scheme [1(win), 0(unknown), -1(loss)] n Efficient for small boards n Becomes useless for strong play on large boards

12 Evaluation Function (2) n Proposed heuristic evaluation function is based on four principles: u Maximizing liberties u Maximizing territory u Connecting stones u Making eyes

13 Maximizing Liberties and Territory n The number of liberties is a lower bound on the number of moves that is needed to capture a stone n Maximizing territory is a long-term goal since it allows the player put more stones inside his own territory (before filling it completely)

14 Connecting and Making Eyes n Why should connect stones to a larger group? u A small number of larger groups is easier to defend than a large number of small groups n Making eyes is derived from normal Go. u After a player has run out of alternative moves, he might be forced to fill his own eyes

15 Implementation n Use bit-boards for fast computation of the board features n Territory is estimated by a weighted sum of the number of first-, second- and third- order liberties

16 Implementation (2) n Connections and eyes are more costly to calculate than the liberties n Use Euler number to estimate the connections and eyes The Euler Number of a binary image is: u The number of objects minus the number of holes

17 Euler Number n Minimizing the Euler Number thus connects stones as well as creates eyes E = = - 16 E = = - 17

18 Outline n Introduction n Motivation n Method Summary n Results and Analysis n Conclusions

19 Results and Analysis n The program solved the empty square boards up to 5x5

20 First Play First Win? n 2x2 board: no n 3x3 board: yes n 4x4 board: no n 5x5 board:yes n 6x6 board:don’t know yet! Test on 6x6 board took a few weeks (before system crash), the solution is at least 24-ply deep!

21 Experiment Results n The table shows the winner, the depth (in plies) of the shortest solutions, the number of nodes, time and the effective branching factor

22 6x6 board n Two alternative way are used for testing:

23 Another Approach n In 2002, Cazenave solved Atari-Go on 6x6 with crosscut starting n Use Gradual Abstract Proof Search (GAPS) algorithm, which is an combination of alpha-beta with a clever threat-extension scheme n Proved a win at depth 17 in around 10 minutes

24 Comparison n The authors’ algorithm found the shortest win at depth 15 in a comparable time frame n Using the same search enhancements into GAPS, Cazenave also found the solution at depth 15 in 26 seconds

25 6x6 board with Stable Starting n Still too difficult! (estimates that about one month of computation time!) n Prove the black win (at the depth of 31) by manually playing the first move

26 Solutions for Non-empty 6x6 board

27 Impact of Search Enhancements n Experiment results show that, on larger boards the enhancements become increasingly effective

28 Comparison of Evaluation Functions n Authors’ heuristic evaluation function performs better!

29 Program Performance n Against Rainer Schutze’s freeware “AtariGo 1.0” in 10x10 board, won most of the game n After adding an implementation about extending ladders, won all! n Against an amateur 1D in a 9x9 board, sometimes the program was able to win, but most of the games was lost!

30 Future Work n Solve the empty 6x6 board and solving the 8x8 board with crosscut starting n Since search extensions for ladders are essential for strong play on larger board, future work will focus on selective search- extensions n Test the algorithm in Go!

31 Conclusions n Authors‘ conclusions: u solved Atari-Go on the 3x3, 4x4, 5x5 and some non-empty 6x6 boards u the combination of enhancements and the heuristic evaluation fucntion is effective n My conclusions: u Focusing on enhancements, or trying to solve larger board one by one might not be a right direction u We need something different!