Solving Ponnuki-Go on Small Board

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 • Introduction • Motivation • Method Summary • Results and Analysis • Conclusions

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

4. Motivation Why we study Atari-Go? • 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? • A good benchmark for testing the performance of algorithms • Successful algorithms in small board Atari-Go might be useful for computer Go

6. Outline • Introduction • Motivation • Method Summary • Results and Analysis • Conclusions

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

8. Transposition Table • Use the two-deep replacement scheme: • 225 (32M) double entries

9. History Heuristic • 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 • First the transposition move is tested • Second are the killer moves • Third the rest of the moves are ordered by the history heuristic

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

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

13. Maximizing Liberties and Territory • The number of liberties is a lower bound on the number of moves that is needed to capture a stone • 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 • Why should connect stones to a larger group? • A small number of larger groups is easier to defend than a large number of small groups • Making eyes is derived from normal Go. • After a player has run out of alternative moves, he might be forced to fill his own eyes

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

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

17. Euler Number • Minimizing the Euler Number thus connects stones as well as creates eyes E = 3 - 19 = - 16 E = 1 - 18 = - 17

18. Outline • Introduction • Motivation • Method Summary • Results and Analysis • Conclusions

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

20. First Play First Win? • 2x2 board: no • 3x3 board: yes • 4x4 board: no • 5x5 board: yes • 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 • 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 • Two alternative way are used for testing:

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

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

25. 6x6 board with Stable Starting • Still too difficult! (estimates that about one month of computation time!) • 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 • Experiment results show that, on larger boards the enhancements become increasingly effective

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

29. Program Performance • Against Rainer Schutze’s freeware “AtariGo 1.0” in 10x10 board, won most of the game • After adding an implementation about extending ladders, won all! • 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 • Solve the empty 6x6 board and solving the 8x8 board with crosscut starting • Since search extensions for ladders are essential for strong play on larger board, future work will focus on selective search-extensions • Test the algorithm in Go!

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