A Library of Eyes in Go

1. A Library of Eyes in Go Author: Thomas Wolf, Matthew Pratola Dept of Mathematics Brock University Presented by: Xiaozhen Niu

2. Outline • Introduction • Bend-4-in-the-corner Positions • Position Generation • Position Evaluation • Unusual Positions • Influence of External Liberties • Summary

3. Introduction • Eye evaluation is important for life and death problem • Goal: generate a library of monolithic eyes positions with stored solutions

4. Bend-4-in-the-corner Positions • Definition • Bend-4-type positions • Characterization

5. Definition • If for a regular position, passing belongs to the best moves of both sides then either: • Position is unconditional dead, seki or alive, called settled • Position is not settled, having a ko status, now it is called calm

6. Bend-4-type Positions • Three properties: • Initial position does not have a ko-forbidden point • Defender is not unconditionally dead • Passing is the single best move for defender

7. Best First Moves • Bend-4-type positions have more than one solutions: • If playing elsewhere is beneficial then the single best move for attacker is to pass • If playing elsewhere is not beneficial, or attacker is asked to kill, then it should play the move leads to ko

8. Characterization • Four types of bend-4 positions:

9. Characterization (2)

10. Position Generation • Unique representations of three cases of eyes: • In the corner • On the edge • In the centre • Norm-locating procedure use shift, rotation and reflection repeatedly for a store position when encountering a new position

11. Generation • Use 64 bit hash code (Zobrist) to identify a monolithic eye position (made from White and Black is the attacker) • Start with three eyes of size 1 and increasing successively (around 200 million positions of size up to 11)

12. Generation (2) • The generation includes 4 steps: • At first all internally empty eyes • All possible legal combinations of white stones are generated • All possible legal combinations of black stones are generated • Positions including one or more ko-fights will be generated once for each empty potential ko point

13. Observations • Growth factors (Log (Number of positions)) are remarkably constant according to size • The number of positions only grow little by adding white stones • The number of extra positions due to initial ko grows faster then other numbers, and it’s highest for corner eyes

14. Position Evaluation • Use program GoTools, performs binary alpha-beta search • To answer whether a position is at least “worth one eye” so that it could live if it had another “eye”, evaluate two cases • The eye on its own • One extra external 1-point eye being attached • Status of eyes often depend on the number of external liberties • Computations are done for 0 to 31 external liberties for two cases

15. Observations

16. Observations • A large proportion of all eyes are unconditionally alive (do not need to store). Only store a small fraction of all eyes • The larger the eye, the more likely the status and/or the best move depend on the number of external liberties and the more records are needed for the eye

17. Optimizations • In evaluating, eye size n either keep its size, or it shrinks. Any smaller eye can be evaluated by using stored information • Evaluate eyes with many interior stones first and then empty ones later • In theory each case would be evaluated 128 times (2x2x32), but it can always be sped up • Consistency check: White must not do worse when having more external liberties or extra external eyes

18. Efficiency • Monolithic eye database is a single read-only file • A module in GoTools norm-transforms any monolithic eye encountered during a computation by performing a rotation, reflection and shift • Generates a hash code, look up the entry and decode the information (status and moves) • Time saving, but loading a large database is slow!

19. Unusual Positions • Calm positions: a position is not settled, and passing is one of the best move for both sides • Include: • Bend-4-type • Eyes of size 5, 6, 7…

20. Eyes of Size 5 • 4 such positions • All have exactly 1 external liberty and the status is that White needs 1 external ko-threat to live

21. Eyes of Size 6 • Two types of calm positions of size 6, one is in diagram 8 where White needs 1 ko-threat to live. In second type Black needs one ko-threat to kill

22. Eyes of Size 7 • Black needs exactly 1 external ko-threat to kill

23. Eyes of Size 8 • Black needs exactly 1 external ko-threat to kill

24. Eyes of Size 9 • 14: Black need 2 ko-threats to kill. 15: White needs an extra eye and 1 ko-threat to live. 16: with 2 or 0 external liberties it is calm (Black needs 2 or 1ko-threats). But with 1 external liberty it’s not calm for 1 ko-threat

25. Eyes of Size 10 • 17,18: Black needs 2 ko-threats; 19: Black needs 1 ko-threat; 20: White needs 1 ko-threat

26. Eyes of Size 11 • 21: Black needs 3 ko-threats to kill. 1 external liberty it’s not calm, 0 external liberty is calm with Black needs 2 ko-threats; 22: Black needs 2 ko-threats; 23: Black needs 1 ko-threat; 24: White needs 1 ko-threat

27. Influence of External Liberties • Solution strategy depends on the number of external liberties • The more external liberties, the more ambitious aim White can try, i.e. a complete win instead of a ko

28. Choices of Best Moves • For 1 or 2 external liberties, Black plays a1 or d3 is equivalent (both needs 1 ko-threat)

29. Choices of Best Moves (2) • With 3 external liberties: d3 is better (needs 1 ko-threat) than a1 (needs 2 ko-threats) • Values of passes: d3 (1 ko-threat and 4 passes of White) or a1 (2 ko-threat and 2 passes of White)?

30. Choices of Best Moves (3) • 4 external liberties: Black d3 is unconditional loss whereas a1 at least leads to a ko (needs 3 ko-threats)

31. Summary • Cover monolithic eyes of size up to 11 • All possible internal configurations of stones of both sides are considered • Each eye is evaluated and the best status/moves are provided • If results in ko, the number of ko-threats are determined