1. A Solution to the GHI Problem for Best-First Search D. M. Breuker, H. J. van den Herik, J. W. H. M. Uiterwijk, and L. V. Allis Surveyed by Akihiro Kishimoto

2. Outline What is the GHI problem? Some other solutions to GHI BTA (Base-Twin Algorithm) Experimental Results Conclusions

3. Introduction Transposition table –Enhances search algorithms Chess factor of 5 Checkers factor of 10 –Detect “transpositions” Example Save result in TT Reuse result

4. GHI Problem (1 / 2) Transposition table contains a flaw with repetitions –Path to reach a node is ignored E.g. Alpha-Beta + TT if (TTlookup(&n) == OK && n.LBOUND >= beta) { return n.LBOUND; }

5. GHI Problem (2 / 2): Example Assumptions: –Repetition == draw What is F’s score? –Win or draw? A B 1 st player2 nd player DC F G E W: win for 1 st player W W ?

6. Solutions Case: Depth-First Search GHI rarely happens in Alpha-Beta search Ignored in practice –E.g. alpha-beta search int alphabeta(node n, int alpha, int beta) { if (cycle (n) == TRUE) return 0; ……… }

7. Solutions: Case: ISshogi’s Tsume-Solver (1 / 2) Shogi: –Mate with 4 repetitions  Loss GHI really happens –Can’t solve some problems in Shogi Zuko take an ad hoc approach to avoid

8. Solutions: Case: ISshogi’s Tsume-Solver (2 / 2) Do not store a loss because of a repetition –Save the threshold of proof numbers If (Cycle(n) == true) 1.(n == root)  return a loss 2.Check if n is really a losing position Example 1 st player2 nd player Loss

9. Beta-Twin Algorithm (1 / 8) Solution for proof-number search [c.f. Allis:94] Idea: two identical positions can have different scores –Base node Can expand deeper –Twin node Link to base node

10. Beta-Twin Algorithm (2 / 8) A B DC F G E W W 1 st player2 nd player f c Uppercase: Beta node Lowercase: Twin node

11. Beta-Twin Algorithm (3 / 8) Prepare new concepts –Possible-draw: draw because of a repetition Can be a win –Draw: real draw Mark possible-draw and keep the depth (of the ancestor) if in a repeated position Select the most proving node among the nodes marked neither possible-draws nor draws

12. Beta-Twin Algorithm (4 / 8) A B C 1 st player2 nd player F DE de c =2 Mark as a possible draw Depth = 0 Depth = 1 Depth = 2 Depth = 3 Depth = 4 Depth = 5

13. Beta-Twin Algorithm (5 / 8) If all the children are marked as either possible-draw or draw: –Mark node under way as a possible draw –Back up the minimal possible-draw depth –Delete all the possible-draw marks from the children draw possible-draw

14. Beta-Twin Algorithm (6 / 8) A B C 1 st player2 nd player F DE de c Mark as a possible draw =2 Delete possible draw mark =2=3 =2 Take minimal possible-depth =2

15. Beta-Twin Algorithm (7 / 8) If (depth of node == possible draw depth) –Guaranteed to be a draw –Store a draw A BC possible-draw a =0 draw

16. Beta-Twin Algorithm (8 / 8) A B C 1 st player2 nd player F DE de c =2 Depth = 2 Store a draw

17. Experimental Results (1 / 2) Game: Chess problems –117 positions from Chess curiosities and Win at chess Algorithms: –Tree: Basic proof-number search –DAG: PN-search variant that handles DAG [Schijf:93] –DCG: PN-search variant that handles DCG incorrectly [c.f. Schijf:93] –BTA: Beta-Twin Algorithm

18. Experimental Results (2 / 2) # of positions solved Tree: 99 DAG: 102 DCG: 103 BTA: 107 Total nodes (out of 96) Tree: 4,903,374 DAG: 3,222,234 DCG: 2,482,829 BTA: 2,844,024

19. Conclusions Theirs: –Proposed a solution to GHI –Worked better than pn-search Mine: –Can BTA really achieve much better? In my experience the performance should not improve so much except for some special problems –Needs too complicated implementation