1. Life in the Game of Go David B. Benson Surveyed by Akihiro Kishimoto

2. Outline Motivations Preliminaries Safe Blocks Unconditional Live Unconditional Life Algorithm Conclusions

3. “My” Motivations I need an exact method to detect live stones for my tsumego-solver! Benson’s algorithm can statically detect safe blocks –Cheaper than building game-trees –Reduction of search space Why not surveying this paper?

4. Benson’s Contributions Define the notion of “unconditionally alive” for a set of stones Theoretically guarantee “unconditionally live” stones are never captured

5. Block: a set of stones with the same color connected one another Region: a connected set of points Liberty of b: empty points adjacent to b Preliminaries Block liberty Region

6. Safe Blocks Def. Block b is safe b is not captured even if the opponent makes infinite sequences of moves def

7. Example: Safe or Unsafe? SafeUnsafe

8. Unconditional Life (1 / 4) Small x-enclosed region R 1.No x stone in R 2.R is surrounded by x 3.Internal of Region must be an empty set or filled in x-stones Small black-region def Black block Border of Region White stoneInternal of Region

9. Unconditional Life (2 / 4) R is healthy for b 1.R: small x-enclosed region 2.Every empty vertex in R must be in liberty of b R healthy for black block def Black block Border of Region Internal of Region

10. Examples: Healthy or Not? Black-enclosed but not healty for either block Healthy for both blocks

11. Unconditional Life (3 / 4) R is vital to b in X 1.R is healthy for b 2.All the x-blocks neighboring R are in X R is vital to b1 in {b1, b2, b3} def b1 b2 b3

12. Unconditional Life (4 / 4) X is unconditionally alive All b in X have 2 vital regions R1 and R2 –C.f. we need two eyes to be alive. def

13. Examples: Unconditionally Alive or Not? Unconditionally aliveNot unconditionally alive

14. Relations between Safe & Unconditional Life Theorem1 –X: unconditionally alive all the blocks in X is safe. Theorem2 –X: set of all safe x-blocks X is unconditionally alive

15. How to Find Safe Blocks? 1.Compute the transitive closure of “joined” blocks Z = {b1,…,bk} 2. Compute x-enclosed S= {R1,…, Rk} neighboring these blocks 3.Update Z = {b | b has two healthy regions} 4.Update S to x-enclosed regions neighboring b in Z 5.Continue 3 & 4 to converge b1 and b2 is joined there’s x-enclosed region R neighboring b1 and b2 def x-enclosed b1 b2

16. Conclusions I’ll write Benson’s algorithm for my tsumego solver! –Better than doing a search –Not hard to implement –Good starting point to look at the weakness of the tsumego solvers