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Benson's Algorithm
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Benson's algorithm is a rigorous and static (i.e. no search is needed) method for recognizing stones that are uncapturable even if the attacker is allowed to play an infinite number of times in a row (i.e. the defender always passes). This is called unconditional life in Benson's paper [1], a confusing name for something that has no equivalent in traditional Go terminology.

The algorithm is only used by a few specific life-and-death programs and modules.

Perhaps you could describe the algorithm? I have a rough mental idea of how it would work, but I'm not sure... -- Karl Knechtel

jvt: I don't want to paraphrase, so here is a completely rewritten and much simpler version.

"c" stands for Black or White, "-c" for the opposite color. Each point of the board is, of course, either black, white or empty.

A c chain is a maximum connected set of c points.

A c region is a maximum connected set of -c and empty points.

A c chain B is unconditionally alive iff it is adjacent to at least two c regions R that satisfy:

  1. all empty points in R are adjacent to B,
  2. all chains adjacent to R are unconditionally alive, barring circularity [2].

Note: proof depends on suicide being forbidden by the rules.


Sketch of Proof

(1) prevents the attacker from filling all liberties of an unconditionally alive chain in 2 adjacent regions, because of the no-suicide rule. (2) prevents the attacker from making liberties by capturing another chain adjacent to the same region. And two liberties ensure the chain will never be in atari.


Disclaimer

I don't claim these definitions are the same as in the original article [1] by David Benson. Besides I don't define small c-enclosed regions, healthy regions, nor vital regions.

The requirement about a c region's interior being filled with -c stones has been intentionally removed because of my different definition of regions.


Examples

[Diagram]
Diag.: 1: the white chain is unconditionally alive.

[Diagram]
Diag.: 2: none of the chains is unconditionally alive.


References

[1] Benson, D.B., Life in the Game of Go. Information Sciences, Vol.10, pp.17-29, 1976. Reprinted (with corrections) in

   Computer Games II
   David N. L. Levy (ed.)
   Springer-Verlag, 1987
   ISBN 0-387-96609-9

[ext] David Dyer's eye shape library


[2] Jan: Doesn't this introduce a circularity? Take this example:

[Diagram]
Diag.: Should be unconditionally alive, right?

There is only one chain, and three regions - one large one and two of one point each. However when I apply the definition, I get the following:

The large region doesn't count (not all empty points are adjacent to the chain) so we turn our attention to the two small regions. The first requirement checks out OK, but for the other one, we need to know whether our chain is unconditionally alive. But that was the question we were asking in the first place.

I think the second requirement needs to be reworded like this: all other chains adjacent to R are unconditionally alive Did I miss anything here?

jvt: You are right. But there is still some indirect circularity remaining because you can have c1 - R1 - c2 - R2 - c1. (c = chain, R = region).

Benson uses "transitive closure" to convey the meaning of recursivity without circularity. I am not sure how to express this more clearly. Is "barring circularity" clear enough?

Using these criteria "backwards" as an elimination test may seem more logical. A graph of regions and chains of the same color is generated, then it's like pulling a thread. False eyes don't satisfy the criteria and get eliminated. That's the way I did it in my Java programming experiments, anyway.

See also the Eyes Collection of pages.



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This is a copy of the living page "Benson's Algorithm" at Sensei's Library.
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