A Library of Eyes in Go
Author: Thomas Wolf, Matthew Pratola
Dept of Mathematics
Brock University
Presented by: Xiaozhen Niu
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Outline
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Introduction
Bend-4-in-the-corner Positions
Position Generation
Position Evaluation
Unusual Positions
Influence of External Liberties
Summary
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Introduction
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Eye evaluation is important for
life and death problem
Goal: generate a library of
monolithic eyes positions with
stored solutions
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Bend-4-in-the-corner Positions
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Definition
Bend-4-type positions
Characterization
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Definition
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If for a regular position, passing belongs to
the best moves of both sides then either:
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Position is unconditional dead, seki or alive, called
settled
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Position is not settled, having a ko status, now it is
called calm
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Bend-4-type Positions
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Three properties:
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Initial position does
not have a koforbidden point
Defender is not
unconditionally dead
Passing is the single
best move for
defender
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Best First Moves
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Bend-4-type positions have more than one
solutions:
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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
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Characterization
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Four types of bend-4 positions:
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Characterization (2)
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Position Generation
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Unique representations of three cases of eyes:
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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
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Generation
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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)
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Generation (2)
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The generation includes 4 steps:
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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
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Observations
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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
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Position Evaluation
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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
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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
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Observations
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Observations
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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
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Optimizations
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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
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Efficiency
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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!
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Unusual Positions
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Calm positions: a position is not settled, and
passing is one of the best move for both sides
Include:
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Bend-4-type
Eyes of size 5, 6, 7…
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Eyes of Size 5
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4 such positions
All have exactly 1
external liberty and
the status is that
White needs 1
external ko-threat
to live
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Eyes of Size 6
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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
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Eyes of Size 7
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Black needs exactly 1 external ko-threat to kill
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Eyes of Size 8
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Black needs exactly 1 external ko-threat to kill
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Eyes of Size 9
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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
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Eyes of Size 10
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17,18: Black needs
2 ko-threats; 19:
Black needs 1 kothreat; 20: White
needs 1 ko-threat
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Eyes of Size 11
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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
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Influence of External Liberties
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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
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Choices of Best Moves
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For 1 or 2 external liberties, Black plays a1 or d3 is
equivalent (both needs 1 ko-threat)
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Choices of Best Moves (2)
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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)?
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Choices of Best Moves (3)
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4 external liberties: Black d3 is unconditional loss whereas a1
at least leads to a ko (needs 3 ko-threats)
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Summary
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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
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