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General Eye Definition
Path: EyesCollection   · Prev: EyeInTheBelly   · Next: EyeLiberties
    Keywords: Life & Death

Here is an attempt to give a general (not specific) definition of what is an eye (or sometimes called real eye). It is slightly modified from the 2/4 - 1/2 - 1/1 rule. A rigorous definition is attempted in the Eye Definition Discussion.

If a group has two or more real eyes, then it is a living group.

Note that the concept of miai is deliberately left out in this definition.

Also see the two-headed dragon where a living group is formed using two false eyes.


General Definition of an Eye

To qualify as an eye, the group must satisfy the following two conditions. If only the first condition is satisfied but not the second, then it is deemed as a false eye.

  • The group must surround at least one empty point.
  • The group must posess the following properties described below.

Confused: Here's an alternative wording for second condition which should work better:

  • No subset of the group can be captured without capturing the whole group.

Jasonred OOOH!!! I find that really brilliant, man! Not only is it brief, exact, and clear, but you can actually program it into a computer! Now you can explain eyes and two eyes easily. Dude, you might have just revolutionized Go programming! (well, it'll help if I want to make a go program anyhow, much easier than the these points for sides, these points for corners, etc. thing.

Bill: Confused is right. I do not know of any way to distinguish real eyes from false eyes in general without reference to play. There are many examples that can be distinguished without reference to play, but a general algorithm requires determining whether a cutting stone can be captured without reading.


(The properties described below will use black eyes, please switch the colours around if we are discussing white eyes.)

[Diagram]
An eye in the centre

For a Black eye in the center, three out of the four marked points must be unavailable to White. The corollary to this is that Two Corners Kills The Eye.


[Diagram]
An eye at the side

For a Black eye at the side, both marked points must be unavailable to White.


[Diagram]
An eye at the corner

For a Black eye at the corner, the marked point must be unavailable to White.

Matt Noonan: Although this is a decent heuristic for false eyes, i think is is important to remember that it is flat-out wrong. There are groups for which the only eyes have all four diagonals controlled by white. This shows that the problem of finding out if an eye is false is really not a local one. In fact, as Bill pointed out above it is even worse -- in general it can't be done without reading.

InstantEyeTester has a diagram of a black group with two real eyes, both of which have all the diagonals controlled by white. Here is an example in the corners, where the only diagonals are controlled by white but the black group still lives.


[Diagram]
Black lives, two "false" eyes

Down with the TwoCornersKillsTheEye myth!



Examples

[Diagram]
Real eye

This is a real eye. There is no need for Black to play at a to secure this eye.


[Diagram]
False eye

This is a false eye because White occupied the two marked points.


[Diagram]
Unsettled half eye

This is an unsettled half eye. If Black plays at a, it is an eye. If White plays at a, it becomes a false eye.


[Diagram]
Two real eyes

This group has two real eyes. Note that White can never get to occupy the point a to make b a false eye, as this move is suicide. Similarly, playing at b is prohibited, so a is a real eye.


[Diagram]
Two real eyes

Two real eyes at a and b.



Proof that a False Eye is not an Eye

Here is a proof that a false eye is not an eye.

[Diagram]
False eye

We consider this false eye.


[Diagram]
External liberties filled up

Say White fills up external liberties by playing at the marked points. Now the three marked black stones are under atari.


[Diagram]
Black connects

If Black connects at 1, the first condition fails and so there is no eye here.


[Diagram]
White captures

If Black allows White 1 to capture the three marked stones, there is obviously no eye here.

Therefore, we proved that a false eye is not an eye.



Proof that a "False" Eye might be an Eye after all

[Diagram]
Two False Eyes?

Matt Noonan: Constructive proofs are nice. :) Also, see Instant Eye Tester for a position in which Black has two living eyes, but all the diagonal points of those eyes are occupied by White.



For some even simpler examples of the real eyes with points occupied:

[Diagram]
Not false eye.

[Diagram]
An eye at the corner

[Diagram]
An eye at the side

Just don't ask me how these shapes came to occur though....



Jasonred : What in the world are these eyes? I count at least one true eye and at least one false eye... what are the rest?

[Diagram]
corner

For interest sake, there's how the position occured. C was a white stone before 3 captured.



Contributors:

  • unkx80
  • (Add your name here.)

This page is part of the Eyes Collection.



Path: EyesCollection   · Prev: EyeInTheBelly   · Next: EyeLiberties
This is a copy of the living page "General Eye Definition" at Sensei's Library.
(OC) 2003 the Authors, published under the OpenContent License V1.0.