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False Eye
  Difficulty: Beginner   Keywords: Shape, Life & Death, Go term

[Diagram]
Diag.: Black group with false eye

A false eye is (surprise! surprise!) something which looks like an eye, but really isn't an eye. Usually false eyes involve diagonal connections.

Black's structure to the left has a false-eye at 'a', because the marked black stone isn't really connected to the unit.

'b' is of course, the key point, because if B can play there, he will make two true eyes and will live, but if W plays there, she will kill black.


BillSpight: I think it is better to regard a as a HalfEye? rather than as a false eye.

How about this for an example?


[Diagram]
Diag.: False eye at ''a''



[Diagram]
Diag.: Another common false eye

Here's another false eye which you will encounter frequently in games. Even if the marked stone gets captured there is no real eye here for black. --ArnoHollosi
(See also ThrowIn) --MortenPahle



There was some discussion on rec.games.go a few months ago about ways to instantly tell if an eye is false or not. It turns out you can, by using my magical Instant Eye Tester!

    -- Matt Noonan



[Diagram]
Diag.: A false eye?

:-) The false eye discussion is always fun.

I've seen other 'definitions' which, related to the diagram on the left, say that, if 2 or more of the squared points are taken by the opponent, your potential eye at the circle is false.

E.g. in Arnos example above, white occupies 2 of these spaces, and the eye is false.

I often found it useful to think along the lines of 'can I be forced to fill this point by e.g. my opponent putting some stones into atari'.

--MortenPahle

The problem is, that definition isn't always true.. see the beast at the end of Instant Eye Tester for a spectacular failure of it. :)

On the other hand, what you could say is that the eye could not be false if less than two of those points are taken away, so you could even avoid doing a path test in many cases.

Can anybody think of a more mundane situation where the 2/8 test fails? I seem to remember them existing, but I'm not positive.

-- Matt Noonan


[Diagram]
Diag.:

Here's a completely improbable situation that really violates the 2/8 rule.

-- FCS

The only time when the 2/8 test does not work is in the case of a 'two-headed-dragon' (at least for stones in the center of the board). Your example is a bit over-complex because you made the eyes 'super-false', removing all four rather than just two of the points.

-- Andre Engels


[Diagram]
Diag.: 2/8 Failure

2/4 and still alive without go stone yoga. :)

-- Matt Noonan (said 2/8 again at first, but I guess if one of the four non-diagonal points are filled, it's pretty obvious that it isn't an eye..)

BillSpight: The above does not violate the 2/4 rule, since the opponent occupies only 1 of the 4 points in question for each eye.

-- MortenPahle - These reminded me of an article I once found on the IGS ftp archive (taken from rec.games.go). See Maximum number of live groups

JanDeWit writes: The version of the 2 out of 4 test goes something like this: when checking an eye for falseness, put 'fake stones' at an open intersection at the corner when the opponent can't play there (when this is suicide for example). Then, if all directly adjacent points and at least three of the corner points are yours, the eye is real. This at least rules out MattNoonan's situation above. But I think it breaks down here:


[Diagram]
Diag.: Another 2/4 failure

Here, white can play at the circled points without a problem for black.



Here is an attempt to cover the subject:

The 2/4 - 1/2 - 1/1 rule


[Diagram]
Diag.: An eye in the centre

An eye in the centre is false, if 2 out of the four marked corner points are occupied by a stone (belonging to a living group) of the opponent.


[Diagram]
Diag.: An eye in the centre

This is already a real eye, because A and B are miai for occupying 3 of the 4 corner points.


[Diagram]
Diag.: An eye in the centre

This is already a false eye, because A and B are miai for occupying 2 of the 4 corner points.


[Diagram]
Diag.: An eye in the centre

This eye is still unsettled: Black playing any of A, B or C turns the eye into a real eye. White playing on either of A, B or C makes it false. Of course, the eventual status of the stones depends on many other conditions.


[Diagram]
Diag.: An eye at the side

An eye at the side is false, if one of the two marked corner points are occupied by a stone (belonging to a living group) of the opponent.


[Diagram]
Diag.: An eye in the corner

An eye in the corner is false, if the marked corner point is occupied by a stone (belonging to a living group) of the opponent.



A logical definition

As stated above, this theory breaks down for the - very rare - examples included in InstantEyeTester or TwoHeadedDragon.

The best definition I've heard, covering all possibilities, is this one:

  • A virtual eye is an empty space, reached by chains of the same color.
  • A group lives, if all its chains reach the same two virtual eyes.

(sidemark: If chains that don't reach the virtual eyes can connect to the chains that do, without filling any of the virtual eyes chosen, they can be included in the living group).

Reach is meant as in the Tromp/Taylor rules. Virtual eye because it deviates from the concept of 'eye' we are accustomed to.

Although this is theoretically very nice, and probably fits the computer algoritm, the human preference for visualization makes the 2/4 - 1/2 - 1/1 definition much more practical, and correct for 99,99% of the occasions.

--DieterVerhofstadt

Actually, this works great for humans! This is exactly the method I personally use (and last I checked I was a human). It is easy to quickly visually check if all your chains can reach both eyes. If one of your chains can only reach one eye, and can't be joined to one of the other chains (that reaches both) - then it isn't part of the live group. Furthermore, if such a chain is forming one of the eyes - then that eye is false. This is exactly and clearly the case with the chains above the point a in the first two figures on this page.

This brings up two points:

  1. Does it make sense to define the concept of "false" eye out of the context of finding two live eyes?
  2. This discussion uses the term "chain" where I've seen other literature use "string". And the InstantEyeTester uses "string" to mean something else.

I suppose that's why this whole sequence needs ToBeMasterEdited!

--MtnViewMark




[Diagram]
Diag.: corner

The previous definition does not cover this simple example. These chains do not reach the same virtual eyes.

My proposition for the definition is:

  • A virtual eye is an empty point, reached by chains of the same color.
  • A chain is about to live, if it reaches two virtual eyes that are not falsified.
  • A virtual eye becomes false, if it is reached by a chain that is not about to live.
  • A chain lives unconditionally, if it is still about to live when all the appropriate eyes are falsified.

A chain is said to be alive, if the unconditional life can be obtained with alternate play. Seki is not taken into account here.


[Diagram]
Diag.: corner

If one would like to use the definition with larger eyes, he should be careful. This would be alive using the definition, if virtual eyes larger than one would be blindly accepted.

--TapaniRaiko




[Diagram]
Diag.: corner

If you include the sidemark of my definition, your example meets it. The point A is then a connection point, and not a virtual eye.

(TapaniRaiko: Sorry, missed it. (Or there is a slight inaccuracy in the definition: no chain reaches the two virtual eyes in the example, so nothing can connect to such a chain either.) My definition is more complicated, but it does not require the selection.)



About your definition:

  • The concept of a living chain relies on the concept of a false eye.
  • The concept of a false eye relies on the concept of a living chain.

The definitions are thus self-recurrent.

(TapaniRaiko: Think of it as an algorithm. Virtual eyes are implicitely not false until the algorithm changes them to be. I did not want to write it mathematically, since it is less readable.)



About the purpose of a definition for life.

There is a difference between a giving correct definition and including theorems in your definition.

Of course other configurations of stones (chains) are said to be alive even if they don't meet the definition. That is because we know that the groups can always evolve to the status mentioned in the definition. Examples:


[Diagram]
Diag.: A live group

This group isn't alive according to any definition. It's alive because it will always be able to make two eyes. One can't include such groups in definitions. Rather you make a "theorem" saying that this group is alive, and give a proof.

(TapaniRaiko: Having two eyes is a shortcut to know that the group can not be captured. You can end reading, when black gets two eyes. "A chain has two liberties that the opponent cannot remove" is a nice definition in itself and these two just clarify, what the unremovable liberties would look like. A "better definition" could cut the reading shorter. In fact a lot of Go knowledge is shortcuts. (like: TheLGroupIsDead.))


[Diagram]
Diag.: The sansan invasion is alive

(in abscence of outside walls)

--DieterVerhofstadt



DieterVerhofstadt:

How about this:

Definition:

  1. A virtual eye is an empty point, reached only by chains of the same color.
  2. A collection of chains lives with eyes, if all the chains reach the same two virtual eyes.

Addition:

  1. A group of stones is alive with eyes, if for any sequence of moves by the opponent, there is a sequence of answers that turns the group into a collection of chains complying with definition 2.

Remarks:

  1. The definition reads "if" and not "if and only if"
  2. There is no question of bigger eyes. A virtual eye is one empty point reached only by chains of one color (so no other empty points).
  3. The definition does not include seki
  4. The addition gives a method to bring down the status of a group to the definition. The addition is not included in the definition, since there are groups of stones for which no pro, let alone a computer program, can decide whether they are alive or not.


BillSpight: Howard Landman, in his "Eyespace values in go" paper in Games of No Chance, defines topological life this way: A group is topologically alive iff the chains of the group surround more than one one-point eye, and each chain reaches more than one of those eyes.
The definition given above, which required all chains to reach the *same* two eyes, is not so good. Each chain just has to reach two eyes, not necessarily the same ones.


Proposal for seki

Defintion:

  1. A mutual liberty is an empty point that reaches two chains of different color.
  2. Two chains of different color live in simple seki if they reach no empty points but the same two mutual liberties.

Addition:

  1. A group of stones is alive in simple seki, if for any sequence of moves by the opponent, there is a sequence of answers that turns the group into a chain which is part of a couple of chains, complying with definition 2.

[Diagram]
Diag.: Seki

(TapaniRaiko: The proposal is not enough, as these two examples show. I saw an example where 35x35 board was filled with small groups all living in a single seki.)


[Diagram]
Diag.: Not a seki




This is a copy of the living page "False Eye" at Sensei's Library.
(C) the Authors, published under the OpenContent License V1.0.