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Eye Definition Discussion
Path: EyesCollection · Prev: EyePotential · Next: EyeDefinitionDiscussionToo
Difficulty: Intermediate
Keywords: Life & Death, Shape, Go term
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
:-) 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 Arno's 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. -- Morten Pahle 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
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
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.) Bill Spight: The above does not violate the 2/4 rule, since the opponent occupies only 1 of the 4 points in question for each eye. Morten Pahle: 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 Jan de Wit: 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 Matt Noonan's situation above. But I think it breaks down here:
Here, White can play at the circled points without a problem for Black. A Logical DefinitionAs stated above, this theory breaks down for the - very rare - examples included in Instant Eye Tester or Two Headed Dragon. The best definition I've heard, covering all possibilities, is this one:
[1] (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 algorithm, the human preference for visualization makes the 2/4 - 1/2 - 1/1 definition much more practical, and correct for 99.99 percent of the occasions. 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:
I suppose that's why this whole sequence needs ToBeMasterEdited!
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 chain is said to be alive, if the unconditional life can be obtained with alternate play. Seki is not taken into account here.
If one would like to use the definition with larger eyes, they should be careful. This would be alive using the definition, if virtual eyes larger than one would be blindly accepted. -- Tapani Raiko
If you include the side remark[1] 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 definitions are thus self-recurrent. (Tapani Raiko: 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 LifeThere is a difference between giving a 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:
This group is alive but not according to the strict definition of alive. 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. (Tapani Raiko: 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: The L group is dead.))
(in absence of outside walls) How about this: Definition:
Addition:
Remarks:
Bill Spight: Howard Landman, in his "Eyespace Values in Go" paper in Games of No Chance, defines topological life this way: A group is topologically alive if and only if 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. Karl Knechtel: This is my attempt at definitions. It has an aesthetic advantage in that there is no distinction between chains/strings/groups/whatever. I'll call them strings. Define a path as a set of stones including a and b such that every stone is orthogonally adjacent to some other stone in the set. Define a string as a path which is not a subset of a larger path. (This is the easiest way I can think of to specify the things formally.) Define an eyespace as a set of points on the board satisfying the above definition for string, except with "stones" replaced with "empty points". Define a liberty of a string (just to make sure we're clear on everything) as a point which is
Define an eye as an eyespace, such that every string which has a point in the eyespace as a liberty is of the same colour. These strings are said to enclose the eye. Define a simple eye as an eye, such that every point of the eye is a liberty of every string enclosing it. (A 3x3 eye is not "simple", therefore: this is important to the definition of unconditional life which follows, because with infinite passes a group with two such eyes could be captured. See Two Eyes Can Die.) Next, apply the following algorithm to define true eyes and false eyes:
A true eye is any simple eye which is marked true after application of the algorithm; the simple eyes marked false are false eyes. (Large spaces are still not categorized as eyes; the idea is that further play can settle them into simple eyes which will almost certainly be true, but sufficient passes allow the other player to capture.) Then: [3] (Bookmark) A string is unconditionally alive if and only if its liberties include two true eyes. It is alive in sente if and only if,
It is alive in gote if and only if,
It is unconditionally dead if and only if there does not exist a set of points such that occupying those points with same-colour stones would render the group unconditionally alive. It is dead in gote if none of the above apply, and it is not one of a pair of strings alive in seki: Two strings are alive in seki if and only if
(Slight change here. "There exists" was the wrong criterion for the shared liberties in seki. If playing in one liberty makes a living shape and playing in the other makes a dead shape - that's no seki.) Finally: The game is completed when
I think this is self-consistent, and sufficient to cover all cases. Comments? Saesneg: I think I can simplify your definition, though I consider it doomed from the outset! Every point on the board has the colour black, white or empty. If a point is not empty it is filled. Two points are hard-connected if they are the same colour, and there exists an orthogonal path from one to the other via other points of the same colour. A string is a set of points of the same colour such that every pair of members is hard-connected, and none is hard-connected to any non-members. Note that by this definition a string may consistent of empty points. Two strings are adjacent if any member of one is adjacent to any member of the other. Two adjacent strings are called neighbours. Note that neighbours are, by definition, coloured differently. A simple eye is a small empty string whose neighbours are all the same colour ('small' will be discussed later!). Two filled strings are soft-connected if they are the same colour, and there is a path from one to the other via simple eyes and strings of the same colour. A group is a set of filled-strings such that each pair of members is soft-connected, and none is soft-connected to any non-members. Every simple eye adjacent to a group is a true eye, except those with a neighbour which is adjacent to only one true eye. Is this recursive statement logically well-formed? A group with two true eyes is alive. Unfortunately these attempts to formalise the definition of life cause as many problems as they solve. I do think they are useful as long as you continue to apply intuition and 'common sense' (which can feel very uncommon to a beginner). Firstly a sufficiently large eye is not really an eye in any meaningful sense. Secondly an eye can in practice be adjacent to strings of either colour, when it contains dead stones. Any attempt to define a dead stone rigorously is not for the faint-hearted. If there really were a simple formal definition of true eyes Computer Go would be a lot stronger than it is. jvt: Semantics... in my opinion you can't define a single global true eye rigorously but it's easy to define two (or more) eyes. There is a simple formal definition of true eyes no one mentioned yet: a stone or group of stones is said to have two eyes if there is a way for the defender (even if the attacker plays first) to reach a configuration (including these stones) which Benson's algorithm recognizes as unconditionaly alive. That's all. Simple isn't it? This algorithm is used in Computer Go by some life and death programs / modules. Search + Benson as a cutoff test recognize all sort of shapes that are alive with two eyes, including any two-headed dragon, etc. Benson's algorithm is static (i.e. without search). There is no way to recognize two eyes generally without search. However many programs use an eye shape library for the most usual shapes. Dieter: Due to a question I received by a reader who felt like not disturbing this community - (we want to be bothered more!) - I feel compelled to add that the subject of the discussion are groups of stones of different colour that are both alive (can't be captured), yet do not fall under the usual forms of co-existing life we encounter on the board:
One of the examples under discussion displays a group with two (or more) eyes that we ordinarily recognize as being false, yet the group is alive. The whole issue is rather of formal-algorithmic kind than crucial to our understanding of Go.
This is the curious situation I proposed. There have been several definitions of eyes and alive / dead groups in this discussion. In this situation the small white group is unconditionally alive (black can't do anything to capture it) and the black group is dead (white can capture it at any time if needed). I was simply wondering how well the definitions of an alive group presented here can handle this situation. - Juha Nieminen? Here's another position with 'eyes' that don't easily fit the usual sterotypes.
There are no eyes on the black group, so the implication is that if the position is a seki, there aren't any on the two-stone white groups either.
Here's the same two-stone corner group, but here it's an eye. The black group has an eye, so for seki, the white stones do too. Dieter: It is harder to read what's on this page already than to think of something new. Well, here it comes: Preamble: the Tromp-Taylor rules DefinitionsA stone is a point not colored empty. A chain is a number of stones of the same color that all reach eachother. A liberty to a chain is an empty point adjacent to that chain. A group is a number of chains. A simple eye belonging to a group is a liberty to at least two chains of that group, which is no liberty to opponent chains, and which can only be filled by the opponent after all liberties of all chains to which it is a liberty, are filled by the opponent. Theorem
I think this covers all examples of groups alive with one-space eyes, which form the basis of the rest of life and death theory. Should eyes own groups?nstone Conceptually, it might be easier to think of spaces owning contiguous groups, rather than vice versa. Consider: A contiguous group of spaces is an ownerspace if it is continuously surrounded by stones of a single color, and/or a section of the board's boundary. Ownerspaces own all boundary stones and their chains, Any ownerspace within an ownerspace is owned by the inner space. It can be other-colored. A living group is a collection of stones made entirely of multiply-owned chains, all sharing owners. Assurance of life is equivalent to assurance of multiple-ownership. Notice that when an ownerspace is entirely filled from the inside by an inner ownerspace, it ceases to exist, and its boundary groups may longer be multiply-owned and can be killed. unkx80: Discussion moved from General Eye Definition. 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. 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.
Confused: Here's an alternative wording for second condition which should work better:
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. 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. Instant Eye Tester has a diagram of a black group with two real eyes, both of which have all the diagonals controlled by white. (The properties described below will use black eyes, please switch the colours around if we are discussing white eyes.)
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.
For a Black eye at the side, both marked points must be unavailable to White.
For a Black eye at the corner, the marked point must be unavailable to White.
RobertJasiek: You all fail because you do not define and refer to hypothetical-strategy and you are too careless about eye spaces. Here you find the intermediate result of 4 months of research: Defining such terms for Greedy Ancient Rules is much easier. Likewise, considering subboards given by pass-alive (Benson alive) strings is. See also: Path: EyesCollection · Prev: EyePotential · Next: EyeDefinitionDiscussionToo This is a copy of the living page "Eye Definition Discussion" at Sensei's Library. ![]() |