Half-eye
A half-eye is an unfinished eye: a potential eye that can be completed or destroyed depending on who plays first. Typically, the moves at a half-eye are locally small for both players, and are only played in order to make a group live or die, or to keep it on the run.
It is so called because two half-eyes are effectively a whole eye, as they are miai.
A three-quarter-eye – where forming one eye threatens to form a second – is sometimes also referred to as a half-eye; both are instances of fractional eyes, whose values need to combine to ``geq 2`` for a live group.
Basic example
How half-eyes combine
One and a half eyes is unsettled
This unsettled black group has a half-eye at a.
The key point is of course b, because Black needs to play there to complete his second eye.
Two half-eyes make a whole eye
Half-eyes can be used to count up to two eyes.
In this example, White has four half-eyes at a, b, c and d, which is equivalent to 1/2 + 1/2 + 1/2 + 1/2 = 2 eyes, so White lives.
Common half-eye shapes
Once one understands the idea of half-eyes, it is useful to be able to recognise them in common situations. In particular, this helps one solve some life and death problems.
(See Landman^{[2]} for many examples of fractional eye shapes, including many corridors.)
Half eyes at the edge of the board
Assume in the following examples that Black connects through to the rest of his group. All White’s stones are assumed alive.
Half-eyes in Combinatorial Game Theory
The methods of CGT may be used to treat a half-eye as a sort of combinatorial game with the result:
- 1 eye for Black, if Black goes first.
- 0 eyes for Black, if White goes first.
This could be written as ``{1_e|0_e}``, with the subscript ``""_e`` to show that ``1`` and ``0`` stand for the number of eyes and do not have their usual meaning in CGT^{[1]} (nor do they stand for the resulting score).
This approach applies to other fractional eyes as well, and has been worked out by Howard Landman^{[2]}. This yields fractional values which must add up to at least ``2`` for a group to live, with one exception: one might expect ``{2_e|0_e}`` to be equivalent to ``1`` eye, but ``{2_e|0_e}+{2_e|0_e}`` and ``1_e+1_`` live^{[3]}, while ``{2_e|0_e}+1_e`` is unsettled.
- To do: make an article on eyespace values? explaining the main points of Landman’s paper as a more theoretical counterpart to fractional eyes.
See also
- Fractional eye — Eyespace regions that can be considered to be ⅓ eye, ⅔ eye, ¾ eye, 1¼ eye and 1½ eye. Also refers to Landman’s paper^{[2]}.
- Eyes collection — A collection of articles on all aspects of eyes.
- Farmer's hat half-eye — A particular half-eye shape at the edge.
Notes and references
[1] In CGT, ``0`` and ``1`` are respectively defined as the games ``{|}`` and ``{0|}`` (i.e. ``{{|}|}``).
[2] Eyespace Values in Go by Howard Landman: http://www.msri.org/publications/books/Book29/files/landman.pdf
[3] We assume here that the incomplete eyes occur within the same group – if they can be disconnected
Discussion
So the result of this game is
- 2 eyes for Black, if Black goes first.
- 1 eye for Black, if White goes first.
(This is denoted { 2 | 1 } in Combinatorial Game Theory, I believe)
-- Jan de Wit
Bill Spight: See “Eyespace Values in Go” by Howard Landman: http://www.msri.org/publications/books/Book29/files/landman.pdf
Jan de Wit: Another nice reference is Martin Mueller’s Ph.D. thesis “Computer Go as a Sum of Local Games: An Application of Combinatorial Game Theory” which can be found at ftp://ftp.inf.ethz.ch/pub/publications/dissertations/th11006.ps.gz. This also has the most accessible introduction to Combinatorial Game Theory which I’ve found so far.
- Further discussion moved to Combinatorial Game Theory.