A half eye is either an eye or not an eye, depending on who plays first. It is so called because two half eyes are effectively a whole eye, as they are miai.
The marked point is a half eye for Black...
because by playing first, it becomes an eye; and...
White can remove the eye by playing first.
This black group has a half eye at a.
The key point is of course b, because if Black can play there, Black will make two real eyes and will live; but if White plays there, White will kill Black by making a a false eye.
Charles Assume in these examples that Black connects through to the rest of the group. All White's stones are assumed alive.
(Add similar).
So the result of this game is
(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.
Andre Engels: Half-eyes can be used to count up to two eyes; the rule is that a player lives if he either has two eyes, or 1 1/2 eyes and sente (provided the sente is used correctly).