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11.3 The eyespace as local game

In the abstraction, an eyespace consists of a set of vertices labelled:

 
!  .  X

Tables of many eyespaces are found in the database `patterns/eyes.db'. Each of these may be thought of as a local game. The result of this game is listed after the eyespace in the form :max,min, where max is the number of eyes the pattern yields if `O' moves first, while min is the number of eyes the pattern yields if `X' moves first. The player who owns the eye space is denoted `O' throughout this discussion. Since three eyes are no better than two, there is no attempt to decide whether the space yields two eyes or three, so max never exceeds 2. Patterns with min>1 are omitted from the table.

For example, we have:

 
Pattern 548

 x
x*x!

:2,1

Here notation is as above, except that `x' means `X' or EMPTY. The result of the pattern is not different if `X' has stones at these vertices or not.

We may abstract the local game as follows. The two players `O' and `X' take turns moving, or either may pass.

RULE 1: `O' for his move may remove any vertex marked `!' or marked `.'.

RULE 2: `X' for his move may replace a `.' by an `X'.

RULE 3: `X' may remove a `!'. In this case, each `.' adjacent to the `!' which is removed becomes a `!' . If an `X' adjoins the `!' which is removed, then that `X' and any which are connected to it are also removed. Any `.' which are adjacent to the removed `X''s then become `.'.

Thus if `O' moves first he can transform the eyeshape in the above example to:

 
 ...            or      !...
  .XXX.!                  .XXX.

However if `X' moves he may remove the `!' and the `.'s adjacent to the `!' become `!' themselves. Thus if `X' moves first he may transform the eyeshape to:

 
 !..           or    !..
  .XXX.!              .XXX!

NOTE: A nuance which is that after the `X:1', `O:2' exchange below, `O' is threatening to capture three X stones, hence has a half eye to the left of 2. This is subtle, and there are other such subtleties which our abstraction will not capture. Some of these at least can be dealt with by a refinements of the scheme, but we will content ourselves for the time being with a simplified model.

 
 |- X - X X - - X O X O 
 |X - - - - - X X O O O
 |O X X X X - - X O O O
 |O O O O X - O X O O O
 |1 2 . . O O O O X X O
 |X O . X X X . 3 X O O
 |X O O O O O O O X X O
 |- X X O - O X O - - X
 |X - - X - X X X X X X
 |O X X O X - X O O X O

We will not attempt to characterize the terminal states of the local game (some of which could be seki) or the scoring.


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