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introduction to infinitesimals
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Infinitesimals:

An infinitesimal is a non-zero combinatorial game in which both the Left stop and the Right stop are 0. In CGT you win if your opponent has no move, so play with infinitesimals is concerned with getting the last move (tedomari). A great deal of CGT research has been concerned with play at temperature 0.

The simplest infinitesimal is * (star), also written

                   { 0 | 0 }

As Jan pointed out, this is a dame in go.

I have heard that other infinitesimals have been constructed, that involve seki which alters the parity of the dame on the board, but I have not seen one. There seem to be no other infinitesimals in go.

However, plays with a miai value of 1 may be treated as infinitesimals. Then the CGT theory of infinitesimals may be applied to them. Berlekamp and Wolfe did this in Mathematical Go: Chilling Gets the Last Point.

Some of their problems stumped the pros, even 9 dans! However, the theory makes them fairly easy to solve. A few heuristics should enable the knowledgeable go player to get the last 1-point play if possible, in practically all cases. (See Ongoing Game for an example.) Furthermore, they should help get tedomari at higher temperatures.

See Chilling for more discussion, and the go equivalent of such infinitesimals as

                   ^ = {0 || 0 | 0}     (up)

and

              tiny-2 = {0 || 0 | -2}

Bill Spight


Moved from combinatorial game theory by Charles Matthews



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This is a copy of the living page "introduction to infinitesimals" at Sensei's Library.
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