Atomic Weight

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    Keywords: EndGame

Each infinitesimal is approximately equal to some number of ↑s (UPs). This number is called its uppitiness or atomic weight (Winning Ways, by Berlekamp, Conway, and Guy). Atomic weight is similar to the external liberty count in a semeai. If the atomic weight of an infinitesimal game is ``>= 2``, Black can win; if it is ``<= -2``, White can win. An atomic weight of 1 may not be enough for Black to win. For example, with sente White can win `` "↑"ast `` (UP STAR).

For more on the application of atomic weight to go, see Go infinitesimals, Corridor infinitesimals, and Playing infinitesimals.

Bill Spight, moved by Charles Matthews

The notations of * (star) and ^ (up) are introduced in Chilling. Modern chemistry refers to atomic mass rather than atomic weight.

Charles The definition in Winning Ways is quite subtle (and impressive); so the rather clumsy name is a bit disappointing. How to explain this area to Go players?

Firstly the class of games for which atomic weight is defined is called all small in Winning Ways. For a Go player these all look like dame-filling, that is, no territory involved. But the simple dame in Go are all copies of ``ast``: taking ``n`` copies of ``ast`` in a sum reduces to ``0`` if ``n`` is even, and to ``ast`` if ``n`` is odd. That's not so interesting, just something familiar for the comparison between area scoring and territory scoring.

These games are all in a sense contests for final play (tedomari to go players) - and that's all. They include all impartial games, certainly; but those don't provide interesting examples, because the atomic weight in those is always 0.

What is wanted is to define a notion of 'lead' in a game considered as a race for final play. The definition relies on a normal case ('neck-and-neck') modelled on temperature as opportunity cost: in a level race, the difference one play makes is between 'one ahead' and 'one behind' (cf. capturing races) so is taken to be play at temperature 2. Only in a situation which you could call 'clear-cut' is the atomic weight defined by reference to the opponent's options in playing first, plus two.

The definition of clear-cut for a game G has two parts:

  1. ``G > ast n``, where ``ast n`` is a Nim heap which is 'large enough';
  2. The candidate ``{a(G^L)-2|a(G^R)+2}`` is an integer, where ``a(G)`` denotes the atomic weight of ``G`` and ``G^L``, ``G^R`` run over the left, right options in ``G``.

Obviously this sort of definition can only be justified by the work one can get out of it. Not easily assimilated.

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Atomic Weight last edited by PJTraill on January 23, 2019 - 01:36
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