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 ^* (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 *: taking n copies of * in a sum reduces to 0 if n is even, and to * 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 > *n, where *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(.) denotes atomic weight 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 BillSpight on March 11, 2004 - 10:04
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