Infinitesimals Discussion

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Here are some infinitesimals in chilled go.

[Diagram]

Infinitesimals

Can you tell what they are? What is this position worth?

(Answers in Corridor Infinitesimals, further discussion here. :-))

--BillSpight


Holigor:

The upper one is 2 points for Black if Black moves first and 0 points if White moves first.

So, it is a one-point move, value one point.

The second is 3 (Black)/reduces to the first one (White). As we know that the first was one point then the score is 2 points, and a move is worth one point.

The third is 4 points (Black)/reduces to previous (White). The previous was two points. Thus the score is three points, a move value is one point.

Bill: Yes, indeed. :-)


JanDeWit: Let's see if I have been paying attention :-)

Top position is {2 | 0}, mast value 1, temperature 1, and chills to * (star), I think.

Bill: Right. :-)

Second one is { 2 , 3 | {4 | 0} , {2 | 0} } which is equal to {3 | {2 | 0}} (both first options are dominated by the second). Left stop is 3 with Right to play; right stop is 2 with Right to play. Cooling by one gives {2| 1+*}, so I think the mast value is 5/2 and temperature is 1/2.

Bill: Chilling gives {2 | 2*}. Remember to include the effect of a white play to 1*.

Bottom position is (crunch crunch) { {3|1}, 3 , 5 | {4,5|{6|0},{4|0}} , { {4|2},5 | {6|0},{2|0} } , {2,3 | {4|0} ,{2|0}} } which simplifies to

{ 5 | {5|{4|0}} , { 5 | {2|0}} , {3 | {2 | 0}} which in turn simplifies to { 5 | { 3 | { 2 | 0 }}}. (Modulo the early-morning fudge factor of course!)

Bill: A little too early! ;-) The game simplifies to

                    { 4 ||| 3 || 2 | 0 }     which chills to
                    { 3 | 3^} = 3^^*

Jan: Indeed! Memo to self : 3 + 1 captured stone is not equal to 5; I think I counted the captured stone three times. The rest is wrong as well now... :-(


Juuitchan: I have seen this kind of weird math on several pages so far. How exactly does it work? How do these "infinitesimals" work?

Bill: Infinitesimals are games in Combinatorial Game Theory in which the payoff is getting the last move, or tedomari. In their book, Mathematical Go, Berlekamp and Wolfe? show how certain go positions with a miai value of 1, such as the ones at the top of this page, act like infinitesimals. When tedomari is the issue, knowing something about infinitesimals may help.


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