Zero in CGT terms

Path: <= CGT path =>
    Keywords: EndGame, Theory

How do you recognise go positions that are equal to 0, in the sense of CGT? Since the idea of number appropriate in CGT corresponds to territory in go, zero games should comprise positions recognisable as 'no net territory for anyone'.

However, the term is usually used in chilled go, where it is used for any go position or combination of positions that has the same integer score with correct play, regardless of who plays first, not just positions where that score is 0. In that context it is important to distinguish zeroes from go infinitesimals.


Here is a 0 in go:

[Diagram]
Zero (Seki)  

One way to write it is

                    { 6 | -10 || 8 | -8 }

If Black plays first White can reply and get 10 points (White scores are negative); if White plays first Black can reply and get 8 points. Neither player can afford to play. Since neither player will play, we can simplify this to { | }, where neither player has any option. { | } = 0 by definition.

Here is another 0, this time a clear case of miai:

[Diagram]
Zero  

We can write it

                    { 2 | 0 || 0 | -2 }.

In this case neither player can gain by playing here.

Migeru Does this mean that the only go positions that are numbers involve miai or seki?
Bill: No. In general, territory and captured stones correspond to numbers. But making a correspondence to plays requires some doing. See Mathematical Go.




Zero in Chilled Go

[Diagram]
Zero  



[Diagram]
White plays first  



[Diagram]
Black plays first  



Whoever plays first, the second player gets the last one point play, for the same score. When Black plays first at B1, W2 prevents Black from saving the black+circle stone because of damezumari.


Note on the diagrams: When we're getting picky, it's important to indicate where the plays are. So instead of the usual convention of surrounding positions by open points, I (Bill Spight) have surrounded them by empty space or by nothing. The stones that frame the position cannot be captured. See assumptions of yose problems.


The most basic game in Conway's theory is the "empty game" { | } where neither player can legally move. An example in go is

[Diagram]
{ | } = 0  

Any position consisting entirely of (any number of) minimally alive groups (of either colour) is an example of { | }. I suspect that this is the reason why mathematical go is equivalent to go scoring with a group tax.


It is easy to see that { | } = 0. If you want, you can check that 0 + G = G + 0 = G for any game G, using the definition of addition. - Migeru

Robert Pauli: Disagree. Filling your second to last eye may be stupid, but it isn't illegal ;-). If I got the abstract nonsense, what you show is {-24 | 22}, which, yes, should lead to the same result as { | } (provided no triple-digit kyus are messing around). Here's my suggestion for zero (1x1 board):

[Diagram]
{|}=0  


Migeru: Well, as a matter of fact {-24|22}=0. There is a rule stating (roughly) that if {A|B} is a number, then it is equal to the simplest number between A and B, but I didn't want to get into that when I wrote my contribution to this page.

Robert Pauli: Guess you're right. Let's see if I can prove it (to myself). The question is (according to Equality of Games), whether {-24 | 22} >= 0 and 0 >= {-24 | 22} ? The second term would be the same as 0 - {-24|22} >= 0 (assuming G - G = 0 for the moment) or 0 + {-22|24} >= 0 (according to Negative of a Game) or {-22|24} >= 0 (since 0 provides no option). So, both proofs are essentially the same. Let's proof the left term. The question (according to Ordering of Games) is, whether Left can win {-24|22} if Right starts? I'm trying hard, but just can't make it: Right passes and the most Left gets is a tie. Where's my error (in this trivial endavour)? Should the definition with >= be "can win or tie"

Bill: There are no ties in CGT.

Migeru: There are no passes either, strictly speaking. with right to move, the score is 22 so left wins. With left to move, the score is -24 and right wins. In the case of {|}, with right to move, they can't move so left wins; with left to move, they can't move so right wins. In either case, "the player whose turn it is loses", which is the CGT meaning of 0={|} and shows why {-24|22}=0.

Bill, on a related note, resigning a game is the same as moving into * (1st player win) with your opponent to move. Is it then possible to add the stipulation that * is always a valid move? In that case,

{-24|22} => {-24,*|22,*}

What is the value of this new game? Is it still zero?

RafaelCaetano: Yes, it is still zero, since whoever plays first loses. But why would you want to do that?

fractic: Another way to see that {-24|22} (and {-24,*|22,*}) is equal to zero is the fact that both options are reversible


Original page by Bill Spight, moved and edited by Charles Matthews.


Path: <= CGT path =>
Zero in CGT terms last edited by fractic on July 29, 2006 - 13:17
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