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Sub-page of Chilling

xela: Reading Mathematical Go, I'm having trouble understanding chilling and cooling.

The book gives a defintion of chilling (page 53) which looks similar to, but not quite the same as, cooling. However, the discussion which follows seems to prove that, in the specific context of go positions, chilling turns out to be exactly the same as cooling. So why do we need the two separate concepts? Why not just use cooling?

It gets more confusing when specific go positions are discussed. For example, on page 17 we are told that a two-point gote chills to *, for example:


If Black plays first, a captures a white stone, making two points; if White plays first, there are no points, so the position is {2|0}. This much makes sense to me.

Now, the definition of cooling by one point is to subtract 1 from all the left options, and add 1 to the right options.[101] So {2|0} should cool to {1|1}, which is the same as 1* (1 plus star) if I understand rightly. And if chilling is the same as cooling, then {2|0} should cool to {1|1}. Right?

No. It seems that {2|0} chills to *. The reason is that before chilling, marks are added to the diagram "to indicate a point that Black has earned in the unchilled and unmarked game" (page 57), and each black mark subtracts a point. "The reader may require some time to get used to the markings"--how true!

We're carrying out analysis to determine how many points Black will end up with--and the starting point is to mark the diagram with how many points Black is going to end up with. Doesn't this look a bit circular?

I can see how having the games chill to infinitesimals is convenient for the sake of the theory, but adding arbitrary constants in order to get a convenient answer isn't my idea of rigorous mathematics.

Of course, it isn't actually arbitrary--the authors do get results which end up making sense, so there must be some systematic principle at work. But this is never explicitly addressed.

So, can anyone give an explicit definition of "marking and chilling" that actually describes the procedures of Mathematical Go?

Bill: Chilling vs. cooling: {2 | -1} has a mean value of 1/2. If we chill it we get {1 | 0}, which also has a mean value of 1/2. The same is true if we cool in by 1 point. To chill a game we tax each move by 1 point. To generalize that, suppose that we tax each move by 4 points. The result for {2 | -1} is {-2 | 3}, which equals 0, not 1/2. But if we cool {2 | -1} by 4 points, we get 1/2, since in cooling we stop taxing once we reach a number.

xela: Are you saying that {2 | -1} chilled by 4 points is {-2 | 3}?

Bill: No. Chilling is specifically a tax of 1 point.

walleye: So chilling is a tax of 1 point on every move even if the game is a number already, right? I've got a serious problem with that. The rules they use in the book is nopass go with prisoner return. If you've got a prisoner you can give it to the opponent as a legal move. Since every move is taxed by 1 point, returning a prisoner will be taxed as well. This is essentially equivalent to the opponent getting one extra move. So he'll make that move. It will be taxed. You get an extra move. You play it. It is taxed too. And so on and on and on. The game will never end.

xela: My current understanding (work in progress!) is that chilling reduces the temperature by one point. If the game is already a number, then the temperature is zero, so chilling does nothing--i.e, a chilled number is just the same as the original number.

Bill: My bad. Chilling indeed maps a number to itself. It doesn't exactly reduce the temperature by one point. Cooling does that. (But the difference is academic in the context of go and chilled go.)

Bill: Now, the main page does not offer a formal definition of chilling, but I does say that it is a tax on a board play (not on returning a stone). Suppose that all the dame are filled and that Black is ahead by one point, which he fills. Since that is a board play, it is taxed (by handing over a prisoner). White returns the prisoner, and Black is stuck. So I will amend that statement.

walleye: The main page seems to consider chilling along with Japanese rules, whereas the book applies chilling to games played according to mathematical rules, ie nopass go with prisoner return. So, are we talking about two different chilling procedures?

Bill: We are only talking about one procedure. The rules discussions in the appendages are separate from the main text. The only form of territory scoring in common use is the Japanese/Korean. Chilling applies to it (as well as other forms, such a Lasker-Maas rules or Spight rules).

walleye: Japanese rules do not produce a combinatorial game. Therefore, the theory does not apply to normal go directly. Cooling as well as chilling are defined for a combinatorial game. Nopass go with prisoner return and no kos qualifies as a combinatorial game. Most developments in Mathematical Go are concerned with such a combinatorial game. Of course the authors mention that the results apply to normal go as well. On SL you just take normal go and apply chilling to it, although it is not entirely clear what you mean by that. So I see it as quite different from the theory.

Bill: Chilling does not apply to positions with ko. Otherwise, it applies to Japanese rules, no problem.

walleye: I haven't got the book with me but as far as I recall they apply chilling to games played with mathematical rules. My impression was that cooling, chilling, heating and so on apply in a proper sense to combinatorial games only. So is there a page somewhere on SL that explains how Japanese rules come into consideration? I think this is an important point that unfortunately got swept under the rug it seems.

Bill: You were perhaps misled by this in Appendix A: "The various rulesets which are most tractable for pursuing formal and rigorous proofs . . . are based upon the mathematical foundations of combinatorial game theory. All such rulesets essentially prohibit passing, and transform conventional score-counting into a contest to see who can get the last legal move." But since Mathematical Go is aimed at go players as well as mathematicians, the lessons of chilled go are meant to apply to real go with territory scoring, i. e., Japanese/Korean rules.

Bill: As Berlekamp and Wolfe say in Appendix B, "It is not hard to engineer an exhaustive methodology. . . . One begins by considering the set of all terminal Go positions, from which Black and White gurus would both choose to pass. One accepts whatever score their rules specify. This integer-valued score is then transated to an integer-valued combinatorial game. . . ." By this simple means the lessons of chilled go apply to any territorial ruleset, including Japanese/Korean rules, which are by far the most commonly used.

Chilling and marking: {2 | 0} chills to {1 | 1}, which equals 1 + *. (We can verify that by playing 1 + * out.) Now, we play 1 + * exactly the same way we play * (by the number avoidance theorem). We play in *, not 1. Therefore, when we are interested in how to play chilled go, we ignore the number, and treat *, 1 + *, 2 + *, -3 + *, etc., all the same. Marking is a way of adjusting things so that we ignore the number.

So we start with a go position with the game tree, {2 | 0}, and subtract 1 point, which we indicate by marking. That gives us {1 | -1}, which chills to *. Voila!

Marking is a way of adjusting the mean value of a position so that it is < 1 and > -1.

As for getting used to markings, I ignore them. ;) I ignore the integer, too, so it comes to the same thing in the end. :)

xela: Aha! So really you chill first, notice that the result is "number plus infinitesimal", and add marks for the (integer part of) the number?

Bill: No, you basically mark each point of territory. Dead stones get two marks, half dead stones get one mark. You can even do fractional marks, but Mathematical Go doesn't get into that. I find marks a bother, myself. ;)

xela: "Each point of territory"--how do you determine what is going to be territory and what isn't? And I'd noticed that "dead" stones get either one or two marks--but whether or not something is dead depends, like the size of the territory, on the subsequent play. So it still looks to me like a circular argument: you need to analyse the position to determine how many marks to place so you can analyse the position.

Bill: The point of markings is not to analyze the position, but to normalize the mean value of a position. The integral part of the mean value is set to 0. To be sure, it helps to know the mean value first.

Why normalize? For convenience. And I think it has to do with the mathematical mindset.

Linguistically, I am happy to have two definitions of *.

1) {0 | 0}
2) {0 | 0} + n, where n is a number

But the mathematical mindset cringes at the ambiguity of having a second definition. However, it is convenient to refer to go positions with game trees, {2 | 0} and {0 | -2}, as stars (after chilling), because that refers to their salient feature. So you normalize them via marking to {1 | -1}.

Note that in this case, and many others, you can normalize without analysis. Only {1 | -1} has a left stop greater than 0 and a right stop less than 0. (The mean value of non-ko positions lies between the stops.)

The prototypical go position in Mathematical Go is a corridor. How to mark a sufficiently long corridor with two empty points at the end?


The points with xs are empty or occupied by a White stone.

White stop  

By go knowledge we know that the enclosed area contains territory or dead White stones.

Black stop  

The result is the same except that a is territory.

To normalize, we mark the area with xs as territory or dead stones. We have a choice of marking a or not. Because this position may be White's sente, we do not mark a.

Now let's mark this corridor:

Corridor 2  

White stop  

Black stop  

Clearly, we mark the xs as territory or dead stones. white+circle is dead after the Black stop, live after the White stop, so we give it one mark (half dead). We can leave a unmarked.

Note that this marking requires go knowledge. It tells us that the xs are territory or dead, and it tells us how to play the corridors. Neither player starts at a, for instance. ;)

I hope this makes things a bit clearer. :)

xela: Getting there... I've been scribbling lots of symbols and diagrams onto bits of paper; I don't have time right now to turn it into something coherent enough to post here, but I'll try again later. (Work in progress at cooling/examples.)

Is the "mean value" the same as "mean" as described in the definition of "cooling" (Mathematical Go, page 50--51)? It seems reasonable that the mean is somewhere between the left stop and the right stop (although I haven't yet tried to make a formal proof of that fact), but not necessarily the average of those two numbers. In your example 1 above, the left stop is 5, the right stop is 4, so the mean is somewhere between 4 and 5, hence 4 markings. (Actually I think the mean is 4 1/32--details later.) In your example 2, the left stop is 7, the right stop is 4, so there's still room for confusion. I think I've worked out that the second position is hotter than the first--temperature between 1 and 2 (it doesn't have to be an integer, right?), mean 5 1/32--details later.

[101] Bill: No, that's the definition of chilling.

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Chilling/discussion last edited by Bill on November 13, 2007 - 16:03
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