# Chilling

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Chilling is a technique used to make situations in go easier to analyse with Combinatorial Game Theory (CGT). The reason is that CGT is easier to apply in cool games, where players are reluctant to move, while go is generally hot, i.e. there is an advantage to moving.

Main author: BillSpight

 Table of contentsThe relationship with scoring Why chill a game? Numbers Infinitesimals STAR (*) UP (↑) & DOWN (↓) Confused games UP STAR (↑*) Double UP STAR (↑↑*) See also Notes Discussion Table of diagramsDame = * = {0|0}

Two notions are introduced:

• The chilled value of a move: 1 less than its normal value.
• The chilled game (and playing the chilled game): breaking the board up into chilled regions and analysing these using CGT.

## The relationship with scoring

In general, the difference between Territory and Area scoring is that stones on the board are counted in area scoring but not in territory scoring. Suppose that you play by area scoring but each board play costs one point. You would actually be playing by territory scoring, since each board play results in a stone on the board or captured. (Japanese and Chinese scoring also differ about whether to count points in seki, but that is another matter.)

It is a somewhat curious fact that, aside from exceptional cases involving seki or ko, winning play by territory scoring is winning play by area scoring. The difference between the two forms of scoring has very little effect on optimal play.

Similarly, if you impose a tax on territory scoring of 1 point per board play[1], you get a form of scoring such that, aside from exceptional cases involving seki or ko, winning play by it is winning play by territory scoring. In Mathematical Go Berlekamp and Wolfe refer to imposing this tax as chilling and solve endgame problems by "playing the chilled game". Chilling is almost identical to cooling by 1.[2]

## Why chill a game?

Because for certain go positions how to play the chilled game is well known. Those not involving ko or seki with a miai value less than 1 chill to numbers. While it may be correct in some overall positions to make a play that is not the largest, numbers should not be played before hotter plays, and hotter numbers should be played before cooler ones. Also, plays with a miai value of 1 chill to infinitesimals, and correct play with many infinitesimals is known. The object is to get the last play (tedomari) at temperature 1. Getting that tedomari may not matter, but it doesn't hurt. :-)

## Numbers

Some games (local go positions) that are not numbers chill to numbers. Conventionally, we just call them numbers. If there are no kos involved, it is never optimal to play in a number instead of a hotter play, and it is never wrong to play in the hotter of two numbers. With no kos involved, go numbers always have a power of 2 in the denominator. The ones with larger denominators are hotter; play in them has a miai value of 1 - 1/D, where D is the denominator.

The article Numbers shows examples of go positions that chill to numbers.

## Infinitesimals

Non-ko games with a miai value of 1 chill to infinitesimals. The point of play in infinitesimals in combinatorial game theory is to get the last move (tedomari). A good deal is known about best play with infinitesimals that carries over to go through chilling.

The following sub-sections describe a number of infinitesimals, all of which, like many other infinitesimals, arise as chilled go positions.

### STAR (*)

The simplest infinitesimal is STAR (ast), defined by:

 ast = { 0 | 0 }

STAR is a first player win. In territory scoring, a dame is a ast.

Let's see why. In Zero in CGT terms I have a zero position that I will use as standard. Then,
Dame = * = {0|0}
The marked point is dame. It is the only legal play for either player, and results in zero. – Migeru

Over to Bill.

Two asts are miai. In CGT terms,

 ast + ast = 0

Zero is a second player win.

Considering territory scoring as chilled area scoring, in area scoring a dame is worth

 { 1 | -1 }

So if the number of dame is odd, getting the last one (tedomari) is worth 1 point.

### UP (↑) & DOWN (↓)

An important infinitesimal is UP (uarr), defined as:

 uarr = { 0 | ast }

UP is positive, which means that Black (Left) wins, no matter who plays first.

UP is an ambiguous game, but is like a White sente. Black has the advantage because of the possibility of playing the reverse sente.

The negative of UP is DOWN (darr).

### Confused games

Two games are said to be confused if their difference game is a first player win, meaning that neither is greater than or equal to the other.

### UP STAR (↑*)

UP and STAR are confused. To see this, observe first that

 uarr - ast = uarr + ast  ; since ast + ast = 0 , so that ast is its own negative.

Now consider play in uarr + ast, also written  "↑"ast  and known as UP STAR.

Black moves in ast, leaving uarr, which wins, even though White has sente.
White moves in uarr to ast + ast, which equals 0, so White wins, having gotten tedomari.

So we see that  "↑"ast is a first player win, and hence that uarr is confused with ast.

### Double UP STAR (↑↑*)

Unlike UP STAR, Double UP STAR ( uarr + uarr + ast , written  "↑↑"ast ) is positive. Black moves to "↑↑", which is positive, and wins. White can do no better than to play to  uarr + ast + ast = uarr + 0 = uarr, which is also positive, and loses.

• Infinitesimals — for more on infinitesimals in chilled go and how to play them

## Notes

[1] At least until the dame stage. At that point you stop taxing.

[2] The difference between chilling and cooling: Cooling by 1 point is the same as chilling, except that if cooling a game, G, by less than 1 point produces a value infinitesimally close to a number, x, the cooled game is x. For go positions the difference between chilling and cooling by 1 is academic. See /discussion

## Discussion

Wouldn't it make sense to move the definition of the various infinitesimals to infinitesimals and move the content of infinitesimals to infinitesimals in chilled go? or infinitesimal example?? - Migeru

I think it would and it might help Evpsych who commented that the discussion of Combinatorial Game Theory was difficult to follow. However, I think I will leave this to someone who has a more comprehensive understanding than I do. I have noticed that BillSpight seems to know a lot about this topic. – Randall

Charles Bill is an expert. We'd all like to see him connect up his contributions here into something more readable.

Patrick Traill: An expert indeed: his work on thermography in go is cited in Winning Ways – but whether he has the time and inclination for the editorial work here is another matter! This page can say how infinitesimals work with cooling, but definitions and examples belong on infinitesimals.

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Chilling last edited by PJTraill on June 4, 2019 - 22:59