Corridor

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

Corridors are go positions that are effectively one-dimensional. They yield numbers and infinitesimals in chilled go, in which they are analyzed in terms of combinatorial game theory (CGT).

In the endgame, there are cases where one should (surprisingly) attack shorter corridors before longer ones, as illustrated in Mathematical Go Endgames, 1.2 (p. 3).

Table of contents Table of diagrams
Closed corridor of length 3, empty
Open corridor of length 4, empty
Gold corridor (star)
Gold corridor (star, alternative version: the gold is extra empty space rather than a prisoner)
Gold corridor (rich)
Example 1: closed corridor of length 2
Example 2: closed corridor of length 3
Example 3: closed corridor of length 4
Example 4: open corridor of length 3
Example 4a: What is 4×?
Example 5: open corridor of length 4
Example 5a: What is 2×⅝?
Example 5b: White first
Example 5c: Reversal
Equivalent
white to play

Types of corridor

Corridors can be open or closed. They can be empty or can contain "gold" at the end (terminology from Endgame 2 - Values). A gold corridor is "rich" if the gold is worth more than two points; otherwise Jasiek calls it a "star corridor".

[Diagram]
Closed corridor of length 3, empty  
[Diagram]
Open corridor of length 4, empty  
[Diagram]
Gold corridor (star)  

N.B. This corridor is properly a double up star, written ``"↑↑"ast``.

[Diagram]
Gold corridor (star, alternative version: the gold is extra empty space rather than a prisoner)  

N.B. This corridor is also double up star.

[Diagram]
Gold corridor (rich)  


Counts and values of empty corridors

For closed corridors, a move by either player is gote, sente, or ambiguous, so we can work out the counts and move values by straightforward miai counting. For open corridors, there are some complications. If you want to skip the working out and just see the answers, go to the summary.


Simple empty closed corridors

[Diagram]
Example 1: closed corridor of length 2  

This is the smallest interesting example. A single move settles the position. If black plays a then there's a point of black territory. If white plays a then there's no territory. So the count is and the move value is .

[Diagram]
Example 2: closed corridor of length 3  

If black plays a here, there's two points of territory. If white plays a, there's an unsettled position, so we have to think about the follow-up moves. But luckily we've already done that: we know from example 1 that it's a half-point position. So the count is the average of 2 points and point: that's 1 points. And a black move changes the count to , meaning that the value of a black move is point. As you'd expect, a white move has the same value.

[Diagram]
Example 3: closed corridor of length 4  

By now you might be starting to see a pattern. Black a: three points. White a: see previous example. Count = 2⅛; move value = ⅞.

General pattern: Corridor of length n has count ``n - 2 + 1/2^{n-1}`` and move value ``1 - 1/2^{n-1}``.

If you like writing down equations, you can prove this using a [ext] recurrence relation. Let ``a_n`` be the count of a closed corridor of length n. Then for ``n>=2``, taking the average of black and white follow-up moves gives ``a_n = (n-1+a_{n-1})/2``.

Notice that the move value is a little bit less than 1, and gets closer to 1 for long corridors. To put it another way, if you push into a simple, empty, closed corridor, it gets shorter and the value of follow-up moves is less. So playing in a simple, empty, closed corridor is always gote.


Open corridors

This gets interesting. You might think you can apply the same method with an extra step. If white pushes into an open corridor, you get a shorter open corridor. If black plays first, it turns into a closed corridor. Both moves look like gote, so you should be able to average the results, right? Let's take a look.

[Diagram]
Example 4: open corridor of length 3  

If white plays a, what's left is a bamboo joint with no territory. If black plays a, we've created the position of example 1 above, so it's half a point. And both options may appear to be gote. On average, then, you might think black has a point of territory. We can check this with the method of multiples: four of these corridors should add up to a full point.

[Diagram]
Example 4a: What is 4×?  

Oops, black didn't get any territory! It looks like an open corridor of length 3 is actually worth zero points. Except for ko threats, it is equivalent to a dame.

Try again with a longer corridor:

[Diagram]
Example 5: open corridor of length 4  

Black a gives us a closed corridor that we know is 1 points. White a gives us an open corridor of length 3. We now know that the shorter corridor is worth zero points, so if Black a is gote, then this one should be worth half of 1, or ⅝ of a point.

So two of these should be worth 1. By the principle that the first player can "round to an integer" we should get 2 points if black goes first, or 1 point if white goes first.

[Diagram]
Example 5a: What is 2×⅝?  

No, there's only one point for black here. You can try a few different move orders, same result. Or try four or eight multiples, it won't help. This corridor is exactly half a point.
What's gone wrong here is that we've been treating all the moves as gote, and trying to average the follow-up positions. Perhaps B1 in these corridors is actually sente, which is how we have played them so far.

Let's see what happens when White plays first.

[Diagram]
Example 5b: White first  

The result is the same, one point for Black. The problem is that the reverse sente, W1, should gain something by comparison with when Black plays first with sente, as in the previous diagram. But it doesn't.
So what is going on here? The next diagram supplies the answer.

[Diagram]
Example 5c: Reversal  

The sequence, B1 - W2, B3 is gote, gaining on average point. B1 temporarily raises the local temperature to , so W2 replies, and then B3 continues. We say that B1 reverses. We consider the sequence through B3 as a unit.


Blocking a simple empty open corridor reverses through White's reply.

What this means is that these two corridors are (except in terms of ko threats) equivalent.

[Diagram]
Equivalent  


And that is so no matter what is the length of the closed corridor.

Except for ko threats, a simple empty closed corridor is equivalent to a simple empty open corridor two spaces longer.

Summary

Closed corridor of length ``n``: count = ``n - 2 + 1/2^{n-1}`` and move value = ``1 - 1/2^{n-1}``.

Open corridor of length ``n``: treat as a closed corridor of length ``n-2``.

Counts and values for small corridors:

small corridors
closed length open length count move value
1 3 0 0
2 4
3 5 1
4 6 2⅛
5 7 ``3 1/16`` ``15/16``


Example problem

[Diagram]
white to play  

Should white play a attacking a closed corridor of length 4, or b attacking the open corridor of length 5? Using the theory above, you should be able to calculate the best move and the final result.


See also


(Sebastian:) Besides their use for theoretical considerations, can there anything interesting be said about them for practical play, other than that the bamboo joint is an open corridor?

Bill: The whole point of chilling is practical play. :-) It enables already known results in CGT to be applied to go.


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Corridor last edited by 107.210.159.110 on December 25, 2019 - 16:01
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