Gérard TAILLE wrote:
Bill Spight wrote:
Here is an sgf file for Three-Points-Without-Capturing, assuming no ko threats.
I am completly lost Bill. Let's come back to the beginning
Bill Spight wrote:
Here is the game tree: {-3|-3||-2}.
How do you find this game tree?
Why not {-3|-2} or {-3|-3||-2|-2} or some longueur notation to take into account black and white sequences?
Sorry. I have revised the game tree to show why the result is {-3|-3} when Black plays first.
The reason is that there is a dame left which either player can fill. (When White plays first there is no dame which either player can fill.) Normally, go players ignore the dame for territory scoring, because they do not alter the final score. For thermography they do not alter the result at or above temperature 0, so it is convenient to ignore them as well. However, they do matter when the original position is itself a number (score). Which means that they do matter for subterranean thermography.
We could show a large game tree with branches at each turn, but the long sequences down to the dame form units. We could show that fact by backing up the game tree, but that would be tedious.
Quote:
Bill Spight wrote:
This is a standoff, as neither player wishes to play first
OK Bill, no problem.
Seeing that neither player wishes to play first is the key to seeing that the long sequences are units.
Quote:
Bill Spight wrote:
In CGT this reduces to { |-2}
Why not {-3|-2}
Because {-3|-3} does not reduce to -3. Sure, go players normally ignore dame for territory scoring, but they do matter for CGT. In terms of CGT the prototypical value of -3 is when White has 3 Black prisoners, with prisoner return. I'll come back to prisoner return shortly. In this case Black has 2 prisoners and White has 5, for a net result of 3 prisoners for White. That fact is represented by the number, -3. With prisoner return does Black have a play? Yes, Black can return a prisoner so that Black has 1 prisoner and White has 5, for a net score of -4. Similarly, White can return a prisoner for a net score of -2. I.e.,
-3 = {-4|-2}
which is different from {-3|-3}.
Now, the simplest tree for -3 is { |-2}. For instance, suppose that White has 3 prisoners, but Black has none. Then Black has no play, but White can return 1 prisoner for a score of -2. I.e.,
-3 = { |-2}
The question is then
{-3|-3||-2} =?= { |-2} = -3 ; i.e.:
{-3|-3||-2} + 3 =?= 0
In CGT 0 is defined as a game in which the second player wins. As with Nim, a player wins when her opponent has no play. In its simplest form
0 = { | }
I.e., neither player has a play, so the first player loses and the second player wins.
Let's play the left side out. If the second player wins, it is equal to 0.
1) Black to play plays to
{-3|-3} + 3
Then White replies to
-3 + 3 = 0
which is a win for White, the second player.
2) Black returns a prisoner, for
{-3|-3||-2} + 2
Then White replies to
-2 + 2 = 0
and wins.
Now let White play first.
3) White to play plays to
-2 + 3 = 1
Which, OC, is a Black win.
So the second player wins and
{-3|-3||-2} + 3 = 0 , and
{-3|-3||-2} = { |-2}
Quote:
Bill Spight wrote:
To handle such games Berlekamp extended thermography downward to temperature -1, which he called subterranean thermography. Below temperature 0 the vertical blue scaffold shows up.
I do not see how such games can be handled without more information than the game tree.
To take another example, what will be the game score of the game tree {-20|-2}? You cannot caculate it can you?
Sure.
{-20|-2} = -3
Let's play out
{-20|-2} + 3
1) Black first can play to
-20 + 3 = -17
which is obviously a White win.
2) Black first can play to
{-20|-2} + 2
Then White plays to
-2 + 2 = 0
and wins.
3) White first can play to
-2 + 3 = 1
which is a Black win. So
{-20|-2} + 3 = 0 , and
{-20|-2} = -3
Edit: The thing is, {-20|-2} is already has a score. Neither player has to play if the game is equal to a number, i.e., a score. Hence, Three-Points-
without-Capturing. White does not have to capture to get 3 points.
Interesting that go players also said that
{-3|-3||-2} = -3
At least, until 1989.
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Gérard TAILLE
