If L(G) > R(G) then G is not a number.

If L(G) = R(G) != G then G is not a number

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Oops it does not help very much Bill.

Definition :Let G be a short game. We define numbers L(G) and R(G), the left and right stops of G, as follows.
If G is (equal to) a number, we set L(G) = R(G) = G. Otherwise,
L(G) = max(R(GL) and R(G) = min(L(GR).

With this definition, in order to calculate L(G) and R(G), I need first to know if G is a number! Obviously I cannot use L(G) and R(G) to know if G is a number, can I?

I understand how that definition came about. It assumes that you know what a number is. In that case, anything that does not fit the definition of a number is not a number. Easy.

But what if we do not know whether G is a number or not. How then can we use the first if clause? So let's use the second clause. A game, G, is not a number iff

max(R(GL)) > min(L(GR)) or

max(R(GL)) = min(L(GR)) != G

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

I am sure you know it is not quite satisfactory because the problem occurs at each step of the induction : to calculate R(GL) and L(GR) I still need to know if GL or GR are numbers etc.

I am wondering if I can simply play the games "black to move" and "white to move" and compare the two results (which are here only integers and not any number). If the score "black to move" is greater than the score "white to move" is it true that it could not be a number ?

Yes, I thought about that, and I'm not sure. OC, in go it is not a problem. And if G is reduced to its simplest terms, it's not a problem. Certainly the test for whether G is a number may require reducing G to its simplest terms. I don't think we can get away from the possibility that that may be necessary.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Eventually your suggestion of guessing that a game G is equal to a number x seems a good solution.
1) if G=x then G is a number
2) if G||x then G is not a number
Otherwise try another number!
Surely with experience the convergence will be quick won'it be?

Well, as you can tell, I have little experience with that question. It is not difficult with go.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

How do we have to adapt CGT to Go game with area counting?
What about infinitesimals : do we have to replace chilling by cooling by 2.
In addiion, do we have to use special technics to get the last dame point which is relevant with area counting?

We can just plug in area scores.



With the ko caveat, chilled area scoring is a form of territory scoring and chilled territory scoring is chilled go. Then optimal play by chilled go is also optimal play by territory scoring, which is also optimal play by area scoring. (You simply bypass the Japanese rules about points in seki.)



No. If, for instance, Black takes the last dame for a net score of +1 by area scoring, the net score by (non-Japanese) territory scoring is 0. But White has the move and loses under CGT. OC, if ties are allowed there will be a difference.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

As you may know french players use area scoring.

Click Here To Show Diagram Code

[go]$$B
$$ -----------------
$$ | O . O . . . . |
$$ | X X O . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------[/go]

With territory scoring the top-left subgame looks like {2|0} which chills in {1|1} = 1*.
With area scoring this subgame is equal to {4|0}. In order to get an infinitesimal I have to use cooling by 2 haven't I?
BTW it looks like good news because in semeai cooling by 2 has also to be used!

When I convert from area scoring I normalize the positions by subtracting the original number of stones.

Click Here To Show Diagram Code

[go]$$B Three Points
$$ -----------------
$$ | C W X X . . . |
$$ | B B O X . . . |
$$ | B B O X . . . |
$$ | O O O . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------[/go]

For instance, assuming no ko fight in area scoring the 6 marked points are worth 0, since each player will have 3 live stones there. Subtracting 4 points for the original stones and adding 1 point for the original stone gives -3 points.

Click Here To Show Diagram Code

[go]$$B Seki
$$ -----------------
$$ | W W W X X . . |
$$ | C C B O X . . |
$$ | B B B O X . . |
$$ | O O O O X . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------[/go]

By area scoring the corner is worth +1 point. But the CGT value is {9|-9||9|-9} = 0, which is also the territory value. Normalizing the area score yields the CGT value.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Let's concentrate on infinitesimals

Click Here To Show Diagram Code

[go]$$B
$$ -----------------
$$ | W C C O . . . |
$$ | X X X O . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------[/go]

Here the concerned area is made of the three marked points and, depending of the play, the resulting score will stand between +3 and -3.
The game looks like {+3||+1|-3} and here again I have to use cooling by 2 to reach the infinitesimal {+1||+1|+1}.

And {+1||+1|+1} = 1↑.

OC, when we are focused on getting the last play we simply ignore the numerical part.

I remember the first environmental go game between Jiang Jujo and Rui Naiwei was played by Ing rules, but after the game we found out that they were counting the endgame using territory scoring, which screwed up the environmental values on the cards from the standpoint of area scoring. That is the usual practice among Chinese pros, it seems. OC, when we published out analysis of the game we used territory scoring.

If we normalize the area scores we get {4||2|-2} at the start and 2↑ at the end. FWIW.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

At least it is clear you use cooling by 2 in order to transform {4||2|-2} into 2↑.

Considering the normalization issue I do not understand clearly why you use this way of normalizing a game in area scoring. Isn't it more natural to use as the reference score the mean value?
BTW don't you use the mean value to normalize a game in territory scoring?
With the mean value and the diagram above I find :
territory scoring : G = {+1||0|-2} which chilled to {0||0|0} = {0|*} = ↑
area scoring : G = {+2||0|-4} which with cooling by 2 gets again G = {0||0|0} = {0|*} = ↑

Well, as I indicated with Three Points without Capturing and the seki, simply plugging area scores into CGT can give the wrong mean value. The discrepancy with Three Points without Capturing is indeed 3 points, unless you normalize the scores.



As it turns out, the main normalization you need with
territory scoring is adjusting for the group tax.



Yes, that form of normalization yields any infinitesimal.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Now it's quite clear for infinitesimals let's take sekis

You wrote {9|-9||9|-9} = 0
Assume the game G = {13|-5||13|-5}
Neither black nor white want to play in this game. How do you calculate the value of such game for a CGT point of view?

{13|-5||13|-5} = 0

By definition, 0 is a second player win.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Click Here To Show Diagram Code

[go]$$B Seki
$$ -----------------
$$ | W W W X X C C |
$$ | C C B O X X X |
$$ | B B B O X C C |
$$ | O O O O X X X |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------[/go]

This result {13|-5||13|-5} = 0 proves that describing a game by {13|-5||13|-5} fails to show the referenced score used (normalization?)
Maybe we have to be carreful when transforming a go game into a CGT game.
You hightlighted the Three Points without Capturing and the sekis which are zugswang positions and cannot be scored by CGT. My view is the following. When you encounter a zugswang position it cannot help to go further and show a sub game tree because CGT will not be able to put a score on the position. In that case the go player have to describe the zugswang simply by the number that have to be eventually allocated to the position due to go player knowledge.

Click Here To Show Diagram Code

[go]$$B Seki
$$ -----------------
$$ | W W W X X . . |
$$ | C C B O X . . |
$$ | B B B O X . . |
$$ | O O O O X . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------[/go]

Maybe it is a mistake to define this seki by the game {9|-9||9|-9}.
In territory scoring this position is simply the number 0 and in area scoring it is the number +1.

Click Here To Show Diagram Code

[go]$$B Three Points
$$ -----------------
$$ | C W X X . . . |
$$ | B B O X . . . |
$$ | B B O X . . . |
$$ | O O O . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------[/go]

Here it is the same:
In territory scoring this position is simply the number -2 (2 more prisonners for white) and in area scoring it is the number 0.

That way the CGT game corresponding to the go game seems far better.

By territory scoring (sans group tax) we have this sum.

{8|-8||10|-6} + 4 != {12|-4||14|-2} = 0

You can only add a number to each option in a game if the game is in its simplest form.

{8|-8||10|-6} + 4 = 0 + 4 = 4




Yes, that's what I am talking about.



It is not that this Three Points without Capturing position is worth -2 by territory scoring, it is worth -2 by the Japanese 1989 rules. By other territory rules it is worth -3, as advertised.

Except for the group tax, non-ko area scores may be normalized to CGT scores by accounting for the original number of stones.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Yes Bill. Assume I am using J89 rules. How I translate to the CGT game the fact that the Three Points without Capturing position is worth -2 rather than -3?

Concerning seki let's take a more general position with eyes and dead stones:

Click Here To Show Diagram Code

[go]$$B Seki
$$ -----------------
$$ | C B W C W W C |
$$ | B C W W X W B |
$$ | W W W X X X W |
$$ | X X X . X W W |
$$ | . . . X X W C |
$$ | . . . . X W B |
$$ | . . . . X W C |
$$ -----------------[/go]

Area counting : if I understand your normalization I count 15 white stones against 4 black stones and I normalized the position to -11.
Thus the area looks like a game of the form -11 + G with G being a seki counted 0.
As a go player I evaluate the position 3 points for black and 21 points for white for a total of -18.

BTW in territory scoring I count this position +4 (at the very end of the gam, after the last dame point, white can force the capture of 4 black stones).

How translate in CGT game such information concerning the rules used?