Click Here To Show Diagram Code

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

Finally this position reduces to {4|*||*} which is somewhere between ↑ and ↑↑.
Taking a game with such infinetisimals as subgame, can CGT help to handle these infinitesimals for gaining tedomari or have we to read the game to choose the best move?
As an example I do not know if I can use the value 1 or 2 for the atomic weight of this infinitesimal and I do not know if I even can use the result of the atomic weight for a game with such subgame.

Well, most go infinitesimals are not as interesting as this one.

Fortunately, playing go infinitesimals is much easier than playing the endgame as whole. In practice, playing them nearly perfectly in actual games should be as easy as falling off a log for an amateur dan player who has studied them. A few years ago I wrote an article, which turned into a pair of articles, for the startup online go magazine, Myosu, now unfortunately defunct, about a very ancient game, the game for a pair of gold-petaled bowls, which has been reviewed a number of times over the centuries. Near the end the White player made the wrong play in a complicated go infinitesimal. I spotted it immediately, and, since the other go infinitesimals were simple, it was easy to read out the whole board to the end. AFAICT, I was the first reviewer to find that mistake. But I had the advantage of having read Mathematical Go.



My original play of the game at temperature 1 showed that it played a lot like an ↑. In a real game I would probably stop there and treat it much like one, delving more deeply only if necessary. OC, in our discussion I have to analyze it thoroughly, and even then I made a couple of slips. But finding its reduced form was quite satisfying and a lot of fun.

In reducing it we discovered a couple of interesting things. It is confused with *, but greater than ↑. OC, ↑ has an atomic weight of 1 and * has an atomic weight of 0, so this game appears to have an atomic weight of 1. I find calculating atomic weight by the definition to be difficult and tedious, but if this infinitesimal is confused with ↑* it will have an atomic weight of 1. Let's see.

{4|*||*} + * + {*|0}

White to play can win by playing to * on the left, since * + * = 0 and {*|0} < 0.

Black to play can win by playing to * on the right, since {4|*||*} > 0.

So {4|*||*} has atomic weight 1, it is greater than ↑ and confused with * and ↑*. That should be enough information to play it nearly flawlessly in a real game without much effort.

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

Visualize whirled peas.

Everything with love. Stay safe.

Thank you Bill, it is a pleasure, OC


Here is a game example

Click Here To Show Diagram Code

[go]$$W White to play
$$ -----------------------
$$ | O X W W . . . O - - |
$$ | . X W X X X X O - - |
$$ | . X X - - - O O - - |
$$ | O O X X X - - - - - |
$$ | . X . . X - - - - - |
$$ | O O X X X - - - - - |
$$ | O O O O X - - - - - |
$$ | O X . . X - - - - - |
$$ | O O O O O O O X - - |
$$ | O . B . . . X X - - |
$$ -----------------------[/go]

The atomic weight of this game is zero. In this case because the threat of white marked stones is greater than the threat of black marked stone white to play normally wins tedomari but here it is not the case is it?

I don't know about normally, but with only the two long corridors White wins. I.e., their sum is negative. But, OC, the other sum is positive, so it's a question of which, if either, is greater.

Click Here To Show Diagram Code

[go]$$W White to play
$$ -----------------------
$$ | O X W W 4 3 1 O - - |
$$ | . X W X X X X O - - |
$$ | 5 X X - - - O O - - |
$$ | O O X X X - - - - - |
$$ | . X . . X - - - - - |
$$ | O O X X X - - - - - |
$$ | O O O O X - - - - - |
$$ | O X . 8 X - - - - - |
$$ | O O O O O O O X - - |
$$ | O . B 7 6 2 X X - - |
$$ -----------------------[/go]

With normal play Black gets the last play. at 6 leaves a positive sum on the table.

Click Here To Show Diagram Code

[go]$$W White to play, variation
$$ -----------------------
$$ | O X W W 6 5 3 O - - |
$$ | . X W X X X X O - - |
$$ | 1 X X - - - O O - - |
$$ | O O X X X - - - - - |
$$ | . X . . X - - - - - |
$$ | O O X X X - - - - - |
$$ | O O O O X - - - - - |
$$ | O X . 8 X - - - - - |
$$ | O O O O O O O X - - |
$$ | O . B 7 4 2 X X - - |
$$ -----------------------[/go]

here leads to a transposition, thanks to the stones.

Click Here To Show Diagram Code

[go]$$B Black to play
$$ -----------------------
$$ | O X W W 5 4 2 O - - |
$$ | . X W X X X X O - - |
$$ | 7 X X - - - O O - - |
$$ | O O X X X - - - - - |
$$ | 8 X . . X - - - - - |
$$ | O O X X X - - - - - |
$$ | O O O O X - - - - - |
$$ | O X . 9 X - - - - - |
$$ | O O O O O O O X - - |
$$ | O . B 6 3 1 X X - - |
$$ -----------------------[/go]

Normal play with Black first allows White to play with sente before replying with , but the positive sum remains on the board, and Black gets the last play.

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

Visualize whirled peas.

Everything with love. Stay safe.

Here is a theoritical question on infinitesimals.

Firstly we know that we have ↑ || * and it may be difficult to compare a move in ↑ and a move in *. However we know also that the atomic weight of ↑ is equal to 1 and the atomic weight of * is equal to 0. As a consequence it may seem preferable for black to play in * rather that ↑ in order to not lose this atomic weight of 1.

Secondly we know that * is a special infinitesimal with the property * + * = 0. Taking a game G made of infinitesimals we can thus always assume that we have only 0 or 1 * in the game (the other * being miai).

Now is my question : Assume a game G + ↑ + * and, because I add here an *, assume there are no * in G. In this context (no * in G) can we say that a black move in * dominates a black move in ↑ ?

Makes sense. That's how Black first wins in ↑*.

But infinitesimals can be tricky. In ↑* the Black play to ↑ reverses to 0, for instance.



Not so special, outside of go. All nim games have that property.



Well, *2 = {0,*|0,*} exists in chilled go. See https://senseis.xmp.net/?MoreInfinitesimals
Also, * can be hidden. For instance, the corridor one longer than ↑ is ↑↑*, but you can't see the *. Also, there is a * in {4|*||*} that is not obvious.



Edited for clarity because I may have misunderstood you:

If, accounting for hidden *s, there is an odd number of *s in G, given ↑*, then the Black move in the ↑ to * is correct, because the * adds to the *s in G to equal 0. Otherwise, you have to take the * into account. If all that is left is ↑*, then the play by Black in the * to ↑ is correct. Domination in CGT is a stronger concept. I believe. David Moews (pronounced Mays) did some work in his master's thesis on that question, among others. I have not read his thesis. I suppose it is on file at UC Berkeley.

_________________
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 look at Gérard's infinitesimal with some White corridors. Look at the single corridors first, then the double corridors.



A bit of a surprise, n'est-ce pas?

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

Visualize whirled peas.

Everything with love. Stay safe.

Oops it is not my question Bill. In the game G + ↑ + *, I do want to know if it is correct for black to play in ↑*.
Taking the game G + ↑ + *, black has three options: black plays in G, or black plays in ↑, or black plays in *.
Assuming there are no * in G (I mean a "pure" *; I accept all hidden *) then does the option black in * dominate the option black in ↑ ?

Yes, I thought I might have misunderstood you and not been clear. I just edited my reply, but let me try again.

First, domination in CGT is, I believe, a stronger concept than domination in von Neumann game theory. I think you mean the latter.

In chilled go, * was the only known impartial infinitesimal when Mathematical Go came out in 1994. Since then Nakamura Teigo discovered one form of *2 = {0,*|0,*} and later I discovered another one. The atomic weight of *2 is also 0. But they rarely come up.

You need to keep track of the parity of the *s to find out if they add to 0. If they do, you don't want to play one by accident. There is nothing special about ↑* in this regard. If and when the time comes when Black faces the question of playing in ↑*. the play in the * is correct unless there are an even number of them.

As for playing in G, I don't have enough information to say. But * has an atomic weight of 0 and is not in general high on the list of candidate plays. And for Black to play in ↑ is like filling your own dame in a semeai, and is almost always at the bottom of the list of candidate plays.

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

Visualize whirled peas.

Everything with love. Stay safe.

Oops now my name is on an infinitesimal ! I am becoming famous !

BTW my idea can be use in various ways:

Click Here To Show Diagram Code

[go]$$
$$ -----------------------
$$ | . X . . . . . . . . |
$$ | O X . . . . . . . . |
$$ | . X . . . . . . . . |
$$ | a X X X . . . . . . |
$$ | O O O X X . . . . . |
$$ | . X X b X . . . . . |
$$ | O O O X X . . . . . |
$$ | . . O O O . . . . . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------[/go]

Here black can choose "a" to reach ↑ in sente or "b" to reach tiny in sente (instead of * and ↑ in my first example). The goal is to allow black to choose between two fuzzy options allowing interesting games. White to move can play "a" and reach ↑ in gote.

Bill, in the link https://senseis.xmp.net/?GoInfinitesimals you said:
I (Bill Spight) have heard that other infinitesimals have been constructed, that involve seki which alters the parity of the dame on the board, but I have not seen one. There seem to be no other infinitesimals in regular go.
What do you mean? It is not very clear for me. Are you in a noraml game or in a chilled game?

That's regular go. OC, since dame do not affect the modern territory score, changing the parity of the dame affects only area score.

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

Visualize whirled peas.

Everything with love. Stay safe.

OK, and what means a seki which alters the parity of the dame ?

Just for fun you case use my idea to build as many options as you want and reach quite difficult positions. You are limited only by your imagination!

Click Here To Show Diagram Code

[go]$$W
$$ ---------------------------------------
$$ | X . . . X . O . . . . . . . . . . . . |
$$ | X O O O O O O . . . . . . . . . . . . |
$$ | X . . X O . . . . . . . . . . . . . . |
$$ | X O O O O . . . . , . . . . . , . . . |
$$ | X . X O . . . . . . . . . . . . . . . |
$$ | X O O O O . . . . . . . . . . . . . . |
$$ | X X O O O . . . . . . . . . . . . . . |
$$ | . O . . O . . . . . . . . . . . . . . |
$$ | X X O O O . . . . . . . . . . . . . . |
$$ | X X X O O . . . . , . . . . . , . . . |
$$ | . O O . O . . . . . . . . . . . . . . |
$$ | X X X O O . . . . . . . . . . . . . . |
$$ | . . X X X . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

The play at temperature 1 (t = 0 in chilled go)

Click Here To Show Diagram Code

[go]$$W White first
$$ -----------------------
$$ | . X . . . . . . . . |
$$ | O X . . . . . . . . |
$$ | 2 X . . . . . . . . |
$$ | 1 X X X . . . . . . |
$$ | O O O X X . . . . . |
$$ | 3 X X 4 X . . . . . |
$$ | O O O X X . . . . . |
$$ | . . O O O . . . . . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------[/go]

Result: +3

Click Here To Show Diagram Code

[go]$$B Black first
$$ -----------------------
$$ | . X . . . . . . . . |
$$ | O X . . . . . . . . |
$$ | . X . . . . . . . . |
$$ | 1 X X X . . . . . . |
$$ | O O O X X . . . . . |
$$ | 2 X X 3 X . . . . . |
$$ | O O O X X . . . . . |
$$ | . . O O O . . . . . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------[/go]

Result: +4 (+3 at chilled go)

The new Gérard infinitesimal is

{6|↑||↑}

It is obviously greater than 0. It is greater than * and confused with ↑.

I think that it's atomic weight is 2.

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

Visualize whirled peas.

Everything with love. Stay safe.

Without ko complications or seki, regular territory values (sans komi) are 6*, or 7, or 8*, etc., with an equal number of plays by each player. So 6* becomes 7 at area scoring, 8* becomes 9, etc. If you have a seki with an odd number of dame when fully played out, those values become 6, 7*, 8, etc., and the area scores become even.

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

Visualize whirled peas.

Everything with love. Stay safe.

The Gérard infinitesimal 2 with some White corridors.



BTW, Gérard, this addresses your last question.

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

Visualize whirled peas.

Everything with love. Stay safe.

In this case it is quite obvious with a simpe miai situation:

Click Here To Show Diagram Code

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

Each player can choose to change or not the parity of dame.

yes Bill, with the experience of your first analyse it is now far simplier

Click Here To Show Diagram Code

[go]$$
$$ -----------------------
$$ | . X . . . . . . . . |
$$ | O X . . . . . . . . |
$$ | . X . . . . . . . . |
$$ | 3 X X X . . . . . . |
$$ | O O O X X . . . . . |
$$ | 2 X X 1 X . . . . . |
$$ | O O O X X . . . . . |
$$ | . . O O O . . . . . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------[/go]

after , reverses and black must play
but by beginning with black can transpose and we can conclude that the option black can be deleted.

If it is white to play

Click Here To Show Diagram Code

[go]$$W
$$ -----------------------
$$ | . X . . . . . . . . |
$$ | O X . . . . . . . . |
$$ | . X . . . . . . . . |
$$ | 1 X X X . . . . . . |
$$ | O O O X X . . . . . |
$$ | a X X . X . . . . . |
$$ | O O O X X . . . . . |
$$ | . . O O O . . . . . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------[/go]

dominates a white move at "a"

Finally, provided some training, the result is {6|↑||↑} is not so hard to discover.

G = {5|0} + {*|-5} = {{5*|0},{5*|*}||{0|-5},{*|-5}}

G <> 0 ; Black can move to {5*|*} and White can move to {*|-5}

G - * <> 0 ; Black can move to {5*|0} + * and White can move to {0|-5} + *

G - *2 <> 0 ; *2 = {0,*|0,*} Whichever option in G Black or White chooses, after the other replies, the first player has the winning option in *2.

In fact, G is confused with every nimber (nim heap). Very nice.

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

Visualize whirled peas.

Everything with love. Stay safe.