Thermography

Bill Spight · Post by **Bill Spight** » Fri Nov 13, 2020 10:18 am

Gérard TAILLE wrote: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.

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

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

Gérard TAILLE wrote:Secondly we know that * is a special infinitesimal with the property * + * = 0.

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

Gérard TAILLE wrote: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).

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.

Gérard TAILLE wrote: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 ↑ ?

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.

Bill Spight · Post by **Bill Spight** » Fri Nov 13, 2020 10:40 am

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?

Gérard TAILLE · Post by **Gérard TAILLE** » Fri Nov 13, 2020 11:57 am

Bill Spight wrote:
Gérard TAILLE wrote: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 ↑ ?
Actually, it is the opposite, which is probably what you had in mind. If, accounting for hidden *s, there is an odd number of *s in G, then the Black move in ↑* is correct, because the * adds to the *s in G to equal 0. 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.

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 ↑ ?

Bill Spight · Post by **Bill Spight** » Fri Nov 13, 2020 12:44 pm

Gérard TAILLE wrote:
Bill Spight wrote:
Gérard TAILLE wrote: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 ↑ ?
Actually, it is the opposite, which is probably what you had in mind. If, accounting for hidden *s, there is an odd number of *s in G, then the Black move in ↑* is correct, because the * adds to the *s in G to equal 0. 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.
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.

Gérard TAILLE · Post by **Gérard TAILLE** » Fri Nov 13, 2020 2:13 pm

Bill Spight wrote:Let's look at Gérard's infinitesimal with some White corridors.

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.

Gérard TAILLE · Post by **Gérard TAILLE** » Fri Nov 13, 2020 2:32 pm

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?

Bill Spight · Post by **Bill Spight** » Fri Nov 13, 2020 2:46 pm

Gérard TAILLE wrote: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.

Gérard TAILLE · Post by **Gérard TAILLE** » Fri Nov 13, 2020 3:01 pm

Bill Spight wrote:
Gérard TAILLE wrote: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.

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

Gérard TAILLE · Post by **Gérard TAILLE** » Fri Nov 13, 2020 3:47 pm

Gérard TAILLE wrote:
Bill Spight wrote:Let's look at Gérard's infinitesimal with some White corridors.
Oops now my name is on an infinitesimal ! I am becoming famous !

BTW my idea can be use in various ways:

$$
$$ -----------------------
$$ | . X . . . . . . . . |
$$ | O X . . . . . . . . |
$$ | . X . . . . . . . . |
$$ | a X X X . . . . . . |
$$ | O O O X X . . . . . |
$$ | . X X b X . . . . . |
$$ | O O O X X . . . . . |
$$ | . . O O O . . . . . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------
Click Here To Show Diagram Code
[go]$$ $$ ----------------------- $$ | . 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.

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]

Bill Spight · Post by **Bill Spight** » Fri Nov 13, 2020 5:18 pm

Gérard TAILLE wrote:
Bill Spight wrote:Let's look at Gérard's infinitesimal with some White corridors.
Oops now my name is on an infinitesimal ! I am becoming famous !

BTW my idea can be use in various ways:

$$
$$ -----------------------
$$ | . X . . . . . . . . |
$$ | O X . . . . . . . . |
$$ | . X . . . . . . . . |
$$ | a X X X . . . . . . |
$$ | O O O X X . . . . . |
$$ | . X X b X . . . . . |
$$ | O O O X X . . . . . |
$$ | . . O O O . . . . . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------
Click Here To Show Diagram Code
[go]$$ $$ ----------------------- $$ | . 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.

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.

Bill Spight · Post by **Bill Spight** » Fri Nov 13, 2020 5:27 pm

Gérard TAILLE wrote:OK, and what means a seki which alters the parity of the dame ?

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.

Bill Spight · Post by **Bill Spight** » Fri Nov 13, 2020 7:18 pm

The Gérard infinitesimal 2 with some White corridors.

BTW, Gérard, this addresses your last question.

Gérard TAILLE · Post by **Gérard TAILLE** » Sat Nov 14, 2020 3:38 am

Bill Spight wrote:
Gérard TAILLE wrote:OK, and what means a seki which alters the parity of the dame ?
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.

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.

Gérard TAILLE · Post by **Gérard TAILLE** » Sat Nov 14, 2020 3:56 am

Bill Spight wrote:
Gérard TAILLE wrote:
Bill Spight wrote:Let's look at Gérard's infinitesimal with some White corridors.
Oops now my name is on an infinitesimal ! I am becoming famous !

BTW my idea can be use in various ways:

$$
$$ -----------------------
$$ | . X . . . . . . . . |
$$ | O X . . . . . . . . |
$$ | . X . . . . . . . . |
$$ | a X X X . . . . . . |
$$ | O O O X X . . . . . |
$$ | . X X b X . . . . . |
$$ | O O O X X . . . . . |
$$ | . . O O O . . . . . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------
Click Here To Show Diagram Code
[go]$$ $$ ----------------------- $$ | . 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.
The play at temperature 1 (t = 0 in chilled 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 . . . . . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------
Click Here To Show Diagram Code
[go]$$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

$$B Black first
$$ -----------------------
$$ | . X . . . . . . . . |
$$ | 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 . . . . . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------
Click Here To Show Diagram Code
[go]$$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|↑||↑}

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.

Bill Spight · Post by **Bill Spight** » Sat Nov 14, 2020 4:57 am

Gérard TAILLE wrote:
Bill Spight wrote:
Gérard TAILLE wrote:OK, and what means a seki which alters the parity of the dame ?
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.
In this case it is quite obvious with a simpe miai situation:

$$B
$$ ---------------------
$$ | . a X O . X X b X |
$$ | O . X O O O O O X |
$$ | X X X O . . . - X |
$$ | . X O O . . . - - |
$$ | X X O . . . . . . |
$$ | O O O . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ --------------------
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.

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.

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