Thermography

Gérard TAILLE · **#201**

Bill Spight wrote:

G = H

says nothing about miai values, which is a go concept, not a CGT concept. Now it is true that

{2|0} + {2|0}

and

2

have the same effective miai values associated with them. That is true for all games that are equal to 2.

https://senseis.xmp.net/?Thermography:
The thermograph of a game is a graphical representation of the value of playing in it at different temperatures. The axes of the thermograph are the temperature (vertical) and the count (horizontal). At each temperature, the left wall of the thermograph shows what Left (Black) gets by playing first, and the right wall shows what Right (White) gets by playing first. At high enough temperatures the thermograph is topped by a vertical mast, which coincides with the left and right walls, and represents the count of the game. The temperature at the base of this vertical mast is the temperature of the game. In go it corresponds to miai value.

Three points concerning this thermograph definition:
1) the thermograph is not colored
2) the temperature of the game is the temperature at the base of the vertical mast
3) "miai value" is not defined in thermography, it is only a notion in go world, corresponding (?) to the temperature of the game in thermography

It is what you said in your post and I agree with you.
In practice the problem is that in many articles about thermography the wording miai value is used instead of temperature of the game.
I understand you proposed also the wording effective miai value to avoid misunderstanding. Isn'it simplier to keep the wording temperature of the game?

In any case what is the defintion of what you call "miai value"?
In the game G = {2|0}+{2|0} the temperature of the game is 0 according to the defintion but you seem to claim that the "miai value" is different. What is the defintion of miai value? How do you calculate it with this game G? Due to the purple color it seems we have here a tally = 0 and it seems we are facing a double sente area which does not exist does it?

Bill Spight · **#202**

Gérard TAILLE wrote:

Bill Spight wrote:

G = H

says nothing about miai values, which is a go concept, not a CGT concept. Now it is true that

{2|0} + {2|0}

and

2

have the same effective miai values associated with them. That is true for all games that are equal to 2.

https://senseis.xmp.net/?Thermography:
The thermograph of a game is a graphical representation of the value of playing in it at different temperatures. The axes of the thermograph are the temperature (vertical) and the count (horizontal). At each temperature, the left wall of the thermograph shows what Left (Black) gets by playing first, and the right wall shows what Right (White) gets by playing first. At high enough temperatures the thermograph is topped by a vertical mast, which coincides with the left and right walls, and represents the count of the game. The temperature at the base of this vertical mast is the temperature of the game. In go it corresponds to miai value.

Three points concerning this thermograph definition:
1) the thermograph is not colored
2) the temperature of the game is the temperature at the base of the vertical mast
3) "miai value" is not defined in thermography, it is only a notion in go world, corresponding (?) to the temperature of the game in thermography

Guilty as charged.

Color is not a defining property of thermographs, but they may be colored. I came up with that idea.

I introduced the term, miai value, to English speaking go players on rec.games.go in the 1990s, in an effort to clarify our thinking about move values. Western players had learned what is called in Japanese deiri values, but most of them assumed that they meant the same thing as miai values. My efforts almost completely failed. Instead, the new term for move values caused more confusion than it alleviated. :sad:

Too much terminology!

I don't know if I was the first to introduce the idea of temperature on rec.games.go, but I certainly used it. However, go players online adopted the term, not to refer the temperature of a position (game) but to the temperature of the whole board. Fine. Language at work.

But that means that the technical meaning of temperature in CGT is different, and using it would cause confusion. So I don't. I have enough trouble with the technical meaning of sente. :lol:

Quote:

It is what you said in your post and I agree with you.
In practice the problem is that in many articles about thermography the wording miai value is used instead of temperature of the game.

Yes, that's probably my attempt to communicate with regular go players, at least those who use the term, miai value. That is my aim on SL. As such, it is informal language. Most people who do use miai value use it to refer to both plays and positions, and to use it where in CGT we would use temperature.

Quote:

I understand you proposed also the wording effective miai value to avoid misunderstanding. Isn'it simplier to keep the wording temperature of the game?

By the time of that writing temperature was used differently by go players. No point in creating confusion. IMX, a surprisingly large number of people want a word to mean only one thing. Most words have multiple meanings, but ambiguity does cause misunderstanding and confusion.

Quote:

In any case what is the defintion of what you call "miai value"?

Well, as a go term, it is not exactly unambiguously defined.

It seems to me that it means how much a gote play or gote sequence as a unit gains, on average. Whether it can be used for how much the first play in a sente sequence gains is unclear. I have not seen it used for that. OC, it can be used that way, I suppose. I have had discussions about whether it can be used for mistakes. Again, I have never seen it used that way in professional writing. At one time I argued against such usage, but now I think it can be useful to do so, as long as you say the play is a mistake, or may be.

Quote:

In the game G = {2|0}+{2|0} the temperature of the game is 0 according to the defintion but you seem to claim that the "miai value" is different.

In my early studies of CGT what I read about the temperature of a game is the temperature at the base of its vertical mast. However, in discussing a miai pair like {2|0} + {2|0}, Berlekamp, who certainly knew what temperature means, said that it was ambiguous, that you could consider the temperature of the sum to be any number up to 1.

Anyway, the miai value of a play in {2|0} is 1. {2|0} is an independent game. So in the sum, {2|0} + {2|0}, which exists only in the abstract, why is it not the same? This argument also applies to temperature.

The CGT text I had read said that the temperature of a sum of games is less than or equal to the maximum temperature of each game in the sum. That does not exactly resolve the ambiguity.

Quote:

What is the defintion of miai value?

One meaning is that it is the average gain of a gote play or sequence of play, as a unit.

It probably has broader meanings, as well.

Quote:

How do you calculate it with this game G? Due to the purple color it seems we have here a tally = 0 and it seems we are facing a double sente area which does not exist does it?

AFAIK, I am the only go writer who calls this miai a double sente. Anyway, we do not talk about the miai value of a sente play sequence, except when we really mean the miai value of the corresponding reverse sente play or sequence. As I said, it's an informal term.

Gérard TAILLE · **#203**

Bill Spight wrote:

Gérard TAILLE wrote:

Bill Spight wrote:

G = H

says nothing about miai values, which is a go concept, not a CGT concept. Now it is true that

{2|0} + {2|0}

and

2

have the same effective miai values associated with them. That is true for all games that are equal to 2.

https://senseis.xmp.net/?Thermography:
The thermograph of a game is a graphical representation of the value of playing in it at different temperatures. The axes of the thermograph are the temperature (vertical) and the count (horizontal). At each temperature, the left wall of the thermograph shows what Left (Black) gets by playing first, and the right wall shows what Right (White) gets by playing first. At high enough temperatures the thermograph is topped by a vertical mast, which coincides with the left and right walls, and represents the count of the game. The temperature at the base of this vertical mast is the temperature of the game. In go it corresponds to miai value.

Three points concerning this thermograph definition:
1) the thermograph is not colored
2) the temperature of the game is the temperature at the base of the vertical mast
3) "miai value" is not defined in thermography, it is only a notion in go world, corresponding (?) to the temperature of the game in thermography

Guilty as charged.

Color is not a defining property of thermographs, but they may be colored. I came up with that idea.

I introduced the term, miai value, to English speaking go players on rec.games.go in the 1990s, in an effort to clarify our thinking about move values. Western players had learned what is called in Japanese deiri values, but most of them assumed that they meant the same thing as miai values. My efforts almost completely failed. Instead, the new term for move values caused more confusion than it alleviated. :sad:

Too much terminology!

I don't know if I was the first to introduce the idea of temperature on rec.games.go, but I certainly used it. However, go players online adopted the term, not to refer the temperature of a position (game) but to the temperature of the whole board. Fine. Language at work.

But that means that the technical meaning of temperature in CGT is different, and using it would cause confusion. So I don't. I have enough trouble with the technical meaning of sente. :lol:

Quote:

It is what you said in your post and I agree with you.
In practice the problem is that in many articles about thermography the wording miai value is used instead of temperature of the game.

Yes, that's probably my attempt to communicate with regular go players, at least those who use the term, miai value. That is my aim on SL. As such, it is informal language. Most people who do use miai value use it to refer to both plays and positions, and to use it where in CGT we would use temperature.

Quote:

I understand you proposed also the wording effective miai value to avoid misunderstanding. Isn'it simplier to keep the wording temperature of the game?

By the time of that writing temperature was used differently by go players. No point in creating confusion. IMX, a surprisingly large number of people want a word to mean only one thing. Most words have multiple meanings, but ambiguity does cause misunderstanding and confusion.

Quote:

In any case what is the defintion of what you call "miai value"?

Well, as a go term, it is not exactly unambiguously defined.

It seems to me that it means how much a gote play or gote sequence as a unit gains, on average. Whether it can be used for how much the first play in a sente sequence gains is unclear. I have not seen it used for that. OC, it can be used that way, I suppose. I have had discussions about whether it can be used for mistakes. Again, I have never seen it used that way in professional writing. At one time I argued against such usage, but now I think it can be useful to do so, as long as you say the play is a mistake, or may be.

Quote:

In the game G = {2|0}+{2|0} the temperature of the game is 0 according to the defintion but you seem to claim that the "miai value" is different.

In my early studies of CGT what I read about the temperature of a game is the temperature at the base of its vertical mast. However, in discussing a miai pair like {2|0} + {2|0}, Berlekamp, who certainly knew what temperature means, said that it was ambiguous, that you could consider the temperature of the sum to be any number up to 1.

Anyway, the miai value of a play in {2|0} is 1. {2|0} is an independent game. So in the sum, {2|0} + {2|0}, which exists only in the abstract, why is it not the same? This argument also applies to temperature.

The CGT text I had read said that the temperature of a sum of games is less than or equal to the maximum temperature of each game in the sum. That does not exactly resolve the ambiguity.

Quote:

What is the defintion of miai value?

One meaning is that it is the average gain of a gote play or sequence of play, as a unit.

It probably has broader meanings, as well.

Quote:

How do you calculate it with this game G? Due to the purple color it seems we have here a tally = 0 and it seems we are facing a double sente area which does not exist does it?

AFAIK, I am the only go writer who calls this miai a double sente. Anyway, we do not talk about the miai value of a sente play sequence, except when we really mean the miai value of the corresponding reverse sente play or sequence. As I said, it's an informal term.

I do not have any experience in the work of convincing go community to a new theory but I imagine it could be very frustrating. Very few people are really open to study new ideas.
Yes for gote area the miai value seems clear.
For sente or double sente (I mean miai gote areas) we have to live with ambiguity haven't we?

Why I prefer using a miai value = 0 for miai gote points? The reason is quite simple.
Let's take a game G = G1 + G2 + G3 + ... each G_i having a miai value m_i<1½ and a score s_i.
Let's suppose score(G) = s1 + s2 + s3 + ... = 2¼
Because the real result of the game is an ordinal number and because the result of the game lies between 2¼ and 2¼+1½ we know for sure that the result of the game should be 3.
Let's now take the game G + H with H = {10|-10} + {10|-10}. If you take as miai value of H the value 10 then you can no more predict the result of the game and that sounds for me as a loss.
What kind of advantage can you see by not taking the miai value 0 for H?

Bill Spight · **#204**

Gérard TAILLE wrote:

Why I prefer using a miai value = 0 for miai gote points? The reason is quite simple.
Let's take a game G = G1 + G2 + G3 + ... each G_i having a miai value m_i<1½ and a score s_i.
Let's suppose score(G) = s1 + s2 + s3 + ... = 2¼
Because the real result of the game is an ordinal number and because the result of the game lies between 2¼ and 2¼+1½ we know for sure that the result of the game should be 3.

Yes, the left (Black) stop of G will be 3, and the right (White) stop will be 1 or 2.

Quote:

Let's now take the game G + H with H = {10|-10} + {10|-10}. If you take as miai value of H the value 10 then you can no more predict the result of the game and that sounds for me as a loss.
What kind of advantage can you see by not taking the miai value 0 for H?

Well, yes, for the purpose of prediction recognizing the miai pair is very important. But there are perhaps practical reasons for remembering the temperature of each of the miai pair is 10 and playing one of them now.

A lot of players, when they first learn about evaluating plays, think that making the largest play at each turn is correct. With a little learning and experience they find out that that is not necessarily so. But it is close enough to the truth that it is a good heuristic, which is known in CGT as hotstrat, the strategy of playing the hottest play. When reading a position, then, we usually start off playing hotstrat and I was doing that with Berlekamp one time. I suggested playing the hottest gote, which was the hottest play, but he said to take a sente instead. OC, the sente raised the global temperature, and was answered, but it removed a potential ko threat. At the same time, it did not risk allowing the opponent to play the reverse sente. Berlekamp did not interpret hotstrat literally.

Now, while recognizing the miai is important for analysis, I have little doubt that Berlekamp would have played next in the miai, breaking it, unless there were a sente for him with a sufficiently large threat. Except in that case, besides the practical matter of avoiding a possible error, the difference in temperature between each of the miai pair and the third hottest play is so great that surely playing one of them dominates all other plays.

Gérard TAILLE · **#205**

Corridors from UP or TINY

Click Here To Show Diagram Code: [go]$$ White to play $$ ----------------------- $$ | X O . . O O . . . . | $$ | X X X X X O . . . . | $$ | X O . . . O O . . . | $$ | X X X X X X O . . . | $$ | X O . . . . O O . . | $$ | X X X X X X X O . . | $$ | X O . . . . . O . . | $$ | X X X X X X X X . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

I understood the infinitesimals of the corridors ending with an UP form an arithmetic serie, the adding value being ↑* (the atomic weight)

Click Here To Show Diagram Code: [go]$$ White to play $$ ----------------------- $$ | X . O . . O O . . . | $$ | X X X X X X O . . . | $$ | X . O . . . O O . . | $$ | X X X X X X X O . . | $$ | X . O . . . . O O . | $$ | X X X X X X X X O . | $$ | X . O . . . . . O . | $$ | X X X X X X X X X . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

What about corridors ending with a TINY? Do we have here also an arithmetic serie?
I tried the atomic weight but it does not work.

Bill Spight · **#206**

Gérard TAILLE wrote:

Click Here To Show Diagram Code: [go]$$ White to play $$ ----------------------- $$ | X . O . a O O . . . | $$ | X X X X X X O . . . | $$ | X . O . . b O O . . | $$ | X X X X X X X O . . | $$ | X . O . . . c O O . | $$ | X X X X X X X X O . | $$ | X . O . . . . d O . | $$ | X X X X X X X X X . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

What about corridors ending with a TINY? Do we have here also an arithmetic serie?
I tried the atomic weight but it does not work.

(Labels added by me.)

No, it's not an arithmetic series. OC, White should play at a first. Mathematical Go says that if the final tinies are equal, then White should play in the shortest corridor first.

As they are not an arithmetic series, the question arises of the comparison between d - c and c - b. The difference between the two is d - 2c + b.

Click Here To Show Diagram Code: [go]$$B Difference game $$ ----------------------- $$ | X . O . . b O O O . | $$ | X X X X X X X X O . | $$ | X . O . . . . d O . | $$ | X X X X X X X X O . | $$ | . X O O O O O O O . | $$ | . X c . . . X . O . | $$ | . X O O O O O O O . | $$ | . X c . . . X . O . | $$ | . X O O O O O O O . | $$ | . X O . . . . . . . | $$ -----------------------[/go]

Click Here To Show Diagram Code: [go]$$B Black first $$ ----------------------- $$ | X . O . . . O O O . | $$ | X X X X X X X X O . | $$ | X . O . . . . 2 O . | $$ | X X X X X X X X O . | $$ | . X O O O O O O O . | $$ | . X 1 . . . X . O . | $$ | . X O O O O O O O . | $$ | . X . . . . X . O . | $$ | . X O O O O O O O . | $$ | . X O . . . . . . . | $$ -----------------------[/go]

After

White makes mirror go and wins jigo.

Click Here To Show Diagram Code: [go]$$W White first $$ ----------------------- $$ | X . O 4 3 1 O O O . | $$ | X X X X X X X X O . | $$ | X . O . . . . 5 O . | $$ | X X X X X X X X O . | $$ | . X O O O O O O O . | $$ | . X 2 6 7 . X . O . | $$ | . X O O O O O O O . | $$ | . X . . . . X . O . | $$ | . X O O O O O O O . | $$ | . X O . . . . . . . | $$ -----------------------[/go]

White wins.

So c - b > d - c.

Edited for correctness.

Gérard TAILLE · **#207**

Bill Spight wrote:

Gérard TAILLE wrote:

Click Here To Show Diagram Code: [go]$$ White to play $$ ----------------------- $$ | X . O . a O O . . . | $$ | X X X X X X O . . . | $$ | X . O . . b O O . . | $$ | X X X X X X X O . . | $$ | X . O . . . c O O . | $$ | X X X X X X X X O . | $$ | X . O . . . . d O . | $$ | X X X X X X X X X . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

What about corridors ending with a TINY? Do we have here also an arithmetic serie?
I tried the atomic weight but it does not work.

(Labels added by me.)

No, it's not an arithmetic series. OC, White should play at a first. Mathematical Go says that if the final tinies are equal, then White should play in the shortest corridor first.

As they are not an arithmetic series, the question arises of the comparison between d - c and c - b. The difference between the two is d - 2c + b.

Click Here To Show Diagram Code: [go]$$B Difference game $$ ----------------------- $$ | X . O . . b O O O . | $$ | X X X X X X X X O . | $$ | X . O . . . . d O . | $$ | X X X X X X X X O . | $$ | . X O O O O O O O . | $$ | . X c . . . X . O . | $$ | . X O O O O O O O . | $$ | . X c . . . X . O . | $$ | . X O O O O O O O . | $$ | . X O . . . . . . . | $$ -----------------------[/go]

Click Here To Show Diagram Code: [go]$$B Black first $$ ----------------------- $$ | X . O . . . O O O . | $$ | X X X X X X X X O . | $$ | X . O . . . . 2 O . | $$ | X X X X X X X X O . | $$ | . X O O O O O O O . | $$ | . X 1 . . . X . O . | $$ | . X O O O O O O O . | $$ | . X . . . . X . O . | $$ | . X O O O O O O O . | $$ | . X O . . . . . . . | $$ -----------------------[/go]

After

White makes mirror go and wins jigo.

Click Here To Show Diagram Code: [go]$$W White first $$ ----------------------- $$ | X . O 4 3 1 O O O . | $$ | X X X X X X X X O . | $$ | X . O . . . . 5 O . | $$ | X X X X X X X X O . | $$ | . X O O O O O O O . | $$ | . X 2 6 7 . X . O . | $$ | . X O O O O O O O . | $$ | . X . . . . X . O . | $$ | . X O O O O O O O . | $$ | . X O . . . . . . . | $$ -----------------------[/go]

White wins.

So c - b > d - c.

Edited for correctness.

Fine proof Bill !

BTW I asked me the following problem: what are the solutions of the equation
G + G = 0
Of course we have the two obvious solutions G = 0 and G = *
Does it exist other solutions or do you have a proof that we have only these two solutions?
For the equation
G + G + G = 0
does it exist another solution than G = 0 ?
This arithmetic is not so easy to handle is it?

Bill Spight · **#208**

Gérard TAILLE wrote:

Bill Spight wrote:

Gérard TAILLE wrote:

Click Here To Show Diagram Code: [go]$$ White to play $$ ----------------------- $$ | X . O . a O O . . . | $$ | X X X X X X O . . . | $$ | X . O . . b O O . . | $$ | X X X X X X X O . . | $$ | X . O . . . c O O . | $$ | X X X X X X X X O . | $$ | X . O . . . . d O . | $$ | X X X X X X X X X . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

What about corridors ending with a TINY? Do we have here also an arithmetic serie?
I tried the atomic weight but it does not work.

(Labels added by me.)

No, it's not an arithmetic series. OC, White should play at a first. Mathematical Go says that if the final tinies are equal, then White should play in the shortest corridor first.

As they are not an arithmetic series, the question arises of the comparison between d - c and c - b. The difference between the two is d - 2c + b.

Click Here To Show Diagram Code: [go]$$B Difference game $$ ----------------------- $$ | X . O . . b O O O . | $$ | X X X X X X X X O . | $$ | X . O . . . . d O . | $$ | X X X X X X X X O . | $$ | . X O O O O O O O . | $$ | . X c . . . X . O . | $$ | . X O O O O O O O . | $$ | . X c . . . X . O . | $$ | . X O O O O O O O . | $$ | . X O . . . . . . . | $$ -----------------------[/go]

Click Here To Show Diagram Code: [go]$$B Black first $$ ----------------------- $$ | X . O . . . O O O . | $$ | X X X X X X X X O . | $$ | X . O . . . . 2 O . | $$ | X X X X X X X X O . | $$ | . X O O O O O O O . | $$ | . X 1 . . . X . O . | $$ | . X O O O O O O O . | $$ | . X . . . . X . O . | $$ | . X O O O O O O O . | $$ | . X O . . . . . . . | $$ -----------------------[/go]

After

White makes mirror go and wins jigo.

Click Here To Show Diagram Code: [go]$$W White first $$ ----------------------- $$ | X . O 4 3 1 O O O . | $$ | X X X X X X X X O . | $$ | X . O . . . . 5 O . | $$ | X X X X X X X X O . | $$ | . X O O O O O O O . | $$ | . X 2 6 7 . X . O . | $$ | . X O O O O O O O . | $$ | . X . . . . X . O . | $$ | . X O O O O O O O . | $$ | . X O . . . . . . . | $$ -----------------------[/go]

White wins.

So c - b > d - c.

Edited for correctness.

Fine proof Bill !

BTW I asked me the following problem: what are the solutions of the equation
G + G = 0
Of course we have the two obvious solutions G = 0 and G = *
Does it exist other solutions or do you have a proof that we have only these two solutions?

Combinatorial games form a group, so for every game G there exists a game, -G. In go, you just switch the colors of the stones.

Quote:

For the equation
G + G + G = 0
does it exist another solution than G = 0 ?

I don't know, but I don't think so. G has to be fuzzy, and it cannot be impartial, since every impartial game is symmetric, and thus its own negative. And since G must be fuzzy, then G + G must also be fuzzy. Let each G be in its simplest form. If G + G in its simplest form is not exactly as simple as G, then it cannot be equal to -G. So G has to thread the needle.

Hmmm. Without loss of generality, let Black play first. Suppose that Black plays to Gb. If White plays in Gb to Gbw, then Gbw = G, which is impossible if G is in its simplest form. So White must play in a different G to Gw. Then Gb + Gw = -G. That is so even if White plays first. So for every Black option in G there is a White option such that the sum of the two equals -G, and vice versa. So each player must have the same number of options. And the options for each player must be incomparable.

If G has depth 0, the only game that meets those criteria is 0.
If G has depth 1, the only possible fuzzy game is {0|0} = *, which is impartial.
If G has depth 2, the only possible fuzzy games are {0,*|0,*} = *2, which is impartial, {1|-1}, {1|0}, and {0|-1}. Doubling each of the last three yields a number, which is not fuzzy.
* * *

I would be surprised if Conway or someone hasn't found a proof, one way or the other.

Quote:

This arithmetic is not so easy to handle is it?

Gérard TAILLE · **#209**

Bill Spight wrote:

Gérard TAILLE wrote:

BTW I asked me the following problem: what are the solutions of the equation
G + G = 0
Of course we have the two obvious solutions G = 0 and G = *
Does it exist other solutions or do you have a proof that we have only these two solutions?

Combinatorial games form a group, so for every game G there exists a game, -G. In go, you just switch the colors of the stones.

[quote]

What do you mean Bill?
My question was about the equation G + G = 0, not the equation G - G = 0.

Gérard TAILLE · **#210**

Bill Spight wrote:

Gérard TAILLE wrote:

For the equation
G + G + G = 0
does it exist another solution than G = 0 ?

I don't know, but I don't think so. G has to be fuzzy, and it cannot be impartial

What does mean impartial Bill?

Bill Spight · **#211**

Gérard TAILLE wrote:

Bill Spight wrote:

Gérard TAILLE wrote:

BTW I asked me the following problem: what are the solutions of the equation
G + G = 0
Of course we have the two obvious solutions G = 0 and G = *
Does it exist other solutions or do you have a proof that we have only these two solutions?

Combinatorial games form a group, so for every game G there exists a game, -G. In go, you just switch the colors of the stones.

Quote:

What do you mean Bill?
My question was about the equation G + G = 0, not the equation G - G = 0.

Well, if G + G = 0 then G = -G.

For all impartial games, which have symmetrical trees, G = -G.

So all impartial games satisfy that equation.

Bill Spight · **#212**

Gérard TAILLE wrote:

Bill Spight wrote:

Gérard TAILLE wrote:

For the equation
G + G + G = 0
does it exist another solution than G = 0 ?

I don't know, but I don't think so. G has to be fuzzy, and it cannot be impartial

What does mean impartial Bill?

Sorry, I think I accidentally cut out some of my earlier answer. Impartial games are those where at each turn, each play has the same options. Each impartial game is its own negative.

So if G is impartial, then G + G = 0.

Gérard TAILLE · **#213**

I try to understand infinitesimals but it is not that easy

Edit

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]

How do you analyse this position?

Gérard TAILLE · **#214**

Gérard TAILLE wrote:

I try to understand infinitesimals but it is not that easy

Edit

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]

How do you analyse this position?

Oops, I believe this position is equal to ↑↑ which is not obvious is it?

Bill Spight · **#215**

Gérard TAILLE wrote:

I try to understand infinitesimals but it is not that easy

Edit

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]

How do you analyse this position?

Well first, let's look at the play at temperature 1 (temperature 0 in the chilled game).

Click Here To Show Diagram Code: [go]$$B Black first $$ ----------------------- $$ | O X . . . . . . . . | $$ | . X . . . . . . . . | $$ | 1 X X . . . . . . . | $$ | O O X X X . . . . . | $$ | 2 X 3 . X . . . . . | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

Black first moves in gote to a position worth +4 (+3 in chilled go.) Black can transpose :b1:

and

.

Click Here To Show Diagram Code: [go]$$W White first $$ ----------------------- $$ | O X . . . . . . . . | $$ | 4 X . . . . . . . . | $$ | 1 X X . . . . . . . | $$ | O O X X X . . . . . | $$ | 3 X 2 . X . . . . . | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

White to play can move to a position worth +3 in sente. Not that this play is White's sente, OC.

This tells us that this go infinitesimal is greater than 0 (from Black's point of view). The numerical score is still +3.

:w1:

is correct. :b2:

and

transpose.

Click Here To Show Diagram Code: [go]$$W White moves to star $$ ----------------------- $$ | O X . . . . . . . . | $$ | a X . . . . . . . . | $$ | 1 X X . . . . . . . | $$ | O O X X X . . . . . | $$ | 3 X 2 . X . . . . . | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

The go infinitesimal after :w1:

-

is star (*). If :w2:

is at a instead, then after :w3:

that go infinitesimal is also *. So the go infinitesimal after :w1:

is also *.

Click Here To Show Diagram Code: [go]$$W White moves to upstar $$ ----------------------- $$ | O X . . . . . . . . | $$ | . X . . . . . . . . | $$ | a X X . . . . . . . | $$ | O O X X X . . . . . | $$ | 1 X b . X . . . . . | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

After

here the go infinitesimal at a is up (↑) and the one at b is *. ↑ + * is written ↑*. (In fact, * plus almost anything may be written with that thing followed by *.

) Now,

↑* > * , i.e.

↑* - * = ↑ > 0 .

So for White * dominates ↑*. That's why :w1:

in this diagram is incorrect in CGT.

When Black plays first things get tricky.

Click Here To Show Diagram Code: [go]$$B Black first, I $$ ----------------------- $$ | O X . . . . . . . . | $$ | . X . . . . . . . . | $$ | . X X . . . . . . . | $$ | O O X X X . . . . . | $$ | 2 X 1 . X . . . . . | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

, OC, raises the local temperature, forcing :w2:

. The resulting infinitesimal is ↑, which is greater than 0. So this looks pretty good for Black. If White does not play :w2:

, then Black at 2 gets a chilled score of 9 - 2 = 7, a gain of 4 points over the original score of 3.

Click Here To Show Diagram Code: [go]$$B Black first, II $$ ----------------------- $$ | O X . . . . . . . . | $$ | . X . . . . . . . . | $$ | 1 X X . . . . . . . | $$ | O O X X X . . . . . | $$ | 2 X . . X . . . . . | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

carries the same size threat as :b1:

in the previous diagram, but the result after :w2:

is different. This way, the infinitesimal is *, which is confused with 0. Since ↑ > 0, we might think that the first diagram is better for Black, but ↑ <> * (up is confused with star). So we can write the original infinitesimal this way:

{{4|↑},{4|*}||*}

Surely we can simplify this.

Especially since we know that it is greater than 0.

We can do so if either of the Black options reverses, so that Black continues play in a unit sequence. That is so if this infinitesimal is greater than or equal to * or ↑. We can guess the answer, but let's prove it.

First let's try *. The question boils down to this: Can White to play win the difference game? If not, the infinitesimal is greater than or equal to *. Subtracting * is the same as adding it, so let's play this game.

{{4|↑},{4|*}||*} + *

White to play wins by playing to * on the left, as * + * = 0.

Now let's try ↑. The negative of ↑ is ↓ = {*|0}. Here is the game.

{{4|↑},{4|*}||*} + {*|0}

We already know that the left game is greater than 0, so White cannot win by playing to 0 on the right. If White plays to * on the left, Black wins by playing to * on the right. White first cannot win. That gives us the following sequence of play.

Click Here To Show Diagram Code: [go]$$B Black first, I $$ ----------------------- $$ | O X . . . . . . . . | $$ | . X . . . . . . . . | $$ | 3 X X . . . . . . . | $$ | O O X X X . . . . . | $$ | 2 X 1 . X . . . . . | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

We may consider :b1:

-

as a unit.

That means that this infinitesimal reduces to

{0,{4|*}||*}

Gérard, you come up with the most interesting positions.

Edited for correctness.

Edit2: Well, almost. I just took another look, and {4|*} > 0, so we can reduce it still further.

{4|*||*}

Bill Spight · **#216**

Gérard TAILLE wrote:

I try to understand infinitesimals but it is not that easy

Edit

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]

How do you analyse this position?

Oops, I believe this position is equal to ↑↑ which is not obvious is it?

It's actually less than ↑↑.

Gérard TAILLE · **#217**

Bill Spight wrote:

Click Here To Show Diagram Code: [go]$$B Black first, I $$ ----------------------- $$ | O X . . . . . . . . | $$ | . X . . . . . . . . | $$ | . X X . . . . . . . | $$ | O O X X X . . . . . | $$ | 2 X 1 . X . . . . . | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

, OC, raises the local temperature, forcing :w2:

. The resulting infinitesimal is ↑, which is greater than 0. So this looks pretty good for Black. If White does not play :w2:

, then Black at 2 gets a chilled score of 9 - 2 = 7, a gain of 4 points over the original score of 3.

Here is the point of my position Bill : the exchange :b1:

in the above diagram is not that good for black because it is in fact a privilege for black
By avoiding this exchange black can keep in reserve the possibility after after :w1:

in the following diagram to not answer at "a" !

Click Here To Show Diagram Code: [go]$$W $$ ----------------------- $$ | O X . . . . . . . . | $$ | . X . . . . . . . . | $$ | . X X . . . . . . . | $$ | O O X X X . . . . . | $$ | 1 X a . X . . . . . | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

You said my position is less than ↑↑ but how you play the difference game?

Click Here To Show Diagram Code: [go]$$W White to play $$ ----------------------- $$ | O X . . . . O O O O | $$ | . X . . . . O X O X | $$ | 1 X X . . . O . O . | $$ | O O X X X . O 4 O 2 | $$ | 3 X . . X . X X X X | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

Bill Spight · **#218**

Gérard TAILLE wrote:

Bill Spight wrote:

Click Here To Show Diagram Code: [go]$$B Black first, I $$ ----------------------- $$ | O X . . . . . . . . | $$ | . X . . . . . . . . | $$ | . X X . . . . . . . | $$ | O O X X X . . . . . | $$ | 2 X 1 . X . . . . . | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

, OC, raises the local temperature, forcing :w2:

. The resulting infinitesimal is ↑, which is greater than 0. So this looks pretty good for Black. If White does not play :w2:

, then Black at 2 gets a chilled score of 9 - 2 = 7, a gain of 4 points over the original score of 3.

Here is the point of my position Bill : the exchange :b1:

in the above diagram is not that good for black because it is in fact a privilege for black
By avoiding this exchange black can keep in reserve the possibility after after :w1:

in the following diagram to not answer at "a" !

Click Here To Show Diagram Code: [go]$$W $$ ----------------------- $$ | O X . . . . . . . . | $$ | . X . . . . . . . . | $$ | . X X . . . . . . . | $$ | O O X X X . . . . . | $$ | 1 X a . X . . . . . | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

You said my position is less than ↑↑ but how you play the difference game?

Click Here To Show Diagram Code: [go]$$W White to play $$ ----------------------- $$ | O X . . . . O O O O | $$ | . X . . . . O X O X | $$ | 1 X X . . . O . O . | $$ | O O X X X . O 4 O 2 | $$ | 3 X . . X . X X X X | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

Click Here To Show Diagram Code: [go]$$W White to play and win $$ ----------------------- $$ | O X . . . . O O O O | $$ | . X . . . . O X O X | $$ | 1 X X . . . O . O . | $$ | O O X X X . O 3 O 2 | $$ | . X . . X . X X X X | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

Gérard TAILLE · **#219**

Bill Spight wrote:

Click Here To Show Diagram Code: [go]$$W White to play and win $$ ----------------------- $$ | O X . . . . O O O O | $$ | . X . . . . O X O X | $$ | 1 X X . . . O . O . | $$ | O O X X X . O 3 O 2 | $$ | . X . . X . X X X X | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

You are right Bill.
I will look at other ideas for positions and I have also to look how to discover more easly equivalent positions.
Thank you again for your help.

Bill Spight · **#220**

Correction. I wrote:

Gérard TAILLE wrote:

I try to understand infinitesimals but it is not that easy

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]

How do you analyse this position?

Moi wrote:

This infinitesimal reduces to

{0,{4|*}||*}

Gérard, you come up with the most interesting positions.

Edited for correctness.

Edit2: Well, almost. I just took another look, and {4|*} > 0, so we can reduce it still further.

{4|*||*}

IOW, this is canonical play.

Click Here To Show Diagram Code: [go]$$B Black first $$ ----------------------- $$ | O X . . . . . . . . | $$ | . X . . . . . . . . | $$ | 1 X X . . . . . . . | $$ | O O X X X . . . . . | $$ | 2 X 3 . X . . . . . | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

Click Here To Show Diagram Code: [go]$$W White first $$ ----------------------- $$ | O X . . . . . . . . | $$ | 2 X . . . . . . . . | $$ | 1 X X . . . . . . . | $$ | O O X X X . . . . . | $$ | 3 X 4 . X . . . . . | $$ | O O X X X . . . . . | $$ | O O O O O . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ | . . . . . . . . . . | $$ -----------------------[/go]

However, at temperature 1 Black does not have to play :b3:

in the first diagram, nor does she have to play :b4:

in the second diagram. The former is why Black can get the last play in this diagram.

Click Here To Show Diagram Code: [go]$$B Black to play and win $$ ----------------------- $$ | O X . . . . . . . . | $$ | . X . . . . . . . . | $$ | 1 X X . . . . . . . | $$ | O O X X X . . . . . | $$ | 2 X . . X . . . . . | $$ | O O X X X - - . . . | $$ | O O O O O O - X X X | $$ | . . . . . . O O O 3 | $$ | . . . . . . . . O . | $$ | . . . . . . . . O X | $$ -----------------------[/go]

Thermography

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