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 Post subject: Re: Thermography
Post #21 Posted: Mon Sep 28, 2020 6:22 am 
Honinbo

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Well, I got carried away with the difference games, eh? :lol:

Constructing the thermograph

Truth to say, difference games are not necessary to thermography at all, and are hardly ever used. However, it was easy to show that the keima sente and the crawl sente are unnecessary. The kosumi sente dominates them. As I said, if they were gote, that would be a different matter. But anyway, all we have to consider are the kosumi sente and the large monkey jump sente. Let's review the reverse sente first.

Click Here To Show Diagram Code
[go]$$Wc Reverse sente, Black continues
$$ -----------------
$$ | C 4 2 3 C C O |
$$ | X X 1 C C C O |
$$ | . X O O O O O |
$$ | . X X X O O C |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

:w1: blocks on the second line. Play can, and usually does, stop there. If Black replies she plays the hane-and-connect for a local score of -5 in the marked region. Each player has made the same number of plays, so the Right wall of the thermograph rises as a vertical line at that point.

Black does not reply above temperature 1, so the wall turns 45° to the left at that temperature rising one degree for each point of territory. Remember, the leftward movement is positive, the reverse the usual convention. The equation of that line is v = -6 + t, where v is the territory and t is the temperature.

Now let's look at the kosumi sente.

Click Here To Show Diagram Code
[go]$$Bc Kosumi sente, Black continues
$$ -----------------
$$ | . . 1 3 4 . O |
$$ | X X 5 2 . . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

The sente actually ends with :w2: but we consider what happens when Black keeps playing. The score after :b5: is -2. Black has made one more move than White, so the line from here angles up and to the right. It's equation is v = -2 - t. How far up does it go?

Click Here To Show Diagram Code
[go]$$Wc Kosumi sente, White follow-up
$$ -----------------
$$ | . . B 1 . . O |
$$ | X X 2 W . . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

If I didn't already know, it would be easy to work out that this is correct play at temperature 1. White could get an equivalent result, except for ko threats, by starting at 2. Each player has made the same number of plays, including the kosumi sente exchange, so the line rises vertically at -3. These two lines intersect at v = -3, t = 1. Above temperature 1 the prospective wall, called a scaffold, rises vertically at -3, below temperature 1 it angles down to -2 at temperature 0.

Now let's look at the monkey jump.

Click Here To Show Diagram Code
[go]$$Bc Monkey jump sente 1
$$ -----------------
$$ | . . 3 4 1 . O |
$$ | X X 2 5 6 . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

This is one of the main lines, with the throw-in, :w4:. White could also get here by starting at 4. That's the other main line. The usual reply is at 6. We'll consider that later.

What if Black continues?

Click Here To Show Diagram Code
[go]$$Bc Monkey jump sente 1, Black continues
$$ -----------------
$$ | . 9 B 7 B 8 O |
$$ | X X W B W . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

1 :wc: prisoner
:b7: connects the Black stones, :w8: makes a second eye, and :b9: saves the four stones. The local score is 0. Since Black has made one more move than White, the scaffold angles up from 0 to the right at one point of territory for each degree of temperature. The equation of the line is v = -t.

How high does it go? To find out, let's look at the White follow-up after the sente.

Click Here To Show Diagram Code
[go]$$Wc Monkey jump sente 1, White follow-up.
$$ -----------------
$$ | . 2 B 3 B 1 O |
$$ | X X W B W . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

1 :wc: prisoner

There is a ko, which White can take and win on the usual assumption of no ko threats. But :w1: comes to the same thing. ;) After :w3: the local score is -4. Since White has made one more move than Black, the scaffold angles up from there to the left. It's equation is v = -4 + t. Note that this is the scaffold for the position after the sente, not the scaffold for the original position. The scaffolds intersect at t = 2, v = -2. The mast rises vertically from there. The scaffold for this sente is v = -2 above temperature 2 and v = -4 + t below temperature 2.

There is another variation to consider.

Click Here To Show Diagram Code
[go]$$Bc Monkey jump sente 1a
$$ -----------------
$$ | . . 3 5 1 6 O |
$$ | X X 2 . 4 . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

:w4: does not play the throw-in, but it threatens it. :b5: prevents that, with sente. As we have seen, with a different order of play, this line is incomparable with the throw-in. After :w6: the local score is -1. The scaffold rises vertically from there. OC, that is worse for White than -2, so this line is not as good as the throw-in above temperature 2. But it is better for White than 0, so we find the intersection v = -t and v = -1 to find where White switches to this line of play. The intersection is at t = 1, v = -1. So for the reply of :w2: the scaffold is vertical at -2 above temperature 2, angling down below temperature 2 to -1 at temperature 1, and vertical below that to -1 at temperature 0. :)

BTW, what if Black does not connect with :b5: in the previous diagram, but tries to kill?

Click Here To Show Diagram Code
[go]$$Bcm5 Monkey jump sente 1a, variation
$$ -----------------
$$ | . . B 2 B 1 O |
$$ | X X W 3 W a O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

Now White cannot play at a because of damezumari. Oh, for a dame!

Click Here To Show Diagram Code
[go]$$Wcm8 Monkey jump sente 1a, variation
$$ -----------------
$$ | 2 1 B 3 B B O |
$$ | X X W B W 5 O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

:b11: at 8
White has no external threats, by assumption, but the play has generated a local threat for White at :w8:. With a dame White would not have needed to play the ko. :)

More later.

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

Visualize whirled peas.

Everything with love. Stay safe.

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 Post subject: Re: Thermography
Post #22 Posted: Mon Sep 28, 2020 7:29 am 
Honinbo

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Gérard TAILLE wrote:
Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. Monkey jump 1
$$ --------------------------------
$$ | . . X b . . O | X c 1 d 3 . . |
$$ | X X . a . . O | X . 2 . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


For the time being I am still not convinced that kosumi is incomparable with the monkey jump.


OK, I guess I made some assumptions that may not hold. Fair enough. :)

Quote:
My conclusion is that it does not exist an environment in which the kosumi is better that the monkey jump.


I suppose that you will grant that there are environments in which the large monkey jump is better. If there is an environment in which the kosumi is better, then the kosumi and the monkey jump are incomparable. (Assuming that both are sente, threatening to kill, OC.)

Let's start at the very beginning.

Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. Monkey jump
$$ --------------------------------
$$ | . . . . . . O | X . . . . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


This mirror position has a value of 0. Whoever plays first, the second player can play mirror go for jigo. (Note: The relevant positions do not have to be on different boards. The point is that they are independent of each other.)

Let Black play first and play the kosumi, and let White reply with the large monkey jump.

Click Here To Show Diagram Code
[go]$$Bc Kosumi vs. Monkey jump
$$ --------------------------------
$$ | . . 1 . . . O | X . 2 . . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

Can Black to play win the difference game from this point? If so, then there is an environment where the kosumi is better than the monkey jump. Namely, the one after :b1:, because White can make jigo by playing the kosumi. :)

Click Here To Show Diagram Code
[go]$$Bcm3 Kosumi vs. Monkey jump, Black first
$$ --------------------------------
$$ | . . X 8 . . O | X 7 O 3 2 0 . |
$$ | X X 9 6 . . O | X . 5 4 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

:b13: captures two stones

Obviously, if :b3: kills on the left, :w4: will kill on the right, for jigo. ;)

:w8: must come back to save the White group on the left. Black wins by 1 pt.

Click Here To Show Diagram Code
[go]$$Wcm8 Kosumi vs. Monkey jump, Black first, variation for :w8:
$$ --------------------------------
$$ | . . X 1 8 9 O | X 4 O 3 O 5 . |
$$ | X X 2 6 7 . O | X . X O X O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

1 :bc: prisoner, :b17: connects

:w8: is clever play. :) :clap: If :b9: plays the atari on the right side board, :w10: will play atari with sente on the left and win. For Black the atari at 13 is tempting, but White will atari and make a mirror position, for jigo. So, faute de mieux, :b9: plays the solid connection.

Now if White connects at 13, Black will atari on the right to transpose to the previous variation and win the difference game. So White saves his stones on the right and Black continues on the left. In the end Black wins by 1 pt. :)

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

Visualize whirled peas.

Everything with love. Stay safe.

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 Post subject: Re: Thermography
Post #23 Posted: Mon Sep 28, 2020 9:59 am 
Lives in gote

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Bill Spight wrote:
Let's start at the very beginning.

Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. Monkey jump
$$ --------------------------------
$$ | . . . . . . O | X . . . . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


This mirror position has a value of 0. Whoever plays first, the second player can play mirror go for jigo. (Note: The relevant positions do not have to be on different boards. The point is that they are independent of each other.)

Let Black play first and play the kosumi, and let White reply with the large monkey jump.

Click Here To Show Diagram Code
[go]$$Bc Kosumi vs. Monkey jump
$$ --------------------------------
$$ | . . 1 . . . O | X . 2 . . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

Can Black to play win the difference game from this point? If so, then there is an environment where the kosumi is better than the monkey jump. Namely, the one after :b1:, because White can make jigo by playing the kosumi. :)

Click Here To Show Diagram Code
[go]$$Bcm3 Kosumi vs. Monkey jump, Black first
$$ --------------------------------
$$ | . . X 8 . . O | X 7 O 3 2 0 . |
$$ | X X 9 6 . . O | X . 5 4 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

:b13: captures two stones

Obviously, if :b3: kills on the left, :w4: will kill on the right, for jigo. ;)

:w8: must come back to save the White group on the left. Black wins by 1 pt.

Click Here To Show Diagram Code
[go]$$Wcm8 Kosumi vs. Monkey jump, Black first, variation for :w8:
$$ --------------------------------
$$ | . . X 1 8 9 O | X 4 O 3 O 5 . |
$$ | X X 2 6 7 . O | X . X O X O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

1 :bc: prisoner, :b17: connects

:w8: is clever play. :) :clap: If :b9: plays the atari on the right side board, :w10: will play atari with sente on the left and win. For Black the atari at 13 is tempting, but White will atari and make a mirror position, for jigo. So, faute de mieux, :b9: plays the solid connection.

Now if White connects at 13, Black will atari on the right to transpose to the previous variation and win the difference game. So White saves his stones on the right and Black continues on the left. In the end Black wins by 1 pt. :)


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


OK Bill, let's consider an environment for which, after :b1:, the sequence :w2: :b3: is best.
I agree in this case that the move :b1: here above is better than the monkey jump (good job for you!). But we have already eliminated the sequence above because of the small keima jump didn't we ?

In other words:
Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | . . a b c . O |
$$ | X X . . . . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]


Well you found an environment for which the move "a" is better than the move "c" but, for this environment, the move "b" is still better than the move "a". Where is the gain Bill?

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 Post subject: Re: Thermography
Post #24 Posted: Mon Sep 28, 2020 10:17 am 
Honinbo

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Gérard TAILLE wrote:
Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | . . a b c . O |
$$ | X X . . . . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]


Well you found an environment for which the move "a" is better than the move "c" but, for this environment, the move "b" is still better than the move "a". Where is the gain Bill?


In this situation, the kosumi at a is better than the keima at b.

Click Here To Show Diagram Code
[go]$$Bc Kosumi vs. Keima, Black first
$$ --------------------------------
$$ | . . 1 9 0 . O | X . 3 2 4 8 . |
$$ | X X a 6 . . O | X . . 5 7 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

:b11: plays at a to win.

Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. Keima, White first
$$ --------------------------------
$$ | . . 2 7 . . O | X . 4 1 5 9 . |
$$ | X X 0 3 . . O | X . . 6 8 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

:b10: makes jigo.

Black first can win, White first cannot. Therefore the kosumi dominates the keima.

The only time that the keima may be better depends on a possible ko.

Edit: The point being that difference games depend on the different positions being independent, and kos can break independence. :)

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

Visualize whirled peas.

Everything with love. Stay safe.

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 Post subject: Re: Thermography
Post #25 Posted: Mon Sep 28, 2020 2:39 pm 
Lives in gote

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Bill Spight wrote:
Gérard TAILLE wrote:
Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | . . a b c . O |
$$ | X X . . . . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]


Well you found an environment for which the move "a" is better than the move "c" but, for this environment, the move "b" is still better than the move "a". Where is the gain Bill?


In this situation, the kosumi at a is better than the keima at b.

Click Here To Show Diagram Code
[go]$$Bc Kosumi vs. Keima, Black first
$$ --------------------------------
$$ | . . 1 9 0 . O | X . 3 2 4 8 . |
$$ | X X a 6 . . O | X . . 5 7 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

:b11: plays at a to win.

Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. Keima, White first
$$ --------------------------------
$$ | . . 2 7 . . O | X . 4 1 5 9 . |
$$ | X X 0 3 . . O | X . . 6 8 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

:b10: makes jigo.

Black first can win, White first cannot. Therefore the kosumi dominates the keima.

The only time that the keima may be better depends on a possible ko.

Edit: The point being that difference games depend on the different positions being independent, and kos can break independence. :)


Surely the kosumi can be better than the keima depending of the environment but this is not my point which is the following:
Because kosumi and monkey jump are incomparable it exists a set of environment E for which the kosumi is strictly better than the monkey jump and now is my point : whatever environment you choose in E the small keima is equivalent or better than the kosumi !
Of course I am not sure at 100% but it is my point.
If this is true we will conclude that the kosumi possibility can be simply ignored.

BTW if it happens that, for each environment in E, the kosumi and the small keima are equivalent then you can also ignore the keima and keep the kosumi. I have to study this point but this is another issue.

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 Post subject: Re: Thermography
Post #26 Posted: Mon Sep 28, 2020 3:34 pm 
Honinbo

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Gérard TAILLE wrote:
Bill Spight wrote:
Gérard TAILLE wrote:
Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | . . a b c . O |
$$ | X X . . . . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]


Well you found an environment for which the move "a" is better than the move "c" but, for this environment, the move "b" is still better than the move "a". Where is the gain Bill?


In this situation, the kosumi at a is better than the keima at b.

Click Here To Show Diagram Code
[go]$$Bc Kosumi vs. Keima, Black first
$$ --------------------------------
$$ | . . 1 9 0 . O | X . 3 2 4 8 . |
$$ | X X a 6 . . O | X . . 5 7 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

:b11: plays at a to win.

Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. Keima, White first
$$ --------------------------------
$$ | . . 2 7 . . O | X . 4 1 5 9 . |
$$ | X X 0 3 . . O | X . . 6 8 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

:b10: makes jigo.

Black first can win, White first cannot. Therefore the kosumi dominates the keima.

The only time that the keima may be better depends on a possible ko.

Edit: The point being that difference games depend on the different positions being independent, and kos can break independence. :)


Surely the kosumi can be better than the keima depending of the environment


I may have been a bit unclear about the environment. It is the rest of the board from the position in question.

For example:

Click Here To Show Diagram Code
[go]$$B The environment
$$ -----------------
$$ | . . a b c . O |
$$ | X X . . . . O |
$$ | T X O O O O O |
$$ | T X X X O O . |
$$ | T T T X O O O |
$$ | T T T X X X X |
$$ | T T T T T T T |
$$ -----------------[/go]

I have marked the empty points in the environment. They are outside the problem area and independent of it. In this case they are Black territory, but that is of zero consequence to the questions being asked.

For the difference game:

Click Here To Show Diagram Code
[go]$$Bc The difference game environment = 0
$$ --------------------------------
$$ | . . . . . . O | X . . . . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | T X O O O O O | X X X X X O T |
$$ | T X X X O O . | . X X O O O T |
$$ | T T T X O O O | X X X O T T T |
$$ | T T T X X X X | O O O O T T T |
$$ | T T T T T T T | T T T T T T T |
$$ --------------------------------[/go]

The environment for the difference game is the sum of the environment on one board plus its mirror opposite. As long as the environment on the original board is not implicated in a ko fight, it does not matter what it is. The sum of it and its opposite is zero.

For example:

Click Here To Show Diagram Code
[go]$$Bc The difference game environment = 0
$$ --------------------------------
$$ | . . . . . . O | X . . . . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | T X O O O O O | X X X X X O T |
$$ | X X X X O O . | . X X O O O O |
$$ | O O T X O O O | X X X O T X X |
$$ | O T O X X X X | O O O O X T X |
$$ | T T O X T T T | T T T O X T T |
$$ --------------------------------[/go]

In this case the environment on the left contains an unsettled group, but the environment on the right contains its opposite. If either player plays in the environment on one side, the other player can mirror that move on the other side and maintain an environment of zero. (You see how ko fights can mess up the logic of difference games. ;)

The position on the left side as the environment of the position on the right

Click Here To Show Diagram Code
[go]$$Bc
$$ --------------------------------
$$ | S S 1 S S S O | X . . . a . . |
$$ | X X S S S S O | X . . . . O O |
$$ | T X O O O O O | X X X X X O T |
$$ | T X X X O O S | . X X O O O T |
$$ | T T T X O O O | X X X O T T T |
$$ | T T T X X X X | O O O O T T T |
$$ | T T T T T T T | T T T T T T T |
$$ --------------------------------[/go]

Let :b1: be the kosumi on the left side. Now our question shifts to the right side. This difference game is played on two boards, but it could be played on a single board. In any event, the environment for the are in question on the right side is the rest of the two boards combined. The regions marked with triangles sum to zero, so we may consider the region on the left side marked with squares as the environment of the playing area on the right side. We could set that up on a single board if we wanted to.

White could make jigo by playing the kosumi on the right side. However, if White plays elsewhere on the right side and Black is able then to win the difference game with correct play, then there exists an environment such that the kosumi is better than that other play.

We don't have to find such an environment. The existence of one is part of the logic of difference games. :)

More later.

Edit: OK, here is a little more.

If there exists a (non-ko fight) environment such that play A is better than play B, and there also exists such an environment such that play B is better than play A, then play A and play B are incomparable. We cannot say that one is better than the other.

However, if there exists such an environment that play A is better than play B, but not the other way around, then play A dominates play B (with the caveat about ko fights or potential ko fights).

Here the kosumi and the large monkey jump are incomparable, but the kosumi dominates both the crawl and the keima (small monkey jump).

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 Post subject: Re: Thermography
Post #27 Posted: Tue Sep 29, 2020 4:04 am 
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With your last post I believe I have found where is the misunderstanding.

The point is quite subtil: considering the environment we both exclude pure ko threats like for example:
Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | X X X O . . . |
$$ | . a X O . . . |
$$ | X X O O . . . |
$$ | . X X O . . . |
$$ | X X O O . . . |
$$ | O O O . . . . |
$$ | . . . . . . . |
$$ -----------------[/go]


But you seem also to exclude from the environment "normal" sente move like for example:

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


Because in my mind I accept such environment which allows to use a sente move as a ko threat our conclusions are logically different

Here is an example for the dicussion kosumi vs keima

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


You can see that after a white move at "a", white may create a sente move at "b" which can be use as a ko threat. That is the point.

As you see, it is not so easy to define what a non-ko fight environment is and, in addition, the restriction to such non-ko fight environment seems really very severe doesn't it?

Anyway Bill, an interesting discussion !

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Post #28 Posted: Tue Sep 29, 2020 7:20 am 
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Gérard TAILLE wrote:
As you see, it is not so easy to define what a non-ko fight environment is and, in addition, the restriction to such non-ko fight environment seems really very severe doesn't it?


Yes and no. In theory, it's quite a severe restriction, although I'm afraid I gave you the wrong impression of the problem, but in practice it is not such a problem.

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

Should White play at a or b?
A difference game will find that they are equivalent. But of course White should play at a, because if White plays at b, Black might reply at a and leave a ko threat behind. There may or may not be a ko or potential ko in the environment, and if there is, this ko threat may not matter, but nothing is lost by avoiding it.

Or consider these sequences.

Click Here To Show Diagram Code
[go]$$Bc
$$ -----------------
$$ | . . 3 5 1 6 O |
$$ | X X 2 . 4 . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

White may play this way in certain situations so as not to leave any local play behind. However,

Click Here To Show Diagram Code
[go]$$Bc Ko
$$ -----------------
$$ | . a 3 6 1 5 O |
$$ | X X 2 7 4 . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

Because of damezumari, Black can play :b5:, forcing White to make a ko for life. White has a local ko threat at a, but it is conceivable that Black could have enough large enough ko threats in the environment to justify this line of play.

However, that would be exceptional, and a caveat to that effect allows us to draw valuable conclusions from difference games for this position. Conway and Berlekamp, who developed difference games, avoided kos in difference games because you can't prove anything in that case. I, however, do not mind defeasible reasoning with exceptions, as long as you mention the caveats. Conway and Berlekamp only applied difference games to non-ko positions, with the general warning that the conclusions only applied to non-ko environments. That was something they had proved. Usually the conclusions also apply to environments with kos or potential kos, as those possibilities are normally irrelevant to any specific comparison.

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Post #29 Posted: Tue Sep 29, 2020 8:35 am 
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OK, Bill let me try a difference game:

Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . . X a . . O | X . . O . . . |
$$ | X X . b . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


If I would try to prove that kosumi does not dominate the keima I have to find a way to win the above diference game with white, right?

I see clearly that I cannot win by beginning at "b" => no choice for my first white move:

Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . . X 1 . . O | X . . O . . . |
$$ | X X a b . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


Now it is black turn. If black plays at "a" white plays at "b" and win => black must ansmer at "a" and the following sequence seems mandatory

Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . . X 1 4 . O | X . 6 O 7 . . |
$$ | X X 3 2 5 . O | X . . 8 . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


then white can simply continue by
Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . 4 X 5 X 1 O | X . X O O 3 . |
$$ | X X O X O . O | X . . X 2 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

winning the game.
Isn't it correct?


This post by Gérard TAILLE was liked by: Bill Spight
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Post #30 Posted: Tue Sep 29, 2020 9:29 am 
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Seeing the difference game in my previous post I built the following very simple position and very simple environment (without ko threat)

Click Here To Show Diagram Code
[go]$$B Kosumi vs. keima
$$ -----------------
$$ | . . . . . . O |
$$ | X X . . . . O |
$$ | . X O O O O O |
$$ | . X X X X O . |
$$ | . X X X X O O |
$$ | . X X X X X O |
$$ | . X . O O . O |
$$ -----------------[/go]


The keima seems better than the kosumi, doesn't it?

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Post #31 Posted: Tue Sep 29, 2020 10:19 am 
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Gérard TAILLE wrote:
OK, Bill let me try a difference game:

Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . . X a . . O | X . . O . . . |
$$ | X X . b . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


If I would try to prove that kosumi does not dominate the keima I have to find a way to win the above diference game with white, right?

I see clearly that I cannot win by beginning at "b" => no choice for my first white move:

Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . . X 1 . . O | X . . O . . . |
$$ | X X a b . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


Now it is black turn. If black plays at "a" white plays at "b" and win => black must ansmer at "a" and the following sequence seems mandatory

Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . . X 1 4 . O | X . 6 O 7 . . |
$$ | X X 3 2 5 . O | X . . 8 . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


then white can simply continue by
Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . 4 X 5 X 1 O | X . X O O 3 . |
$$ | X X O X O . O | X . . X 2 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

winning the game.
Isn't it correct?


Very good. :clap: :clap: :clap:

A couple of notes.

Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . . X 1 6 7 O | X . 3 O . . . |
$$ | X X 2 4 5 . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

:b2: does not threaten to kill.

Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . . X 1 4 6 O | X . a O . . . |
$$ | X X 3 2 5 . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O O . | . X X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

With no dame :b6: makes ko. If :w7: kills Black on the right with a, :b8: fills the ko on the left and wins by 1 pt. for the captured White stone. So White must fight the ko.

As this ko is essential to the difference game, difference games are not appropriate to this comparison.

OC, if there is no damezumari, White can win without a ko fight, and the two plays are incomparable. :)

Click Here To Show Diagram Code
[go]$$Wc Kosumi vs. keima
$$ --------------------------------
$$ | . . X 1 4 6 O | X . 8 O 9 . . |
$$ | X X 3 2 5 7 O | X . . 0 . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

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 Post subject: Re: Thermography
Post #32 Posted: Tue Sep 29, 2020 1:46 pm 
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Bill Spight wrote:
Well, I got carried away with the difference games, eh? :lol:

Constructing the thermograph

Truth to say, difference games are not necessary to thermography at all, and are hardly ever used. However, it was easy to show that the keima sente and the crawl sente are unnecessary. The kosumi sente dominates them. As I said, if they were gote, that would be a different matter. But anyway, all we have to consider are the kosumi sente and the large monkey jump sente. Let's review the reverse sente first.

Click Here To Show Diagram Code
[go]$$Wc Reverse sente, Black continues
$$ -----------------
$$ | C 4 2 3 C C O |
$$ | X X 1 C C C O |
$$ | . X O O O O O |
$$ | . X X X O O C |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

:w1: blocks on the second line. Play can, and usually does, stop there. If Black replies she plays the hane-and-connect for a local score of -5 in the marked region. Each player has made the same number of plays, so the Right wall of the thermograph rises as a vertical line at that point.

Black does not reply above temperature 1, so the wall turns 45° to the left at that temperature rising one degree for each point of territory. Remember, the leftward movement is positive, the reverse the usual convention. The equation of that line is v = -6 + t, where v is the territory and t is the temperature.

Now let's look at the kosumi sente.

Click Here To Show Diagram Code
[go]$$Bc Kosumi sente, Black continues
$$ -----------------
$$ | . . 1 3 4 . O |
$$ | X X 5 2 . . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

The sente actually ends with :w2: but we consider what happens when Black keeps playing. The score after :b5: is -2. Black has made one more move than White, so the line from here angles up and to the right. It's equation is v = -2 - t. How far up does it go?

Click Here To Show Diagram Code
[go]$$Wc Kosumi sente, White follow-up
$$ -----------------
$$ | . . B 1 . . O |
$$ | X X 2 W . . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

If I didn't already know, it would be easy to work out that this is correct play at temperature 1. White could get an equivalent result, except for ko threats, by starting at 2. Each player has made the same number of plays, including the kosumi sente exchange, so the line rises vertically at -3. These two lines intersect at v = -3, t = 1. Above temperature 1 the prospective wall, called a scaffold, rises vertically at -3, below temperature 1 it angles down to -2 at temperature 0.

Now let's look at the monkey jump.

Click Here To Show Diagram Code
[go]$$Bc Monkey jump sente 1
$$ -----------------
$$ | . . 3 4 1 . O |
$$ | X X 2 5 6 . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

This is one of the main lines, with the throw-in, :w4:. White could also get here by starting at 4. That's the other main line. The usual reply is at 6. We'll consider that later.

What if Black continues?

Click Here To Show Diagram Code
[go]$$Bc Monkey jump sente 1, Black continues
$$ -----------------
$$ | . 9 B 7 B 8 O |
$$ | X X W B W . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

1 :wc: prisoner
:b7: connects the Black stones, :w8: makes a second eye, and :b9: saves the four stones. The local score is 0. Since Black has made one more move than White, the scaffold angles up from 0 to the right at one point of territory for each degree of temperature. The equation of the line is v = -t.

How high does it go? To find out, let's look at the White follow-up after the sente.

Click Here To Show Diagram Code
[go]$$Wc Monkey jump sente 1, White follow-up.
$$ -----------------
$$ | . 2 B 3 B 1 O |
$$ | X X W B W . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

1 :wc: prisoner

There is a ko, which White can take and win on the usual assumption of no ko threats. But :w1: comes to the same thing. ;) After :w3: the local score is -4. Since White has made one more move than Black, the scaffold angles up from there to the left. It's equation is v = -4 + t. Note that this is the scaffold for the position after the sente, not the scaffold for the original position. The scaffolds intersect at t = 2, v = -2. The mast rises vertically from there. The scaffold for this sente is v = -2 above temperature 2 and v = -4 + t below temperature 2.

There is another variation to consider.

Click Here To Show Diagram Code
[go]$$Bc Monkey jump sente 1a
$$ -----------------
$$ | . . 3 5 1 6 O |
$$ | X X 2 . 4 . O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

:w4: does not play the throw-in, but it threatens it. :b5: prevents that, with sente. As we have seen, with a different order of play, this line is incomparable with the throw-in. After :w6: the local score is -1. The scaffold rises vertically from there. OC, that is worse for White than -2, so this line is not as good as the throw-in above temperature 2. But it is better for White than 0, so we find the intersection v = -t and v = -1 to find where White switches to this line of play. The intersection is at t = 1, v = -1. So for the reply of :w2: the scaffold is vertical at -2 above temperature 2, angling down below temperature 2 to -1 at temperature 1, and vertical below that to -1 at temperature 0. :)

BTW, what if Black does not connect with :b5: in the previous diagram, but tries to kill?

Click Here To Show Diagram Code
[go]$$Bcm5 Monkey jump sente 1a, variation
$$ -----------------
$$ | . . B 2 B 1 O |
$$ | X X W 3 W a O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

Now White cannot play at a because of damezumari. Oh, for a dame!

Click Here To Show Diagram Code
[go]$$Wcm8 Monkey jump sente 1a, variation
$$ -----------------
$$ | 2 1 B 3 B B O |
$$ | X X W B W 5 O |
$$ | . X O O O O O |
$$ | . X X X O O . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ -----------------[/go]

:b11: at 8
White has no external threats, by assumption, but the play has generated a local threat for White at :w8:. With a dame White would not have needed to play the ko. :)

More later.


Here under the thermograph corresponding to basic sente sequences identified by Bill's post above.

My conclusion is that, under ideal environment, monkey jump is better than kosumi. BTW it was clearly identified when Bill, using difference games, shows that monkey jump is most of the time better than kosumi


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 Post subject: Re: Thermography
Post #33 Posted: Tue Sep 29, 2020 2:12 pm 
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Click Here To Show Diagram Code
[go]$$B keima best move
$$ -----------------
$$ | . . . . . . O |
$$ | X X . . . . O |
$$ | . X O O O O O |
$$ | . X X X X O . |
$$ | . X X X X O O |
$$ | . X X X X X O |
$$ | . X . O O . O |
$$ -----------------[/go]

In the position above keima is strictly better than both kosumi and monkey jump.

Click Here To Show Diagram Code
[go]$$B monkey jump best move
$$ -----------------
$$ | . . . . . . O |
$$ | X X . . . . O |
$$ | . X O O O O O |
$$ | . X X X X O . |
$$ | . X X X X O O |
$$ | . X X X X X O |
$$ | . X . O O O O |
$$ -----------------[/go]

In this one monkey jump is strictly better than both kosumi and keima.

My question is now the following : does it exist a non-ko environment for which kosumi is strictly better than both keima and monkey jump?

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Post #34 Posted: Tue Sep 29, 2020 2:31 pm 
Honinbo

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Gérard TAILLE wrote:
With your last post I believe I have found where is the misunderstanding.

The point is quite subtil: considering the environment we both exclude pure ko threats like for example:
Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | X X X O . . . |
$$ | . a X O . . . |
$$ | X X O O . . . |
$$ | . X X O . . . |
$$ | X X O O . . . |
$$ | O O O . . . . |
$$ | . . . . . . . |
$$ -----------------[/go]

Quote:

Actually, the existence of a ko threat in an environment is not a problem, per se. The basic question is this.

Click Here To Show Diagram Code
[go]$$B Zero?
$$ ------------------------------
$$ | X X X O . . . |. . . X O O O |
$$ | . a X O . . . |. . . X O . . |
$$ | X X O O . . . |. . . X X O O |
$$ | . X X O . . . |. . . X O O . |
$$ | X X O O . . . |. . . X X O O |
$$ | O O O O . . . |. . . X X X X |
$$ | . O . O . . . |. . . X . X . |
$$ -------------------------------[/go]

Does the environment and its mirror add to zero? The
existence of ko threats in the environment is not enough for that not to be so. To have a ko (or superko) fight you need a ko.

The logic of difference games has been proven only for non-ko positions and environments. However, I think that they also have a practical value if a ko fight is exceptional. For instance, this board as an environment looks OK to me, but in a difference game, while the players alternate plays, plays do not have to alternate on either board or independent region. The logic of adding to zero is that of playing mirror go, alternating between boards or regions. It would be possible, for instance, for Black to play on the left while White answered on the right, and thus for Black to create a ko on the left, thus invalidating the board on the left as an environment for difference games. While that is a theoretical possibility, it does not keep me up at night. :lol:

Quote:
But you seem also to exclude from the environment "normal" sente move like for example:

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



Not so. See above. Such a ko threat is not a problem in an environment, per se. But to avoid confusion if there is an exceptional ko, best not to include one.

In fact, best to consider environments for difference games as no man's land, where plays are verboten. If a play in an environment matters, it should be part of the position or combination of positions under consideration.

This holds true for an ideal environment of thermography, as well, except for plays at its temperature. Its only significant features should be its temperature, and the fact that the temperature eventually drops in steps so small that they do not matter.

Quote:
Because in my mind I accept such environment which allows to use a sente move as a ko threat our conclusions are logically different

Here is an example for the discussion kosumi vs keima

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


You can see that after a white move at "a", white may create a sente move at "b" which can be use as a ko threat. That is the point.


These different regions are not strictly independent, because they all involve the same string of stones which is not immortal. That may well be of little practical significance, and we may be able to treat them as independent. On a larger board it would be easy to make them independent, as long as there is no ko fight. Kos destroy independence.

It seems to me that the main difference between us is that you want to treat the environment as foreground, and I want to treat it as background. :)

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 Post subject: Re: Thermography
Post #35 Posted: Wed Sep 30, 2020 11:31 am 
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Difference games looks now for me like a magic tool to compare the different moves.
As a consequence it seems also be an help to build thermographs.

Consider the following position

diag.1
Click Here To Show Diagram Code
[go]$$B
$$ ---------------
$$ | X O O O . . .
$$ | X . . O . . .
$$ | X . . O . . .
$$ | X O . O . . .
$$ | X O O O . . .
$$ | X X X X . . .
$$ | . . . . . . .
$$ | . . . . . . .[/go]


And my question is how to build the thermograph of this position ?

BTW, I am not really interested by the best way to build this thermograph but I would like to know if my way of reasoning is correct!

First of all I use a difference game in order to find the best move

Black to play:

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


Firstly of all I prove that it exist an environment for which "a" if better than "b":

Black to play
Click Here To Show Diagram Code
[go]$$B
$$ --------------------------------
$$ | X O O O . . . - . . . X X X O |
$$ | X 1 5 O . . . - . . . X . . O |
$$ | X 4 . O . . . - . . . X 3 2 O |
$$ | X O . O . . . - . . . X . X O |
$$ | X O O O . . . - . . . X X X O |
$$ | X X X X . . . - . . . O O O O |
$$ | . . . . . . . - . . . . . . . |
$$ | . . . . . . . - . . . . . . . |[/go]

and black wins

Secondly I prove that, white to move, white cannot win:

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

and white cannot win.

That proves black "a" is the correct move.

Now white to play:

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


Firstly of all I prove that it exist an environment for which "b" if better than "a":

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

and white wins

Secondly I prove that, black to move, black cannot win:

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

and black cannot win

That proves white "b" is the correct move.

Where are we ?

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


In the diagramm above:
if it is black to blay the sequence begin by :b1: at "a", :w2: at "b"
if it is white to blay the sequence begin by :w1: at "b", :b2: at "a"

Can I deduce the thermogrph, using this result?

diag 2
Click Here To Show Diagram Code
[go]$$B
$$ ---------------
$$ | X O O O . . .
$$ | X X . O . . .
$$ | X O . O . . .
$$ | X O . O . . .
$$ | X O O O . . .
$$ | X X X X . . .
$$ | . . . . . . .
$$ | . . . . . . .[/go]


The thermograph of this diag2 is not difficult to draw and my point is to prove that diag1 has exactly the same than diag2 !

Here is my reasonning:
The miai value for diag2 is equal to 0.75
In diag1, as soon as the temperature drops to 0.75, then in one hand :b1: at "a" in diag1 force the immediat answer by white :w2: at "b", and in the other hand :w1: at "b" in diag1 force the immediat answer by black :b2: at "a".
My conclusion is that miai value if diag1 is also 0.75 and diag1 is completly equivalent to diag2.

Is this correct?


This post by Gérard TAILLE was liked by: Bill Spight
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 Post subject: Re: Thermography
Post #36 Posted: Wed Sep 30, 2020 2:24 pm 
Honinbo

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Gérard TAILLE wrote:
Difference games looks now for me like a magic tool to compare the different moves.
As a consequence it seems also be an help to build thermographs.

Consider the following position

diag.1
Click Here To Show Diagram Code
[go]$$B
$$ ---------------
$$ | X O O O . . .
$$ | X . . O . . .
$$ | X . . O . . .
$$ | X O . O . . .
$$ | X O O O . . .
$$ | X X X X . . .
$$ | . . . . . . .
$$ | . . . . . . .[/go]


And my question is how to build the thermograph of this position ?

BTW, I am not really interested by the best way to build this thermograph but I would like to know if my way of reasoning is correct!

First of all I use a difference game in order to find the best move


Not necessary for deriving the thermograph, but informative. :)

Quote:
Black to play:

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


Firstly of all I prove that it exist an environment for which "a" if better than "b":

Black to play
Click Here To Show Diagram Code
[go]$$B
$$ --------------------------------
$$ | X O O O . . . - . . . X X X O |
$$ | X 1 5 O . . . - . . . X . . O |
$$ | X 4 . O . . . - . . . X 3 2 O |
$$ | X O . O . . . - . . . X . X O |
$$ | X O O O . . . - . . . X X X O |
$$ | X X X X . . . - . . . O O O O |
$$ | . . . . . . . - . . . . . . . |
$$ | . . . . . . . - . . . . . . . |[/go]

and black wins


After :b5: right and left are miai, for 1 pt. for Black. :)

Quote:
Secondly I prove that, white to move, white cannot win:

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

and white cannot win.

That proves black "a" is the correct move.

Now white to play:

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


Firstly of all I prove that it exist an environment for which "b" if better than "a":

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

and white wins


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

I'm sure you saw this. Just for completeness. :)

Quote:
Secondly I prove that, black to move, black cannot win:

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

and black cannot win

That proves white "b" is the correct move.

Very good. :)

Quote:
Where are we ?

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


In the diagramm above:
if it is black to blay the sequence begin by :b1: at "a", :w2: at "b"
if it is white to blay the sequence begin by :w1: at "b", :b2: at "a"

Can I deduce the thermogrph, using this result?

Strictly speaking, no. We know that a is Black's best move, but we have not shown that b is White's best response. OC, it obviously is, and we can prove that (with the ko fight caveat). :)
And, OC, after :w1: at b, a is Black's only response. ;)

Quote:
diag 2
Click Here To Show Diagram Code
[go]$$B
$$ ---------------
$$ | X O O O . . .
$$ | X X . O . . .
$$ | X O . O . . .
$$ | X O . O . . .
$$ | X O O O . . .
$$ | X X X X . . .
$$ | . . . . . . .
$$ | . . . . . . .[/go]


The thermograph of this diag2 is not difficult to draw and my point is to prove that diag1 has exactly the same than diag2 !

Here is my reasonning:
The miai value for diag2 is equal to 0.75
In diag1, as soon as the temperature drops to 0.75, then in one hand :b1: at "a" in diag1 force the immediat answer by white :w2: at "b", and in the other hand :w1: at "b" in diag1 force the immediat answer by black :b2: at "a".
My conclusion is that miai value if diag1 is also 0.75 and diag1 is completly equivalent to diag2.

Is this correct?


If White b is the answer to Black a, and vice versa then they are miai and, indeed, the thermograph for diagram 1 is the same as the thermograph for diagram 2. :) That's an excellent insight. :clap: :clap: :clap:

And, since there are no kos in this position, we can go further. The two positions are equivalent, subject to the ko fight warning.

We can show that with a difference game. Difference games are not just about comparing plays. We can also compare positions.

Click Here To Show Diagram Code
[go]$$Bc Black to play cannot win
$$ --------------------------------
$$ | X O O O . . . - . . . X X X O |
$$ | X B . O . . . - . . . X . 2 O |
$$ | X W . O . . . - . . . X . 1 O |
$$ | X O . O . . . - . . . X . X O |
$$ | X O O O . . . - . . . X X X O |
$$ | X X X X . . . - . . . O O O O |
$$ | . . . . . . . - . . . . . . . |
$$ | . . . . . . . - . . . . . . . |[/go]

If :b1:, :w2: makes a mirror position, for jigo.

Click Here To Show Diagram Code
[go]$$Bc Black to play cannot win
$$ --------------------------------
$$ | X O O O . . . - . . . X X X O |
$$ | X B 1 O . . . - . . . X 4 2 O |
$$ | X W . O . . . - . . . X . 3 O |
$$ | X O . O . . . - . . . X . X O |
$$ | X O O O . . . - . . . X X X O |
$$ | X X X X . . . - . . . O O O O |
$$ | . . . . . . . - . . . . . . . |
$$ | . . . . . . . - . . . . . . . |[/go]

:w4: makes a mirror position, as well.

Click Here To Show Diagram Code
[go]$$Bc Black to play cannot win
$$ --------------------------------
$$ | X O O O . . . - . . . X X X O |
$$ | X B 1 O . . . - . . . X . 2 O |
$$ | X W 4 O . . . - . . . X 3 . O |
$$ | X O . O . . . - . . . X . X O |
$$ | X O O O . . . - . . . X X X O |
$$ | X X X X . . . - . . . O O O O |
$$ | . . . . . . . - . . . . . . . |
$$ | . . . . . . . - . . . . . . . |[/go]

White makes jigo this way.

Click Here To Show Diagram Code
[go]$$Bc Black to play cannot win
$$ --------------------------------
$$ | X O O O . . . - . . . X X X O |
$$ | X B 1 O . . . - . . . X . 2 O |
$$ | X W 3 O . . . - . . . X 4 . O |
$$ | X O . O . . . - . . . X . X O |
$$ | X O O O . . . - . . . X X X O |
$$ | X X X X . . . - . . . O O O O |
$$ | . . . . . . . - . . . . . . . |
$$ | . . . . . . . - . . . . . . . |[/go]

And, finally, another jigo.

White to play cannot win, either.

Click Here To Show Diagram Code
[go]$$Wc White to play cannot win
$$ --------------------------------
$$ | X O O O . . . - . . . X X X O |
$$ | X B 1 O . . . - . . . X 4 3 O |
$$ | X W . O . . . - . . . X . 2 O |
$$ | X O . O . . . - . . . X . X O |
$$ | X O O O . . . - . . . X X X O |
$$ | X X X X . . . - . . . O O O O |
$$ | . . . . . . . - . . . . . . . |
$$ | . . . . . . . - . . . . . . . |[/go]

Black makes jigo.

Click Here To Show Diagram Code
[go]$$Wc White to play cannot win
$$ --------------------------------
$$ | X O O O . . . - . . . X X X O |
$$ | X B . O . . . - . . . X . 1 O |
$$ | X W . O . . . - . . . X . 2 O |
$$ | X O . O . . . - . . . X . X O |
$$ | X O O O . . . - . . . X X X O |
$$ | X X X X . . . - . . . O O O O |
$$ | . . . . . . . - . . . . . . . |
$$ | . . . . . . . - . . . . . . . |[/go]

:w2: makes a mirror position, for jigo.

So Diagram 1 and Diagram 2 are equal, with the ko fight caveat. (White b in Diagram 1 is a possible ko threat.)

Recognizing such miai positions can make the endgame easier in actual games. :)

We may regard each of the following sequences of play as a unit, just as we do the hane-and-connect.

Click Here To Show Diagram Code
[go]$$Wc
$$ ---------------
$$ | X O O O . . .
$$ | X 2 3 O . . .
$$ | X 1 . O . . .
$$ | X O . O . . .
$$ | X O O O . . .
$$ | X X X X . . .
$$ | . . . . . . .
$$ | . . . . . . .[/go]


Click Here To Show Diagram Code
[go]$$Bc
$$ ---------------
$$ | X O O O . . .
$$ | X 1 3 O . . .
$$ | X 2 a O . . .
$$ | X O . O . . .
$$ | X O O O . . .
$$ | X X X X . . .
$$ | . . . . . . .
$$ | . . . . . . .[/go]

Either player has a as a follow-up, to gain ½ pt.

Roughly, the thermograph looks like this.
Code:
               | t = ¾
              / \ t = ½
             |   \
             |    \ t = 0
          ------------
            -1 -1¼ -2


As you say, it's easy to draw, but not everybody knows it.

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

Visualize whirled peas.

Everything with love. Stay safe.

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 Post subject: Re: Thermography
Post #37 Posted: Thu Oct 01, 2020 7:44 am 
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To avoid damezumari, the best is to take your suggestion with the following difference game:

Click Here To Show Diagram Code
[go]$$B Kosumi vs. keima + monkey jump
$$ --------------------------------
$$ | . . a . . . O | X . b c . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


In order to have a still better understanding of difference games I tried to go further with the challenge of proving that kosumi is dominated by the couple (monkey jump + keima).

In other words the challenge is to prove that is does not exist an environment for which kosumi is strictly better than both keima and monkey jump.

Click Here To Show Diagram Code
[go]$$W Kosumi vs. keima + monkey jump
$$ -----------------
$$ | . . X 1 . . O |
$$ | X X 2 . . . O |
$$ | . X O O O O O |
$$ | . X X X O . . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ ----------------[/go]


The first part of the proof is to consider all the environments in which :b2: is (one of) the best move in answer to :w1:
For such environment the kosumi cannot dominate the keima because of the following difference game showing black cannot win:

Click Here To Show Diagram Code
[go]$$B Kosumi vs. keima + monkey jump
$$ --------------------------------
$$ | . . B 4 . . O | X . 1 W 2 8 . |
$$ | X X 5 6 . . O | X . . 3 7 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]



Let's now take all the other environments.

Click Here To Show Diagram Code
[go]$$W Kosumi vs. keima + monkey jump
$$ -----------------
$$ | . . X 1 . . O |
$$ | X X a 2 . . O |
$$ | . X O O O O O |
$$ | . X X X O . . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ ----------------[/go]

For these remaining environments remenber that, if white plays at :w1: black MUST answer at :b2: because a play "a" cannot be the best move

For these remaining environments white chooses the monkey jump and the challenge is again to prove black cannot win

Click Here To Show Diagram Code
[go]$$B Kosumi vs. keima + monkey jump
$$ --------------------------------
$$ | . . B . . . O | X . W a . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


In order to win black must play on the right diagram but what sequence she must choose?
Surprisingly the answer is quite simple because black MUST avoid to play at "a" during the sente sequence as the example here after shows clearly:

Click Here To Show Diagram Code
[go]$$B Kosumi vs. keima + monkey jump
$$ --------------------------------
$$ | . . B 6 . . O | X . W 3 2 . . |
$$ | X X a b . . O | X . 1 4 5 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

and now because a black move at "a" is not allowed, black must play at "b" with an obvious jigo : with the two stones :b3: and :w6: we easily reach a mirror position.

As a consequence the only possibility for black is to play

Click Here To Show Diagram Code
[go]$$B Kosumi vs. keima + monkey jump
$$ --------------------------------
$$ | . . B . . . O | X 3 W . 2 . . |
$$ | X X . . . . O | X . 1 . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


and here again black cannot win because of the following answer:

Click Here To Show Diagram Code
[go]$$W Kosumi vs. keima + monkey jump
$$ --------------------------------
$$ | . . B 4 5 . O | X B W 2 W . . |
$$ | X X 6 1 . . O | X . B 7 3 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


My conclusion:
Though kosumi and keima are incomparable, and kosumi and monkey jump are incomparable, it is possible to prove that the couple (keima + monkey jump) dominates kosumi.

Can you tell me if the reasonning is correct Bill ?

Of course this proof does not mean that we can forget the kosumi move. It proves only that the kosumi move may be the best move ONLY for ko fight reasons which looks a quite interesting information isn'it ?

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 Post subject: Re: Thermography
Post #38 Posted: Thu Oct 01, 2020 11:22 am 
Honinbo

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Gérard TAILLE wrote:
To avoid damezumari, the best is to take your suggestion with the following difference game:

Click Here To Show Diagram Code
[go]$$B Kosumi vs. keima + monkey jump
$$ --------------------------------
$$ | . . a . . . O | X . b c . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


In order to have a still better understanding of difference games I tried to go further with the challenge of proving that kosumi is dominated by the couple (monkey jump + keima).

In other words the challenge is to prove that is does not exist an environment for which kosumi is strictly better than both keima and monkey jump.


Not sure what you mean by that. :scratch:

Quote:
Click Here To Show Diagram Code
[go]$$W Kosumi vs. keima + monkey jump
$$ -----------------
$$ | . . X 1 . . O |
$$ | X X 2 . . . O |
$$ | . X O O O O O |
$$ | . X X X O . . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ ----------------[/go]


The first part of the proof is to consider all the environments in which :b2: is (one of) the best move in answer to :w1:


Ah! The raises the question. Is the position after :w2: is at least as good for White as the original? IOW, can Black to play win this difference game? If not, then the left side is at least as good for White as the right side.

Click Here To Show Diagram Code
[go]$$Bc Can Black to play win?
$$ --------------------------------
$$ | . . B W . . O | X . . . . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


Click Here To Show Diagram Code
[go]$$Bc White makes jigo
$$ --------------------------------
$$ | . 3 B W . . O | X . . 5 4 6 . |
$$ | X X 2 . . . O | X . . . 1 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

The reverse sente, :b1:, does not win. Black has nothing better after :w2:

Click Here To Show Diagram Code
[go]$$Bc Capture :wc:
$$ --------------------------------
$$ | . . B W 3 . O | X 9 4 8 6 . . |
$$ | X X 2 1 0 . O | X . 7 . 5 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

:b1: and :b3: transpose to a position starting with the large monkey jump. To avoid a mirror position, after :w4: and later, Black avoids playing at 8. :b5: holds White to 1 point in the corner.

Click Here To Show Diagram Code
[go]$$Bcm11 Black wins
$$ --------------------------------
$$ | . 3 X 1 X 2 O | X X O O O . . |
$$ | X X O X O . O | X . X . X O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

1 :wc: prisoner
:b11: connects to win by 1 point. The captured stone makes the difference.

Instead of playing the large monkey jump at :w4:, let White play the small monkey jump (keima).

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


Click Here To Show Diagram Code
[go]$$Wcm8 Black wins
$$ --------------------------------
$$ | . 4 X 2 X 3 O | X . X O O . . |
$$ | X X O X 1 . O | X . . X 5 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

The keima does not take away enough territory. Black wins by 2 points.

So in the original zero position for the difference game, if :b1: is the kosumi, the jump attachment for :w2: lets Black win by capturing the :w2: stone. That does not mean that if White makes the jump attachment in later play that Black is forced to capture the stone. If we consider the original position on the right to be an environment for the position on the left after :w2:, that does not mean the correct play with that environment is the same as correct play with a different environment.

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

Visualize whirled peas.

Everything with love. Stay safe.

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Post #39 Posted: Thu Oct 01, 2020 12:34 pm 
Honinbo

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BTW, for the position after :b1: plays the kosumi and :w2: plays the jump attachment, the solid connection first wins the difference game against the atari.

Difference game setup

Click Here To Show Diagram Code
[go]$$Bc Zero position
$$ --------------------------------
$$ | . . B W . . O | X . . B W . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


Click Here To Show Diagram Code
[go]$$Bc Connection first beats the atari
$$ --------------------------------
$$ | . . B W . . O | X 7 4 B W 8 . |
$$ | X X 1 6 . . O | X . 5 2 3 O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

:b9: captures two stones and wins by one point.

Click Here To Show Diagram Code
[go]$$Wcm6 Variation for :w6:
$$ --------------------------------
$$ | . . B W 6 7 O | X 2 O 1 W 3 . |
$$ | X X X 4 5 . O | X . X O X O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]

1 :bc: prisoner

:b13: connects to win by one point.

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

Visualize whirled peas.

Everything with love. Stay safe.

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 Post subject: Re: Thermography
Post #40 Posted: Thu Oct 01, 2020 1:05 pm 
Lives in gote

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It seems you do not really take my point because ... I agree 100% with your last post ! :razz:

Bill Spight wrote:
Gérard TAILLE wrote:
In other words the challenge is to prove that is does not exist an environment for which kosumi is strictly better than both keima and monkey jump.


Not sure what you mean by that. :scratch:


OK let me try to clarify the challenge.

In my previous post https://lifein19x19.com/viewtopic.php?p=260193#p260193 I showed two positions: one for which the only correct move is the keima move, and the other for which the only correct move is the monkey jump move.

After having failed to find a position in which the only correct move would have been the kosumi move I chose to try to prove that it is simply impossible (in absence of ko fight). This is my challenge.

Bill Spight wrote:
Click Here To Show Diagram Code
[go]$$Bc Can Black to play win?
$$ --------------------------------
$$ | . . B W . . O | X . . . . . . |
$$ | X X . . . . O | X . . . . O O |
$$ | . X O O O O O | X X X X X O . |
$$ | . X X X O . . | . . X O O O . |
$$ | . . . X O O O | X X X O . . . |
$$ | . . . X X X X | O O O O . . . |
$$ | . . . . . . . | . . . . . . . |
$$ --------------------------------[/go]


You proved black can win the above difference game but I had no doubts about that!


Click Here To Show Diagram Code
[go]$$W Kosumi vs. keima + monkey jump
$$ -----------------
$$ | . . X 1 . . O |
$$ | X X a . . . O |
$$ | . X O O O O O |
$$ | . X X X O . . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ ----------------[/go]

I never claimed that :w1: is in any case the correct move against kosumi.
I claimed that :w1: is the correct move against kosumi ONLY if black, due to the environment, has to answer with a move at "a".

My strategy with white is to play either the keima or the monkey jump but how do I make the choice?
It is a little subtil: I imagine the following sequence :
Click Here To Show Diagram Code
[go]$$B Kosumi vs. keima + monkey jump
$$ -----------------
$$ | . . 1 2 . . O |
$$ | X X a b . . O |
$$ | . X O O O O O |
$$ | . X X X O . . |
$$ | . . . X O O O |
$$ | . . . X X X X |
$$ | . . . . . . . |
$$ ----------------[/go]

and I find the best move for black (depending of the environment):
if the best move for black is a move at "a" then in the intial position I choose the keima
otherwise I choose the monkey jump.

The purpose of my last post was to analyse this strategy to prove it is correct providing there are no ko fight.

Obviously it is not easy to explain but that is my challenge.

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