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 Post subject: Re: Engame value of ko
Post #41 Posted: Sat Feb 18, 2023 8:37 am 
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RobertJasiek wrote:
Ko thermography is not as straightforward as sente because balloon trajectories need not proceed vertically but can have vertical parts.


Yes I know. My answer was only related to your "hard part":
"The hard part is to justify using the intermediate ko position for evaluating the initial position with two black stones."

OC I agree ko thermography is far more difficult.

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 Post subject: Re: Engame value of ko
Post #42 Posted: Mon Feb 20, 2023 4:44 am 
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Let me propose an other position which is very simple and very very common but however not that easy to evaluate due to the existence of a future ko:
Click Here To Show Diagram Code
[go]$$W
$$ -----------------
$$ | . . a O . . . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |[/go]
When will you play at "a" depending of the temperature?
Is the answer the same for white and black?

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 Post subject: Re: Engame value of ko
Post #43 Posted: Mon Feb 20, 2023 7:19 am 
Judan

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See https://www.lifein19x19.com/viewtopic.p ... 37#p276737 for a hint why this might not be "very simple".

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 Post subject: Re: Engame value of ko
Post #44 Posted: Mon Feb 20, 2023 8:37 am 
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RobertJasiek wrote:
See https://www.lifein19x19.com/viewtopic.p ... 37#p276737 for a hint why this might not be "very simple".

I agree with you to say that the evaluation of the position is not easy.
Concerning the position itself (and not it's evaluation) I consider this position very simple because the sequence to play locally (I mean if ambiant temperature is t = 0) is absolutly obvious.
The difficulty arises as soon as you take into account an ambiant temperature t > 0.
BTW, without giving all the calculation, could you just give us the final result just to motivate the readers to try and find the correct justification?

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 Post subject: Re: Engame value of ko
Post #45 Posted: Mon Feb 20, 2023 9:05 am 
Judan

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I am busy with stage kos... For your position, thermography is an overkill - the method of making a hypothesis will do. To verify the second-most naive assumption of Black's long sente (with White's 3-move sequence), explore the tree, calculate the counts of the followers, derive the gains and compare each to the assumed move value. I do not know if the method of comparing the opponent's branches applies; if it does, it might be faster.

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 Post subject: Re: Engame value of ko
Post #46 Posted: Mon Feb 20, 2023 11:41 am 
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RobertJasiek wrote:
I am busy with stage kos... For your position, thermography is an overkill - the method of making a hypothesis will do. To verify the second-most naive assumption of Black's long sente (with White's 3-move sequence), explore the tree, calculate the counts of the followers, derive the gains and compare each to the assumed move value. I do not know if the method of comparing the opponent's branches applies; if it does, it might be faster.

OK Robert, I understand your are busy.

For the other readers my own result is the following:
Click Here To Show Diagram Code
[go]$$B
$$ -----------------
$$ | . . 1 O 2 . . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |[/go]
The exchange :b1: :w2: is sente for black and for a theoritical point of view this exchange must be played by black when temperature drops between 13/12 and 14/12.
I think in practice any experienced player knows that a local move must be played (by black or white) as soon as temperature gets very near from 1 (I mean 1 + ɛ).

BTW
Click Here To Show Diagram Code
[go]$$W
$$ ---------------------
$$ | . . . . . . . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |[/go]
In this diagramme white (or black) will play hane when t = 1

Click Here To Show Diagram Code
[go]$$W
$$ ---------------------
$$ | . . 2 1 3 . . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |[/go]
after white 1 the temperature increases from 1 to 13/12
then after black 2 the temperature increases again from 13/12 to 14/12.

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 Post subject: Re: Engame value of ko
Post #47 Posted: Tue Feb 21, 2023 9:09 am 
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RobertJasiek wrote:
I am busy with stage kos... For your position, thermography is an overkill - the method of making a hypothesis will do. To verify the second-most naive assumption of Black's long sente (with White's 3-move sequence), explore the tree, calculate the counts of the followers, derive the gains and compare each to the assumed move value. I do not know if the method of comparing the opponent's branches applies; if it does, it might be faster.

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

The method you suggest is fine OC but it looks quite difficult because I guess the tree you consider seems quite big:
-you have probably to evaluate more than ten leaves
-you have then to derive the score of the leaves through all the tree and here again probably through more than ten nodes.

That's true if you consider a player without any knowledge of CGT.

It's the same with tsumego
Click Here To Show Diagram Code
[go]$$W
$$ -----------------
$$ | X X . O . X O . . |
$$ | . X X . . X O . . |
$$ | X O X X X X O . . |
$$ | . O O O O O O . . |
$$ | O O . . . . . . . |
$$ | . . . . . . . . . |[/go]
Any experience player knows that in the position above black is dead, because an experience player knows perfectly what is a false eye and knows perfectly how nakade works. On the other hand for a beginner this position may appear extremely diffcult.
When you try to solve a tsumego problem you will often reach positions which you are able to evaluate without further moves, just by using your knowledge of false eyes and nakade.

Most of the time, when you analysis a position for a CGT evaluation you will encounter "moves increasing/decreasing temperature" and "infinitesimals". Without the basic knowledge of how you can handle "moves increasing/decreasing temperature" and how works "infinitesimals" then The CGT evaluation of a position can be extremely complex.

For me "moves increasing/decreasing temperature" and "infinitesimals" are for CGT what "false eyes" and "nakade" are for tsumego.

As far as I am concern, with these basic knowledge of CGT, I only have to evaluate four leaves and I have to derive these evaluations through only three nodes.

Curiously most of the players have a better knowledge of tsumego comparing to the evaluation of the small yose. Probably because the impact of a tsumego mistake is far more decisice than a mistake in the small yose!

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 Post subject: Re: Engame value of ko
Post #48 Posted: Tue Feb 21, 2023 10:12 am 
Judan

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Gérard TAILLE wrote:
The method you suggest is fine OC but it looks quite difficult because I guess the tree you consider seems quite big:
-you have probably to evaluate more than ten leaves


Do not complain but just do it! Thermography would need ca. 30 times the time. With less robust methods, you cannot be sure to get the right values.

Quote:
That's true if you consider a player without any knowledge of CGT.


Simplifications are allowed if they do apply.

Quote:
Without the basic knowledge of how you can handle "moves increasing/decreasing temperature" and how works "infinitesimals" then The CGT evaluation of a position can be extremely complex.


If you try to shortcut, you might, e.g, use my methods of comparing counts or comparing move values but...

- To use them fast, one has to make many assumptions, of which most probably are false, and prey that no harm is done.
- Comparing counts does not always work because in CGT what you usually can compare is positions (i.e. CGT games, which you might compare in difference games, if there are and can be no kos), especially if they do not involve kos. These two methods often work in practice but there is no guarantee that they produce the correct values. It is just that I have seen them to work for actual positions while maybe non-positional, pathological CGT games might be more problematic. However, it could be that my many test positions were too stereotypical.

Therefore, again, better think about Bill's method of comparing the opponent's branches if you want to be much faster than the method of making a hypothesis.

Quote:
As far as I am concern, with these basic knowledge of CGT, I only have to evaluate four leaves and I have to derive these evaluations through only three nodes.


Wow, nice, please demonstrate! Especially how you verify that you may restrict evaluation to the three or four nodes!

Quote:
Curiously most of the players have a better knowledge of tsumego comparing to the evaluation of the small yose. Probably because the impact of a tsumego mistake is far more decisice than a mistake in the small yose!


Indeed. 100 small yose mistakes losing 7 ranks go unnoticed while 1 L+D mistake losing 3 ranks is noticed;)

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 Post subject: Re: Engame value of ko
Post #49 Posted: Tue Feb 21, 2023 11:14 am 
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[quote="RobertJasiek"]
Wow, nice, please demonstrate! Especially how you verify that you may restrict evaluation to the three or four nodes![quote]

Ok let's try.
I said I used only three intermediate nodes.
OC I have to begin with the deepest one. Here it is:

Click Here To Show Diagram Code
[go]$$W
$$ ---------------------
$$ | . . X . X . . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |[/go]
what is the score of the position what is the value of a move here?

White to play
Click Here To Show Diagram Code
[go]$$W
$$ ---------------------
$$ | . . X a X 1 . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |[/go]
remembering a white prisonner at "a" the score is s1 = -1/3 (black has two points of territory + one prisonner; white has three points of terriory and finally black score is -1/3 due to the ko

Black to play
Click Here To Show Diagram Code
[go]$$B
$$ ---------------------
$$ | . . X . X 1 . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |[/go]
and now we reach an infinitesimal. In that case you can always continue the sequence by adding two moves (one white and one black) providing you continue through infinitesimals.
Then I can continue the sequence :
Click Here To Show Diagram Code
[go]$$B
$$ ---------------------
$$ | . . X 3 X 1 2 . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |[/go]
and the score is s2 = 1 (black has two points of territory and one prisonner, and white has two points of territory.

Click Here To Show Diagram Code
[go]$$W
$$ ---------------------
$$ | . . X . X . . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |[/go]
The score of this position is s = (s1 + s2)/2 = (-1/3 + 1)/2 = 1/3
and the value of a move here is t = (s2 - s1)/2 = (1 - (-1/3)) / = 2/3

Do you agree with this first node evaluation?
Probably you will have doubt with my way of continuing the sequence in an infinitesimal environmement but that is my understanding of how infinetesimals work and I never found a counter example. Do you have one in mind?

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 Post subject: Re: Engame value of ko
Post #50 Posted: Tue Feb 21, 2023 12:24 pm 
Judan

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Where you count one node, I see four nodes;)

Please use the count instead of the score! Score is after successive passes when scoring and determining the winner. The term 'the count' applies to each position.

Apart from basic corridors and a few exceptions, I am not familiar with infinitesimals.

Why is your third diagram an infinitesimal and which?

Why may one add two alternating moves from and along infinitesimals? Must have to do something with CGT reversal.

If so, your calculation is correct.

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 Post subject: Re: Engame value of ko
Post #51 Posted: Tue Feb 21, 2023 2:54 pm 
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RobertJasiek wrote:
Why is your third diagram an infinitesimal and which?


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


I know by heart only very few infinitesimals : corridors when the value of the last move in the corridor as a value v >= 1 and the position above which leads to two other infinitesimals in the following sequence
Click Here To Show Diagram Code
[go]$$W :b2: tenuki in another infinitesimal
$$ ---------------------
$$ | . . X 3 X X 1 . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |[/go]
the initial position and the two positions after white 1 and after white 3 are all infinitesimals

Sure you are able to verify that these positions are infinitesimals.
BTW I do not know if this initial position has a name as infinitesimal.

RobertJasiek wrote:
Why may one add two alternating moves from and along infinitesimals? Must have to do something with CGT reversal.

No there are no link with CGT reversal. Basically CGT reversal has to do with "increasing temperature" but not infinitesimals.

The point is the following : an infinitesimal is played when temperature is t = 1. At this temperature, and assuming an ideal environnement each player can play either in the local infinitesimal or in another infinitesimal in the environment. As a consequence if black (resp. white) can play in the local infinitesimal to reach another infinitesimal and if white (resp. black) can answer to reach another infinitesimal then the local position is still an infinitesimal, the count of the position is the same (because each player has gained one point) and the move played are correct because we assumed a temperature t = 1 in a ideal environment.
You see the advantage of such sequence : two nodes are used but without the need to calculate their count (you know only that they are infinitesimals)

Be aware that this way of proceeding is only true to find the count of a position. In a practical environment be aware that the order in which you play the infinitesimals has to be studied carefully.

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 Post subject: Re: Engame value of ko
Post #52 Posted: Tue Feb 21, 2023 3:07 pm 
Judan

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Part 1/12

Compared to a simple gote or sente, an ordinary ko is 3 times but a stage ko in the corner is 100 times as difficult.

Theory

Every theory of ko evaluation makes some assumptions. The standard theory of ko evaluation is called 'generalised thermography', although its scope of application is not the most general. It applies to so called 'simple' cyclic positions while multiple kos or long alternating cycles require additional or alternative theory. For simple cyclic positions, we can use 'generalised thermography'.

Its major assumptions are as follows. We use territory scoring. Each local play is taxed by T points because we assume a 'rich environment' with arbitrarily many plays elsewhere at every temperature T. They can be imagined as simple gotes or coupons with this move value. Such a rich environment is a model but enables consistent evaluation of many local endgames with kos. Generalised thermography produces move values, counts and gains. Another theory might produce slightly different values but we accept the values generated by generalised thermography because their relations permit a consistent interpretation.

"Definition 3.8. Let G be a loopy game. We say that G is simple if: (i) the only loops in G are kos; and (ii) every subposition of G has at most one Left ko option and at most one Right ko option." Citation reference [18]: https://www.lifein19x19.com/viewtopic.p ... 45#p143245

In this definition, 'game' means 'position', 'subposition' means 'the position itself or a follow-up position', 'ko' means 'basic ko' or 'local position with alternating 2-play cycle', 'loop' seems to have the intended meaning 'positional cycle of plays', 'Left' means 'Black', 'Right' means 'White', 'option' means 'next move', 'ko option' means 'basic ko capture'.

In the definition, 'loop' is ambiguous. In generalised thermography, non-alternating sequences of plays are allowed. In Combinatorial Game Theory, some examples of game graphs show 'simple' loopy games with successive ko options of the same player so that non-alternating long cycles can occur when travelling on game graph. However, below we learn:

- that a player's ko option followed by the opponent's play elsewhere requires the player's local play enabling him to 'win' the ko;

- if the player's ko option is followed by the opponent's local play, it may not recapture this ko.

This implies obeying the basic ko rule that immediate recapture creating a 2-play cycle is prohibited, although condition (i) of the definition of 'simple' loopy game does not declare it. Condition (i) does, however, imply exclusion of the following:

- 0-play loop from G to G, even not if the move would incur the tax T. However, generalised thermography models a coupon stack as the environment. Unless explicitly excluded, a player also has the option of taking a coupon of the value T aka playing elsewhere.

- Suicide cycles even if the rules allow suicide.

- Long cycles without actually played 2-play cycles.

Generalised thermography applied to a simple loopy initial position allows its evaluation without simultaneously evaluating some followers as alternative initial positions. Contrarily, positions with multiple kos, such as a double ko death, might need simultaneous evaluation.

On a player's move, choose his best among the available of these options:

- best local non-ko play

- his ko option followed by the opponent's local play that does not recapture this ko

- his ko option followed by the opponent's play elsewhere and the player's local play

The opponent optimises among the second and third options. Then the player optimises among his first option and the opponent's chosen option. Ignore unavailable options. Optimising means Black maximising and White minimising.

In the theory of generalised thermography, for each local play, Black pays -T while White pays +T. Temperatures start from -1 so that the payment T = -1 for a play is the tax -1 for placing its stone. As to the tax, an equal number of local plays by Black and White cancel each other.

The player's choices do not include his initial play elsewhere. We consider it implicitly by analysing Black's and White's starts and walls and constructing the thermograph. Its mast represents both players' starts by plays elsewhere.

In a rich environment, generalised thermography calculates two pairs of trajectories: the players' walls and scaffolds. The walls are the final trajectories of a thermograph. The scaffolds are the auxiliary trajectories of a thermograph, which help finding the walls. Walls and scaffolds relate counts of a local, for example endgame, position to temperatures. Besides, a wall describes what the player should play at every temperature T while a scaffold describes what happens if the player plays then, regardless of whether he should or should not do so. The basic advice distinguishes local play and play elsewhere.

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 Post subject: Re: Engame value of ko
Post #53 Posted: Tue Feb 21, 2023 3:10 pm 
Judan

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Part 2/12

Positions

The unsettled initial positions and followers are:

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


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


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


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


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


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


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


Each of these positions is 'simple' because each possible alternating cycle of plays is a 2-play cycle and each player has at most one play that is a basic ko capture. In the local endgame, long cycles involve successive black plays and successive white plays while either opponent must play elsewhere or pass.

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 Post subject: Re: Engame value of ko
Post #54 Posted: Tue Feb 21, 2023 3:13 pm 
Judan

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Part 3/12

Click Here To Show Diagram Code
[go]$$B Dia. 1: initial position
$$ -----------
$$ | . O X X .
$$ | O X X . X
$$ | . O X X X
$$ | O O O O X
$$ | . O . O .[/go]


The initial position is evaluated under the presuppositions of having a rich environment but no komaster.

Black Starts

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


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


Click Here To Show Diagram Code
[go]$$B Dia. 4: prisoners = 1, count = 1, result = 1 - T
$$ -----------
$$ | X 3 X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Click Here To Show Diagram Code
[go]$$B Dia. 5: Black 3 elsewhere
$$ -----------
$$ | X 4 X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Click Here To Show Diagram Code
[go]$$B Dia. 6: Black 5 elsewhere, prisoners = 0, count = 0, result = 2T
$$ -----------
$$ | 6 O X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Click Here To Show Diagram Code
[go]$$B Dia. 7: White 2, 4 elsewhere
$$ -----------
$$ | X . X X .
$$ | O X X . X
$$ | 3 O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Click Here To Show Diagram Code
[go]$$B Dia. 8: prisoners = 2, count = 3, result = 3 - 3T
$$ -----------
$$ | X C X X .
$$ | 5 X X . X
$$ | X O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Since Black 1 in Dia. 2 and Black 3 in Dia.7 are ko options and White replies elsewhere, Black's next moves must be local plays. Since White 4 in Dia. 5 is a ko option and Black replies elsewhere, White's next move must be a local play.

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Post #55 Posted: Tue Feb 21, 2023 3:16 pm 
Judan

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Part 4/12

Remaining Ko

Click Here To Show Diagram Code
[go]$$B Dia. 9: prisoners = 1
$$ -----------
$$ | X . X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Dia. 9: This is the position after move 2 in Dia. 3.

Click Here To Show Diagram Code
[go]$$B Dia. 10: prisoners = 1, count = 1, result = 1 - T
$$ -----------
$$ | X 3 X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Dia. 10: Black's non-ko play

Click Here To Show Diagram Code
[go]$$W Dia. 11: Black 4 elsewhere
$$ -----------
$$ | X 3 X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Dia. 11: White's ko option, Black's play elsewhere

Click Here To Show Diagram Code
[go]$$W Dia. 12: prisoners = 0, count = 0, result = 2T
$$ -----------
$$ | 5 O X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Black scaffold: 1 - T

White scaffold: 2T

Equality: 1 - T = 2T <=> 1 = 3T <=> 1/3 = T

Move value: M = T = 1/3

Count: 1 - 1/3 = 2 * 1/3 = 2/3

Black wall:

2/3 if T ≥ 1/3

1 - T if T ≤ 1/3

White wall:

2/3 if T ≥ 1/3

2T if T ≤ 1/3

Strategy: Play elsewhere if T ≥ 1/3. Play locally if T ≤ 1/3.


Last edited by RobertJasiek on Tue Feb 21, 2023 3:41 pm, edited 1 time in total.
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Post #56 Posted: Tue Feb 21, 2023 3:18 pm 
Judan

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Part 5/12

Ko Option and Reply Elsewhere

We analyse backwards.

Click Here To Show Diagram Code
[go]$$B Dia. 13: after move 4, prisoners = 2
$$ -----------
$$ | X . X X .
$$ | . X X . X
$$ | B O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Dia. 13: Since the position occurs after Black's ko option (move 3 in Dia. 7) by playing the marked stone and White's play elsewhere (move 4 in Dia. 7), Black continues in Dia. 8 resulting in 3 - 3T.

Inherited black scaffold: 3 - 3T.

Click Here To Show Diagram Code
[go]$$B Dia. 14: after move 2, prisoners = 1
$$ -----------
$$ | B . X X .
$$ | O X X . X
$$ | . O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Dia. 14: Since the position occurs after Black's ko option (move 1 in Dia. 2) by playing the marked stone and White's play elsewhere (move 2 in Dia. 7), Black continues in Dia. 7 creating Dia. 13 with its black scaffold 3 - 3T.

Inherited black scaffold: 3 - 3T

Click Here To Show Diagram Code
[go]$$B Dia. 15: ko option
$$ -----------
$$ | C O X X .
$$ | O X X . X
$$ | . O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Dia. 15: Black's start with the ko option at the marked intersection and White's play elsewhere create Dia. 14 with its black scaffold 3 - 3T.

Inherited black scaffold: 3 - 3T

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 Post subject: Re: Engame value of ko
Post #57 Posted: Tue Feb 21, 2023 3:20 pm 
Judan

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Part 6/12

White's Choice on Move 2

After Black's ko option (move 1 in Dia. 2),

- White's local reply results in Black's wall 2/3 if T ≥ 1/3 or 1 - T if T ≤ 1/3 (see Dia. 9).

- White's reply elsewhere results in Black's scaffold 3 - 3T (see Dia. 15).

If T ≥ 1/3, compare 2/3 ? 3 - 3T <=> 3T ? 2 1/3 <=> T ? 7/9. On move 2, the minimising White chooses the local reply in Dia. 3 with the result 2/3 if 1/3 ≤ T ≤ 7/9 or the play elsewhere in Dia. 7 with the result 3 - 3T if T ≥ 7/9.

If T ≤ 1/3, compare 1 - T ? 3 - 3T <=> 2T ? 2 <=> T ? 1. Since T ≤ 1/3 is stricter than T ≤ 1, White chooses the local reply in Dia. 3 with the result 1 - T if T ≤ 1/3. The case T ≤ 1/3 and T ≥ 1 does not exist. In summary, the result is

-- 3 - 3T if T ≥ 7/9 and White replies elsewhere,

-- 2/3 if 1/3 ≤ T ≤ 7/9 and White replies locally,

-- 1 - T if T ≤ 1/3 and White replies locally.

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 Post subject: Re: Engame value of ko
Post #58 Posted: Tue Feb 21, 2023 3:22 pm 
Judan

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Part 7/12

Black's Choice on Move 1 in the Initial Position

In the local endgame's initial position, Black does not have any local non-ko play. He only has the ko option, which he must choose. Due to White's choice on move 2, for the initial position on move 1, we have

Black scaffold:

3 - 3T if T ≥ 7/9

2/3 if 1/3 ≤ T ≤ 7/9

1 - T if T ≤ 1/3

We could determine Black's scaffold although

- the result 3 - 3T (T ≥ 7/9 and White replies elsewhere) is given by a black scaffold when, for the initial position, the inherited result includes the tax from move 1 on,

- otherwise the result if White replies locally is given by a black wall when, after move 2, the net tax is 0.

Dias. 16 - 22: We interpret Black's scaffold.

Click Here To Show Diagram Code
[go]$$B Dia. 16: T ≥ 7/9
$$ -----------
$$ | 1 O X X .
$$ | O X X . X
$$ | . O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Click Here To Show Diagram Code
[go]$$B Dia. 17: White 2, 4 elsewhere
$$ -----------
$$ | X . X X .
$$ | O X X . X
$$ | 3 O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Click Here To Show Diagram Code
[go]$$B Dia. 18: prisoners = 2, count = 3, result = 3 - 3T
$$ -----------
$$ | X C X X .
$$ | 5 X X . X
$$ | X O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Dias. 16 - 18: This is the case T ≥ 7/9 of Black's scaffold. White plays elsewhere. The local territory 1 and prisoner difference 2 amount to the count 3. Black's 3 excess plays incur the tax -3T. Therefore, the result is 3 - 3T.

Click Here To Show Diagram Code
[go]$$B Dia. 19: 1/3 ≤ T ≤ 7/9
$$ -----------
$$ | 1 O X X .
$$ | O X X . X
$$ | 2 O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Click Here To Show Diagram Code
[go]$$B Dia. 20: prisoners = 1, count = 2/3, result = 2/3
$$ -----------
$$ | B . X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Dias. 19 + 20: This is the case 1/3 ≤ T ≤ 7/9 of Black's scaffold. After move 2, the players play elsewhere. The marked remaining basic endgame ko has the on-board count -1/3. Together with the prisoner difference 1, we have the count -1/3 + 1 = 2/3. The equal numbers of black and white plays incur the net tax 0. Therefore, the result is 2/3 + 0 = 2/3.

Click Here To Show Diagram Code
[go]$$B Dia. 21: T ≤ 1/3
$$ -----------
$$ | 1 O X X .
$$ | O X X . X
$$ | 2 O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Click Here To Show Diagram Code
[go]$$B Dia. 22: prisoners = 1, count = 1, result = 1 - T
$$ -----------
$$ | X 3 X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Dias. 21 + 22: This is the case T ≤ 1/3 of Black's scaffold. The players play locally. Due to the prisoner difference 1, the count is 1. Black's 1 excess play incurs the tax -T. Therefore, the result is 1 - T.


Last edited by RobertJasiek on Tue Feb 21, 2023 3:44 pm, edited 1 time in total.
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 Post subject: Re: Engame value of ko
Post #59 Posted: Tue Feb 21, 2023 3:24 pm 
Judan

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Part 8/12

White Starts

Click Here To Show Diagram Code
[go]$$W Dia. 23: White starts
$$ -----------
$$ | . O X X .
$$ | O X X . X
$$ | 1 O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Click Here To Show Diagram Code
[go]$$W Dia. 24: White 3 elsewhere, Black 4 at Circle
$$ -----------
$$ | 2 W X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Click Here To Show Diagram Code
[go]$$B Dia. 25: prisoners = 1, count = 1, result = 1 - T
$$ -----------
$$ | X X X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Click Here To Show Diagram Code
[go]$$W Dia. 26: Black 2 elsewhere, count = 0, result = 2T
$$ -----------
$$ | 3 O X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Since Black 2 in Dia. 24 is a ko option and White replies elsewhere, Black's next move must be a local play.

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 Post subject: Re: Engame value of ko
Post #60 Posted: Tue Feb 21, 2023 3:26 pm 
Judan

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Part 9/12

Remaining Ko

Click Here To Show Diagram Code
[go]$$B Dia. 27
$$ -----------
$$ | . O X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Dia. 27: This is the position after move 1 in Dia. 23.

Click Here To Show Diagram Code
[go]$$W Dia. 28: White 3 elsewhere
$$ -----------
$$ | 2 O X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Dia. 28: Black's ko option, White's play elsewhere

Click Here To Show Diagram Code
[go]$$W Dia. 29: prisoners = 1, count = 1, result = 1 - 2T
$$ -----------
$$ | X 4 X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Click Here To Show Diagram Code
[go]$$B Dia. 30: count = 0, result = T
$$ -----------
$$ | 2 O X X .
$$ | O X X . X
$$ | O O X X X
$$ | O O O O X
$$ | . O . O .[/go]


Dia. 30: White's non-ko play


Black scaffold: 1 - 2T

White scaffold: T

Equality: 1 - 2T = T <=> 1 = 3T <=> 1/3 = T

Move value: M = T = 1/3

Count: 1 - 2 * 1/3 = 1/3 = 1/3

Black wall:

1/3 if T ≥ 1/3

1 - 2T if T ≤ 1/3

White wall:

1/3 if T ≥ 1/3

T if T ≤ 1/3

Strategy: Play elsewhere if T ≥ 1/3. Play locally if T ≤ 1/3.


Last edited by RobertJasiek on Tue Feb 21, 2023 3:49 pm, edited 2 times in total.
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