Engame value of ko

[RobertJasiek]

If a position occurs as subposition in an evaluation of another initial position, it does not mean that the evaluation as a subposition would be the same as its evaluation as an initial position! The root position, prisoners and ko bans matter!

To evaluate the other initial position, what we do need from this subposition is exactly and just the black scaffold.

<edit> Since we only need the subposition's black scaffold, we do not need its count as if it were the initial position. For the black scaffold, we also do not need an auxiliary count in this subposition. We need the on-board and prisoners count 3, which is the 3 in the term 3 - 3T, in the settled subposition, from which then we inherit the black scaffold of this unsettled subposition. </edit>

The leaves need their on-board and prisoners count.

When eventually we determine the initial position's thermograph, we can also infer its count (more precisely: its mast value, which for God's sake I call "count"; these additional CGT terms would only confuse us; Berlekamp and Spight have also used the word "count"; recall that the count depends on the assumption of no / a komaster and used model of the environment).

For intermediate subpositions, we need their scaffolds and / or walls. The walls might be equal or unequal to the scaffolds for different temperature ranges and the related trajectory segments. Where they are unequal, move value (aka local temperature) and count (aka mast value) are used to derive the walls from the scaffolds. Like the leaf counts, these counts are auxiliary and only used during the analysis of the initial position.

To determine the intrinsic count of a subposition, consider it as an initial position and evaluate it as such. No prisoners or new ko bans during analysis sequences yet.


Edited.

[RobertJasiek]

Thermograph for All Cases:

Black wall:

2/3 if T ≥ 1/3. Mast.

1 - T if T ≤ 1/3.

White wall:

2/3 if T ≥ 1/3. Mast.

1 - T if T ≤ 1/3.

Mast is defined as long as the walls coincide so this whole ko thermograph is called the mast, whose lower part inclines.

[Gérard TAILLE]

If a position occurs as subposition in an evaluation of another initial position, it does not mean that the evaluation as a subposition would be the same as its evaluation as an initial position! The root position, prisoners and ko bans matter!

To evaluate the other initial position, what we do need from this subposition is exactly and just the black scaffold.

<edit> Since we only need the subposition's black scaffold, we do not need its count as if it were the initial position. For the black scaffold, we also do not need an auxiliary count in this subposition. We need the on-board and prisoners count 3, which is the 3 in the term 3 - 3T, in the settled subposition, from which then we inherit the black scaffold of this unsettled subposition. </edit>

The leaves need their on-board and prisoners count.

When eventually we determine the initial position's thermograph, we can also infer its count (more precisely: its mast value, which for God's sake I call "count"; these additional CGT terms would only confuse us; Berlekamp and Spight have also used the word "count"; recall that the count depends on the assumption of no / a komaster and used model of the environment).

For intermediate subpositions, we need their scaffolds and / or walls. The walls might be equal or unequal to the scaffolds for different temperature ranges and the related trajectory segments. Where they are unequal, move value (aka local temperature) and count (aka mast value) are used to derive the walls from the scaffolds. Like the leaf counts, these counts are auxiliary and only used during the analysis of the initial position.

To determine the intrinsic count of a subposition, consider it as an initial position and evaluate it as such. No prisoners or new ko bans during analysis sequences yet.


Edited.

Now it seems correct to me.
As you know my approach is different : we start both with a tree and an evaluation of leaves but then you use black and white scafolds through the nodes while I work only on the nodes of the tree by handling a count and a value at each node. The advantage of your approach is that you use a pure recursive approach while my approach is basically recursive EXCEPT when I detect that a move increases the temperature (=> expected sente move).
Anyway I completly agree with the fact that a count for a node depends on the initial position. It is perfectly clear through my approach.
Are the following three points a good understanding of your approach ?
1) the count makes sense only on the leaves and the initial position
2) the value of moves make sense only on initial position
3) for intermediate nodes only black and white scafolds make sense.

Beside the "count" issue I see also an issue with the word "position". For intermediate nodes in the tree I prefer to simply use the word "node" instead of "position" because a node contains in my view some historical characteristics and in particular
1) the identity of the last move in case of a ko option
2) the information saying if the last two moves were an option ko followed by a tenuki
3) the number of prisoners
note 1 : concerning the root node I have no problem to call it a "position"
note 2 : concerning a leave node it seems acceptable to use the word "position" providing you put in the "position" wording the number of prisonners.

In your first post (viewtopic.php?p=276756#p276756) you define basic terms :
"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'"
but the basic word "position" is not defined while constantly used. Maybe you can improve your definitions on this point to avoid ambiguity and misunderstanding.

[Gérard TAILLE]

Thermograph for All Cases:

Black wall:

2/3 if T ≥ 1/3. Mast.

1 - T if T ≤ 1/3.

White wall:

2/3 if T ≥ 1/3. Mast.

1 - T if T ≤ 1/3.

are you sur that "black wall = white wall"

Mast is defined as long as the walls coincide so this whole ko thermograph is called the mast, whose lower part inclines.

[RobertJasiek]

Compare the stated black and white walls - they are equal!

It is not my method but Berlekamp's method according to Siegel's description. I am just one who currently applies it.

You mention your method but have you stated it somewhere? If alternative methods exist, fine. However, what is it and what is its (proven) scope? Generalised thermography is proven and its scope is simple loopy games (incl. non-ko games).

For intermediate subpositions, move value and count are often needed but you can call them auxiliary move value and count or, in CGT parliance, local temperature and mast value (for ko thermography, mean value would be the wrong term).

Node is a valid term in graph theory or informatics. Siegel uses the term subposition when he studies CGT. CGT explores many games, not just Go, so most CGT terms do not conform to go terms. Left = Black, Right = White etc. Go players use the term follow-up position and (as I have clarified) follower for an unsettled follow-up position. A leaf is a settled position and related to a CGT number. One can assign states to a node aka subposition aka follow-up position.

It is an exaggeration to say that my first post defined terms; rather, I have given some semi-formal descriptions of terms occurring in Siegel's attempted definition. I have not wanted to repeat Siegel's maths annotation but rather tried to explain it briefly for everybody having some thermographic background.

[Gérard TAILLE]

Compare the stated black and white walls - they are equal!

It is not my method but Berlekamp's method according to Siegel's description. I am just one who currently applies it.

You mention your method but have you stated it somewhere? If alternative methods exist, fine. However, what is it and what is its (proven) scope? Generalised thermography is proven and its scope is simple loopy games (incl. non-ko games).

For intermediate subpositions, move value and count are often needed but you can call them auxiliary move value and count or, in CGT parliance, local temperature and mast value (for ko thermography, mean value would be the wrong term).

Node is a valid term in graph theory or informatics. Siegel uses the term subposition when he studies CGT. CGT explores many games, not just Go, so most CGT terms do not conform to go terms. Left = Black, Right = White etc. Go players use the term follow-up position and (as I have clarified) follower for an unsettled follow-up position. A leaf is a settled position and related to a CGT number. One can assign states to a node aka subposition aka follow-up position.

It is an exaggeration to say that my first post defined terms; rather, I have given some semi-formal descriptions of terms occurring in Siegel's attempted definition. I have not wanted to repeat Siegel's maths annotation but rather tried to explain it briefly for everybody having some thermographic background.

The method I use is not really mine. It is simply a systematic application of tree manipulation you can find under https://senseis.xmp.net/?MiaiValuesList%2FDiscussion
Maybe what is new is the manipulation of a tree in case of a corridor. In this particular case I can reach the result rather quickly. Though it seems to work quite well I have some doubt with your 2-stage ko position.
See https://senseis.xmp.net/?MiaiValuesList%2F000To099#toc2 where you can find the following position

$$W
$$ -----------
$$ | O a . X .
$$ | . O X X X
$$ | O O . X .
$$ | . O . . .
$$ | . O . . .

Click Here To Show Diagram Code

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

It is said that a black move at "a" has a value 1/6 (assuming neither black nor white is komaster).
In my approach a black move at "a" is a weak move and for that the reason I have some doubt on my method.
With your knowledge of this 2-stage ko can you show me how this move can be valued 1/6 ? That way I will know my method is not correct (at least for such kind of positions)

[RobertJasiek]

The 1/6 or 1/9 ko is interesting but I have 30 stage-ko related positions whose evaluation is more urgent because they occur in practice.

The tree manipulation on that webpage

- uses some CGT methods

- was a predecessor mainly by Bill Spight to my method of making a hypothesis

- is somewhat dubios because shorter sequences are checked before longer sequences whether they are worth playing succesdively; making a hypothesis does it the other way round so overriding longer sequences are not overlooked

- is wrong when comparing the tentative initial move value to an alternating sequence's followers' move values; see my counter-example proving that the comparison is not coorect in general; instead, one must compare to the followers' gains, which is more natural anyway and I have proved that one need not compare to the subsequent move value after the traversal sequence to verify its stop

[Gérard TAILLE]

The 1/6 or 1/9 ko is interesting but I have 30 stage-ko related positions whose evaluation is more urgent because they occur in practice.

The tree manipulation on that webpage

- uses some CGT methods

- was a predecessor mainly by Bill Spight to my method of making a hypothesis

- is somewhat dubios because shorter sequences are checked before longer sequences whether they are worth playing succesdively; making a hypothesis does it the other way round so overriding longer sequences are not overlooked

- is wrong when comparing the tentative initial move value to an alternating sequence's followers' move values; see my counter-example proving that the comparison is not coorect in general; instead, one must compare to the followers' gains, which is more natural anyway and I have proved that one need not compare to the subsequent move value after the traversal sequence to verify its stop

$$W
$$ -----------
$$ | O a . X .
$$ | . O X X X
$$ | O O . X .
$$ | . O . . .
$$ | . O . . .

Click Here To Show Diagram Code

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

What'is wrong with my proof that black 1 at "a" is a weak move.
Let's assume the temperature is very small, for example t = 0.01

black to move

$$B
$$ -----------
$$ | O 1 2 X .
$$ | . O X X X
$$ | O O . X .
$$ | . O . . .
$$ | . O . . .

Click Here To Show Diagram Code

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

$$B :b3: tenuki
$$ -----------
$$ | O 4 O X .
$$ | . O X X X
$$ | O O . X .
$$ | . O . . .
$$ | . O . . .

Click Here To Show Diagram Code

[go]$$B :b3: tenuki
$$ -----------
$$ | O 4 O X .
$$ | . O X X X
$$ | O O . X .
$$ | . O . . .
$$ | . O . . .[/go]

count c1 = -2 + t

white to play

$$W :b2: tenuki
$$ -----------
$$ | O 1 . X .
$$ | . O X X X
$$ | O O . X .
$$ | . O . . .
$$ | . O . . .

Click Here To Show Diagram Code

[go]$$W :b2: tenuki
$$ -----------
$$ | O 1 . X .
$$ | . O X X X
$$ | O O . X .
$$ | . O . . .
$$ | . O . . .[/go]

count c2 = -1 + t
c1 < c2 => the move in the first diagram is always wrong (assuming no komaster).

[RobertJasiek]

I have not said that the older tree method would always fail. I have just said that one cannot be sure whether it produces the correct values. Maybe it works for this specific example; I have not read its discussion too carefully yet; I have postponed this until I will have studied the more basic corner kos.

Usually, Black a is not the kind of move one plays. However, under special circumstances, such as Black komonster, it can be a good move.

If neither is even komaster or similar circumstances occur, Black a is wrong indeed.

[RobertJasiek]

$$B
$$ -----------------
$$ | . . 1 O 2 . . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |

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.

$$B
$$ -----------------
$$ | . . . O . . . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |

Click Here To Show Diagram Code

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

If I have not goofed, the following is the evaluation by the method of making a hypothesis. If the 9x3 board is displayed incorrectly here, download and view in an SGF editor.

count = -2
move value = 1
gain of Black 1 = 1 1/6
gain of White 1 = 1 1/6

Now, please justify your temperature between 13/12 and 14/12 remark!

[Gérard TAILLE]

$$B
$$ -----------------
$$ | . . 1 O 2 . . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |

Click Here To Show Diagram Code

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

The exchange 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.

$$B
$$ -----------------
$$ | . . . O . . . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |

Click Here To Show Diagram Code

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

If I have not goofed, the following is the evaluation by the method of making a hypothesis. If the 9x3 board is displayed incorrectly here, download and view in an SGF editor.

count = -2
move value = 1
gain of Black 1 = 1 1/6
gain of White 1 = 1 1/6

Now, please justify your temperature between 13/12 and 14/12 remark!

I completely agree with your calculation Robert.

Concerning the value 13/12 and 14/12 I mentionned earlier, it is simply the result of a calculation when you take as hypothesis that the first move is gote for black.

$$B Black to play => value = -5/6 and M = 7/6
$$ -----------------
$$ | . . 1 O . . . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |

Click Here To Show Diagram Code

[go]$$B Black to play => value = -5/6 and M = 7/6
$$ -----------------
$$ | . . 1 O . . . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |[/go]

$$W White to play => value = -2
$$ -----------------
$$ | . . 1 O . . . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |

Click Here To Show Diagram Code

[go]$$W White to play => value = -2
$$ -----------------
$$ | . . 1 O . . . . . |
$$ | X X X X O O O O O |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |[/go]

IF the first move were gote then the value of temperature would be
t = (-5/6 - (-3)) / 2 = 13/12

Because 13/12 < 7/6 you know that black must play in the initial position if 13/12 < t < 14/12 and you know such black move is sente.
OC it is simply the result of the first phase of analysis and you have to go further in order to have more information.
IOW I gave only the result of the first phase of analyse : it is correct but still incomplete.

More on my method will follow

[Gérard TAILLE]

Code: Select all

My method applies only on "generalized" corridor

               A
              / \
             /   \
            B     C
                 / \
                /   \
               D     E
                    / \
                   /   \
                  F     G
                       / \
                      /   \
                     H     I

Here above is an example of tree corresponding to a corridor.
Nodes A, C, E, G ... are the nodes reached by black when she follows the corridor and white chooses tenuki
Nodes B, D, F, H ... are leaves reached by white when she decides to stop black in the corridor.
I use the word "generalized" corridor because I accept some situation quite similar to common corridor; you will see later what it means really

The most obvious case is the simple gote (I can say a degenerate cooridor ?)

               A
              / \
             /   \
            B     C

let's call "a" the value of node A, "b" the value of node B, etc.
This simple corridor is easy to analyse: because there are two tenuki between B and C, the value of a move from A is (c - b)/2



               A
              / \
             /   \
            B     C
                   \
                    \
                     F

This other simple tree is OC also an acceptable case for me (but not really a common corridor)
You recognize easily a tree corresponding to a ko.
This time we can see three tenuki between B and F, the value of a move from A is (f - b) / 3

With the following trees we have really a corridor:

               A
              / \
             /   \
            B     C
                 / \
                /   \
               D     E

This very important tree will show you the basis of my method:
The point is to know whether of not the temperature increases when the move AC is played. If the temperature increases then the move AC is sente and the move AB is reverse sente. If temperature decreases then AC and AB moves are gote. If temperature does not change then A is ambiguous (you can consider AC as gote or sente but AB is OC always gote).
First step of my method : I assume AC is sente => AB is a reverse sente move => ABvalue = d - b
Second step of my method : between B and E there are three tenuki => an ambiguous situation will occur if 3 * ABvalue = e - b
if 3 * ABvalue < e - b then AC is really sente otherwise the previous assumption is false and I have no other choice than analysing node C.

At this point let's take a (famous) exemple:

$$B
$$ -----------
$$ | . O X X .
$$ | O X X . X
$$ | . O X X X
$$ | O O O O X
$$ | . O . O .

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]

Code: Select all

The corresponding tree as the following form

               A
              / \
             /   \
            B     C
                 / \
                /   \
               D     E
                      \
                       \
                        G

with the following value for the leaves

               A
              / \
             /   \
           1/3    C
                 / \
                /   \
              2/3    E
                      \
                       \
                        3

1) first step : assuming AC sente then ABvalue = 2/3 - 1/3 = 1/3 in reverse sente
2) second step : between B and G there are 4 tenuki => A is ambiguous if 4 * ABvalue = f -b
Obviously we have 4 * 1/3 < 3 - 1/3 => I know for sure that AC is sente and ABvalue = 1/3 in reverse sente.

If now you want to know exactly when black can play AC in sente then you have to analyse the subtree:

                  C
                 / \
                /   \
              2/3    E
                      \
                       \
                        3

and the value of a move here is obviously (3 - 2/3) / 3 = 7/9

You have the all picture:
AB is a reverse sente move value 1/3
AC is a sente move value 1/3 and you can play this move as soon as temperature drops to 7/9.
in my method this position is clearly part of the simpliest corridors.

[RobertJasiek]

The convention is: Black moves leftwards in trees.

You claim a method for corridor-like trees. What kinds of sequences and local endgame types can exist in them?

Is your method invariant under the order of assumptions of sequences worth playing successively and types?

You characterise an AB move as having the move value 1/3 and the move value 7/9. How do you prefer one value to the other?

In the example, you consider the pruned tree.

[Gérard TAILLE]

To answer in detail to your questions will need dozens of pages and I have no time enough to make such job (its on my to do list but not in a near future).
Anyway its quite clear in mind so please if you have some doubt on the process take an example and I will be able to bring you some more information.

The convention is: Black moves leftwards in trees.

OK it will be easy for me to change that according to the usual convention

You claim a method for corridor-like trees. What kinds of sequences and local endgame types can exist in them?

One important point concerning the type of tree I consider is the following : at each node of the tree I consider that the player to play as only one good move. If it exists another move which I cannot characerize as weak then this tree is out of the scope of my method.

Is your method invariant under the order of assumptions of sequences worth playing successively and types?

Take an example because the question is really very large indeed.

You characterise an AB move as having the move value 1/3 and the move value 7/9. How do you prefer one value to the other?

Yes, in the process, the AB value (as well as the AC value) could be 1/3 or 7/9 but later in the process, when I found that AC is sente, then it remains only the value 1/3 for AB. For AC it is a sente move (value 1/3) that can be played as soon as temperature drops to 7/9.

In the example, you consider the pruned tree.

Yes you can see two pruning situations:
1) I eliminate sure weak moves
2) when a simple ko with value 1/3 is reached I prune systematically the subtree of ko, taking the standard count (for a larger ko, I mean a value greater than 1/3 the pruning is not allowed).

[RobertJasiek]

the AB value (as well as the AC value) could be 1/3 or 7/9 but later in the process, when I found that AC is sente, then it remains only the value 1/3 for AB. For AC it is a sente move (value 1/3) that can be played as soon as temperature drops to 7/9.

(So far) your method does not consider ambient temperature so, in terms of your theory, you cannot make a statement like "can be played as soon as temperature drops to 7/9".

However, your analysis generates 7/9 as move value of C, gain of AC, gain of CD, gain of CE and gain of EG. Therefore, you can say that White can play AB with the move value 1/3 and gain 1/3 while Black can play AC with the move value 1/3 and gain 7/9. Not considering preserving ko threats, this is a neat internal characterisation at least for this particular initial position with its pruned tree.