OC but assume I impose these specific black moves (even if they are not the best one). My question concerns ONLY white moves : are these white moves the best one?

1 and 5 not necessarily. If there is an important ko, 1 or 5 at d might better as negative threat. 3 at 4 I have not checked carefully.

(I assume you assume that White cannot invade the black territory.)

What is this position (Francisco Criado, I, Bill Spight)?
If it is the following one I do not see both a black sente option and a white sente option.
[img][/img]

The (initial position of the) general ordinary game tree of depth 2.



In your image, you write three names but we are not the authors of its tree! You are so write your name instead!

As to the contents of the tree, I do not solve all your problems. Analyse your own tree, and then we can discuss your analysis.

According to your post https://lifein19x19.com/viewtopic.php?p=277835#p277835 this tree was created by Francisco Criado and I understood you produced great and surely interested work on this specific position, in cooperation with Bill Spight. For that reason I associated your three names to this position. I mentionned this position only because the sente options and gote options are here quite difficult to handle. I am quite surprised you do not remember this interesting position.

Coming back to the tree I showed in my post https://lifein19x19.com/viewtopic.php?p=280548#p280548 you answer was that the position is not a double sente. I agree with you but that was not my question. I never suggest this position could be a double sente position.
My question was the following: according to your theory are black b and white d both sente options?

Click Here To Show Diagram Code

[go]$$W
$$ ----------------------------
$$ . . . . . . . 6 2 5 . . . |
$$ . . . . . . . 4 3 O O d . |
$$ . . . . . . . X O c X O . |
$$ . . X , . X . X X X . O . |
$$ . . . . . . . . . . O . . |
$$ . . . . . . . . . . O . . |
$$ . . . . . . . . . . . . . |
$$ . . . . . . . . . . . . . |
$$ . . . . . . . . . . O . . |
$$ . . . , . . . . . , . . . |[/go]

This time I agree with you.
If now we ignore the value of a ko threat created locally (I do not know if a theory exists on this subject) then that means that the and are the best answers to black moves (on contrary these black moves may not be necessarily the best black moves!).

I see. There has been my misunderstanding of what tree you referred to. I thought you meant the tree of the theorem of non-existence of local double sente. Now I understand you mean Francisco's example related to making a hypothesis. This example was in text annotation instead of graphical tree representation so I have not recognised it. Now I need to check whether indeed it is. Its analysis is quite difficult so I need to reread its study carefully before I can confirm / reject your specific question about it.

#63

Robert, how apply your theorems for these positions?

Click Here To Show Diagram Code

[go]$$ Black to move. a = -23; b = -22
$$ +---------------------------------+
$$ | O . X X . . b . X . X X X X X X |
$$ | O O . O X . O a O X X X . X O . |
$$ | . O O O . O O X O X . X , X O X |
$$ | . O . O O O . X O X . X X X O O |
$$ | . O X X X O X X O X . X X O O O |
$$ | . O X X . O O O O X . X X O . O |
$$ | O O O O X X X X X X X X X O O . |
$$ +---------------------------------+[/go]

Click Here To Show Diagram Code

[go]$$ Black to move. a = -23; b = -24
$$ +---------------------------------+
$$ | O . X X . . b . X . X X X X X X |
$$ | O O . O X . O a O X X X . X O . |
$$ | . O O O . O O X O X . X , X O X |
$$ | . O . O O O . X O X . X X X O X |
$$ | . O X X X O X X O X . X X O O O |
$$ | . O X X . O O O O X . X X O . O |
$$ | O O O O X X X X X X X X X O O . |
$$ +---------------------------------+[/go]

It's funny that T2 and T4 temperatures are the same, but we get different best moves.



Are T2 and T4 Temperature Regions?

Please clarify: do you mean a late endgame or early endgame position?



While I use temperature for the environment's ambient temperature, I am unsure about your intended meaning of temperature here. Ambient, local or global temperature? Or move value? If you mean the local temperatures of either sum of local endgames, I do not want to apply the definition of "local temperature" with its cooling and infinitesimals.



I have not tried to understand 11/2 yet but as to your question if you mean the ambient temperature T: it is of only the largest region C = {0|-11} and is T = (0 - (-11)) / 2 = 5 1/2 calculated as the gote move value of C.

Either D2 or D4 provides either 1 or 2 as another move value of the environment.

Your attached image in https://www.lifein19x19.com/viewtopic.p ... 71#p280571

has the wrong caption "Francisco Criado, Robert Jasiek, Bill Spight". This example in [22] has been invented by Francisco Criado and analysed by him and me. Bill has nothing to do with it. It is counter-example 1 for making a hypothesis and assumes a rich environment. Citation from [22]:

"At the high ambient temperature T = 4.5, [...] White prefers to move from B to D. At the low temperature T = 1.5 [...] White prefers to move from B to C and Black continues locally [...] For an arbitrary ambient temperature T, we must compare C and D as positions
and must not just compare resulting scores. The choice between C and D cannot be simplified with dominance but depends on the environment."

An analysis of the example requires consideration of the enriched score as in definitions 8 in [22]:

"If Black / White starts, the resulting score of P at temperature T is:
B_T(P) := B(P + Ε_T) - T/2,
W_T(P) := W(P + Ε_T) + T/2."

Here, E_T is a rich environment. See [22] or the CGT literature on details! I do not repeat the entire [22] here.

Oops it seems you missed the point Robert. Your analyse above concerns the subtree from B but this analyse do not really imply any problem.
The difficulty of the position is due to the root A where white can reach a position with only a count -4 which is very disturbing.
Taking your example of temperature:
if T = 4.5 and you assume the sequence AB, BD then black reachs in sente position D with count -5 which is very bad indeed.
if T = 1.5 and you assume the sequence AB, BC followed by a black move then black reachs in gote a position with count -2 and here again this sequence is bad for black.
Robert you have to analyse the position as it was created by Criado i.e. from the root A. Analysing just the subtree from B is not the point.

Sure, as I have done in [22]. It does involve application of CGT techniques to actually calculate the enriched scores, similar to doing it for ideal environments. The T/2 approximation, you know.

Basically your question concerns the impact of the environment on a local position.

Let's take a simplier position to understand the point.

Click Here To Show Diagram Code

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

Click Here To Show Diagram Code

[go]$$W white gote hane : count -4
$$ -----------------------
$$ . . . X . 1 . . . O . |
$$ . . . X X X O O . O O |
$$ . . . . . X X X O O X |
$$ . . . . . . . X X X X |
$$ . . . . . . . . . . . |
$$ . . . . . . . . . . . |[/go]

Click Here To Show Diagram Code

[go]$$B black gote hane : count = -2
$$ -----------------------
$$ . . . X . 3 1 2 . O . |
$$ . . . X X X O O . O O |
$$ . . . . . X X X O O X |
$$ . . . . . . . X X X X |
$$ . . . . . . . . . . . |
$$ . . . . . . . . . . . |[/go]

Click Here To Show Diagram Code

[go]$$B Black sente sagari : count = -3
$$ -----------------------
$$ . . . X . 1 2 . . O . |
$$ . . . X X X O O . O O |
$$ . . . . . X X X O O X |
$$ . . . . . . . X X X X |
$$ . . . . . . . . . . . |
$$ . . . . . . . . . . . |[/go]

What is the best move for black ? black gote hane or black sente sagari?

Let's consider two ideal environments at temperature T = 0.96
environment 1 : 0.32, 0.64, 0.96
environment 2 : 0.24, 0.48, 0.72, 0.96

You can verify the following results:
1) with environment 1 the black best move is to play the sente sagari
2) with environment 2 the black best move is to play the gote hane

As you see the best move depends on the details of the environment; even two ideal environments at the same temperature can give different results.

For that reason you can understand at least my two following points:
1) I do not like very much ideal environments; I prefer a rich environment to get only one result.
2) I do not try to get an accurate result. Typically, instead of given an accurate value like 4 1/3 I prefer 4.33 or even 4.3 if not simply 4+ or 4 ! That way the operations on counts are for me far simplier.

Dany, concerning your example, I think you can easily understand why you can reach different results with two different environments.

Nice!



Good humour!



Both have their merits as has the third tool: the alternating sum of the actual environment of simple gotes on the board.

Since you have not clarified yet, let me guess what you might mean. I think that probably you mean:

- late endgame

- you do not know how to use the word temperature well, wherefore I reply with how I use it: the ambient temperature, that is, the largest move value in the environment

During the late endgame the creator and preventer have different theorems. With Black to move, only the theorems for him as the creator might apply. However, for a local endgame with gote and sente options and the late endgame, the theorems of Bill Spight and me are inapplicable because they presume single plays a) in the gote option, b) in the opponent's sequence, c) for the initial sente option's play and d) for the alternating reply in the sente option. If we try application of theorems nevertheless, it is luck whether they suggest the right answers. (For local endgames without options, the related theorems are more tolerant to long sequences.)

There are theorems for move values and theorems for counts. For a local endgame with gote and sente options and the late endgame, the theorems for move values must not be applied here. The theorems for counts have a greater chance of correctness despite long sequences especially those started by the creator because the theorems prescribe the beginnings of different sequences in the definitions of the counts to be considered. For the creator, we need definition 37,

"Let there by the resulting counts C1 if the creator starts in the environment and the preventer replies locally, C2 if the creator starts locally with the gote option, C3 if the creator starts locally with the sente option." [22],

the remark

"Determination of the correct first move is the only purpose of the test sequences resulting in these counts" [22]

and theorem 128 for any (low or high) temperature,

"The creator starts
- in the environment if C1 ≥ C2, C3,
- locally with the gote option if C2 ≥ C1, C3,
- locally with the sente option if C3 ≥ C1, C2." [22]

Bill Spight suggested the conceptual idea for such a theorem, which I created and proved.

Now, let us try whether we are lucky and the theorem produces the right answer despite long sequences and any remaining basic endgame ko. I end the sequences before playing it out if it occurs and assign its count -1/3 then.



Your first example:

Click Here To Show Diagram Code

[go]$$B Black to move
$$ +---------------------------------+
$$ | O . X X . . . . X . X X X X X X |
$$ | O O . O X . O . O X X X . X O . |
$$ | . O O O . O O X O X . X , X O X |
$$ | . O . O O O . X O X . X X X O O |
$$ | . O X X X O X X O X . X X O O O |
$$ | . O X X . O O O O X . X X O . O |
$$ | O O O O X X X X X X X X X O O . |
$$ +---------------------------------+[/go]

Click Here To Show Diagram Code

[go]$$B start in the environment, C1 = -23 1/3
$$ +---------------------------------+
$$ | O C B B C C C 2 X . X X X X X X |
$$ | O O C O B C O C O X X X . X O 3 |
$$ | . O O O C O O B O X . X , X O X |
$$ | . O . O O O C B O X . X X X O O |
$$ | . O X X X O B B O X . X X O O O |
$$ | . O X X 1 O O O O X . X X O . O |
$$ | O O O O X X X X X X X X X O O . |
$$ +---------------------------------+[/go]

Click Here To Show Diagram Code

[go]$$B local start with the gote option, C2 = -23
$$ +---------------------------------+
$$ | O C B B C C 2 3 X . X X X X X X |
$$ | O O C O B C O 1 O X X X . X O 5 |
$$ | . O O O C O O X O X . X , X O X |
$$ | . O C O O O . X O X . X X X O O |
$$ | . O B B B O X X O X . X X O O O |
$$ | . O B B 4 O O O O X . X X O . O |
$$ | O O O O X X X X X X X X X O O . |
$$ +---------------------------------+[/go]

Click Here To Show Diagram Code

[go]$$B local start with the sente option, correct choice on move 4, C3 = -22
$$ +---------------------------------+
$$ | O . X X . 5 1 3 X . X X X X X X |
$$ | O O . O X . O 2 O X X X . X O 6 |
$$ | . O O O . O O B O X . X , X O B |
$$ | . O C O O O C B O X . X X X O O |
$$ | . O B B B O B B O X . X X O O O |
$$ | . O B B 4 O O O O X . X X O . O |
$$ | O O O O X X X X X X X X X O O . |
$$ +---------------------------------+[/go]

The theorem correctly suggests that the creator starts locally with the sente option if C3 ≥ C1, C2 <=> -22 ≥ -23 1/3, -23:

Click Here To Show Diagram Code

[go]$$B Black's suggested correct start according to the theorem
$$ +---------------------------------+
$$ | O . X X . . 1 3 X . X X X X X X |
$$ | O O . O X . O 2 O X X X . X O . |
$$ | . O O O . O O X O X . X , X O X |
$$ | . O . O O O . X O X . X X X O O |
$$ | . O X X X O X X O X . X X O O O |
$$ | . O X X . O O O O X . X X O . O |
$$ | O O O O X X X X X X X X X O O . |
$$ +---------------------------------+[/go]

Unlike the theorem for only the first move, I interpret its informal application for the first three moves.


Your second example:

Click Here To Show Diagram Code

[go]$$B Black to move
$$ +---------------------------------+
$$ | O . X X . . . . X . X X X X X X |
$$ | O O . O X . O . O X X X . X O . |
$$ | . O O O . O O X O X . X , X O X |
$$ | . O . O O O . X O X . X X X O X |
$$ | . O X X X O X X O X . X X O O O |
$$ | . O X X . O O O O X . X X O . O |
$$ | O O O O X X X X X X X X X O O . |
$$ +---------------------------------+[/go]

Click Here To Show Diagram Code

[go]$$B start in the environment, C1 = -23 1/3
$$ +---------------------------------+
$$ | O C B B C C C 2 X . X X X X X X |
$$ | O O C O B C O C O X X X . X O 3 |
$$ | . O O O C O O B O X . X , X O X |
$$ | . O . O O O C B O X . X X X O X |
$$ | . O X X X O B B O X . X X O O O |
$$ | . O X X 1 O O O O X . X X O . O |
$$ | O O O O X X X X X X X X X O O . |
$$ +---------------------------------+[/go]

Click Here To Show Diagram Code

[go]$$B local start with the gote option, C2 = -23
$$ +---------------------------------+
$$ | O C B B C C 2 3 X . X X X X X X |
$$ | O O C O B C O 1 O X X X . X O 5 |
$$ | . O O O C O O X O X . X , X O X |
$$ | . O C O O O . X O X . X X X O X |
$$ | . O B B B O X X O X . X X O O O |
$$ | . O B B 4 O O O O X . X X O . O |
$$ | O O O O X X X X X X X X X O O . |
$$ +---------------------------------+[/go]

Click Here To Show Diagram Code

[go]$$B local start with the sente option, correct choice on move 4, C3 = -24
$$ +---------------------------------+
$$ | O . X X . 5 1 3 X . X X X X X X |
$$ | O O . O X . O 2 O X X X . X O 6 |
$$ | . O O O . O O B O X . X , X O B |
$$ | . O C O O O C B O X . X X X O B |
$$ | . O B B B O B B O X . X X O O O |
$$ | . O B B 4 O O O O X . X X O . O |
$$ | O O O O X X X X X X X X X O O . |
$$ +---------------------------------+[/go]

The theorem correctly suggests that the creator starts locally with the gote option if C2 ≥ C1, C3 <=> -23 ≥ -23 1/3, -24:

Click Here To Show Diagram Code

[go]$$B Black's suggested correct start according to the theorem
$$ +---------------------------------+
$$ | O . X X . . 2 3 X . X X X X X X |
$$ | O O . O X . O 1 O X X X . X O . |
$$ | . O O O . O O X O X . X , X O X |
$$ | . O . O O O . X O X . X X X O X |
$$ | . O X X X O X X O X . X X O O O |
$$ | . O X X . O O O O X . X X O . O |
$$ | O O O O X X X X X X X X X O O . |
$$ +---------------------------------+[/go]

Unlike the theorem for only the first move, I interpret its informal application for the first three moves.

Note that your only two counts per example in both examples overlooks the possibility of Black's start in the environment. My theorem also takes it into account.

Furthermore, note how we have been lucky with the theorem because accidentally move 3 of the gote option is correct. The definition and theorem do not consider branching at that moment. This is the danger of long sequences.


EDIT: added last paragraph.

You claimed that a local double sente position cannot exist, providing both player have only simple follow-ups. To prove that a local double sente position cannot exist you must first define what you would call local double sente position. What is this defintion for you? Is it sufficient to say that both players has a sente option or do you need other conditions in order to say that a position is a local double sente?

Thank you for the detailed and interesting answer.



Yes, late endgame. But what is "late endgame"?

My temperature is CGT temperature.



I don't understand what the point of the theorem is. I don't see what prediction the theorem makes.

I calculated the count for each move to the end of the game. Then I chose the move that led to the best result. I don't see what the theorem gives new. It repeats the same calculations that I did (C1, C2, C3). And to choose the best result (and first move) among C1, C2, C3, the theorem is not needed.

I clearly don't understand something. Maybe the game is still going on after "test sequences" and C1, C2, C3 not the final results?

"Presuppositions

We score due to a ruleset.
Let BB, BW, WB, WW ∈ ℤ,
G := {BB|BW||WB|WW},
B := (BB + BW) / 2, W := (WB + WW) / 2,
MGOTE := (B - W) / 2,
MB,SENTE := BW - W,
MW,SENTE := B - WB.
Suppose BB > BW, WB > WW, G is without reversible plays.
Suppose an environment without ko now or later.

Theorem 20 [non-existence of a local double sente]

G with MGOTE > MB,SENTE, MW,SENTE does not exist." [22]

Proof see [22]. Definition of reversible see [22] or the CGT literature.

https://www.lifein19x19.com/viewtopic.p ... 45#p143245

We can, in principle, solve the position by reading and counting because the necessary amount of time is not too large.



No, because CGT knows (at least) three different kinds of temperatures: local, global, ambient. You need to specify which if you want to be understood. And beware: CGT definitions are nasty because they rely on cooling and infinitesimals.



The theorem takes as input a) an arbitrary local endgame tree with one player's gote option and sente option, single play sequences, and arbitrary resulting counts, and b) an arbitrary environment of arbitrarily many simple gotes without follow-ups with arbitary move values. The theorem always says which first move is correct. Hence, it solves an infinite number of such example positions.



What you (pretend to) do is the method of reading and counting applied to one particular example. What the theorem does is

- an acceleration of that method because only three (not more) sequences and counts are considered while a careful application of only the method needs more sequences, counts and decisions,

- the generalisation to all eligible example positions,

- the generalisation independent of whether the temperature is low or high,

- the clarification that the theorem's value comparisons apply to the creator (while a different theorem with other value comparisons applies to the preventer),

- the abstraction of either Black and favouring larger values, or White and favouring smaller values, as one unified player (the creator),

- together with related theorems using move values, this theorem using counts enables further insight including that for the early endgame.



You do not appreciate that fundamental maths research is sometimes as boring as explaining again but formally what is known to some informally. Such maths confirms what some of us already suspected to be right and paves the way towards advanced insight about previously unknown things, such as my theorem for the early endgame with its insight that the gote move value is sufficient and sente option must be ignored at high temperature. Before that theorem, this was unknown or at best suspected by some strong players. "Boring" fundamental maths enables powerful advanced maths and can do so very quickly while centuries of informal go theory overlooked certain things.

In this defintion I see an important condition : "G is without reversible plays".
Could you tell us if this condition (no reversible plays) exists also in the defintion of a sente OPTION?