Gote move vs sente move in yose

[Gérard TAILLE]

"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."

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.

[RobertJasiek]

The difficulty of the position is due to the root A [...] you have to analyse the position as it was created by Criado i.e. from the root A.

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.

[Gérard TAILLE]

#63

Robert, how apply your theorems for these positions? ...

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

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

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

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]

$$W white gote hane : count -4 $$ ----------------------- $$ . . . X . 1 . . . O . | $$ . . . X X X O O . O O | $$ . . . . . X X X O O X | $$ . . . . . . . X X X X | $$ . . . . . . . . . . . | $$ . . . . . . . . . . . |

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]

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

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]

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

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.

[RobertJasiek]

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

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]

Nice!

environment 1 : 0.32, 0.64, 0.96
environment 2 : 0.24, 0.48, 0.72, 0.96

Good humour!

1) I do not like very much ideal environments; I prefer a rich environment to get only one result.

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

[RobertJasiek]

#63

Robert, how apply your theorems for these positions?

$$ Black to move. a = -23; b = -22 $$ +---------------------------------+ $$ | O . X X . . b . X . X X X X X X | $$ | 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 | $$ | O O O O X X X X X X X X X O O . | $$ +---------------------------------+

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]

$$ Black to move. a = -23; b = -24 $$ +---------------------------------+ $$ | O . X X . . b . X . X X X X X X | $$ | 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 | $$ | O O O O X X X X X X X X X O O . | $$ +---------------------------------+

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]

Code: Select all

C:= {0|-11}; D2:= {0|-2}; T2:= C + D2; D4:= {0|-4}; T4:= C + D4

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

Code: Select all

> T2.Temperature
11/2
> T4.Temperature
11/2

Are T2 and T4 Temperature Regions?

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:

$$B Black to move $$ +---------------------------------+ $$ | O . X X . . . . X . X X X X X X | $$ | 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 | $$ | O O O O X X X X X X X X X O O . | $$ +---------------------------------+

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]

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

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]

$$B local start with the gote option, C2 = -23 $$ +---------------------------------+ $$ | O C B B C C 2 3 X . X X X X X X | $$ | 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 | $$ | O O O O X X X X X X X X X O O . | $$ +---------------------------------+

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]

$$B local start with the sente option, correct choice on move 4, C3 = -22 $$ +---------------------------------+ $$ | O . X X . 5 1 3 X . X X X X X X | $$ | 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 | $$ | O O O O X X X X X X X X X O O . | $$ +---------------------------------+

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:

$$B Black's suggested correct start according to the theorem $$ +---------------------------------+ $$ | O . X X . . 1 3 X . X X X X X X | $$ | 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 | $$ | O O O O X X X X X X X X X O O . | $$ +---------------------------------+

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:

$$B Black to move $$ +---------------------------------+ $$ | O . X X . . . . X . X X X X X X | $$ | 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 | $$ | O O O O X X X X X X X X X O O . | $$ +---------------------------------+

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]

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

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]

$$B local start with the gote option, C2 = -23 $$ +---------------------------------+ $$ | O C B B C C 2 3 X . X X X X X X | $$ | 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 | $$ | O O O O X X X X X X X X X O O . | $$ +---------------------------------+

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]

$$B local start with the sente option, correct choice on move 4, C3 = -24 $$ +---------------------------------+ $$ | O . X X . 5 1 3 X . X X X X X X | $$ | 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 | $$ | O O O O X X X X X X X X X O O . | $$ +---------------------------------+

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:

$$B Black's suggested correct start according to the theorem $$ +---------------------------------+ $$ | O . X X . . 2 3 X . X X X X X X | $$ | 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 | $$ | O O O O X X X X X X X X X O O . | $$ +---------------------------------+

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.

[Gérard TAILLE]

The theorem and proof of non-existence of local double sente are for a local endgame with both players' simple follow-ups, i.e., a tree of depth 2 with 4 leaves. In the theory for a local endgame with gote and sente options, the reverse sente is a single move, wherefore a local double sente cannot exist. For practical purposes, however, we must consider longer sequences so that then a local double sente is a principal possibilty because we have no proof for that. If you (or anybody) know a double sente example with gote and sente options, show it!

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?

[dany]

Thank you for the detailed and interesting answer.

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

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

My temperature is CGT temperature.

"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]

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?

[RobertJasiek]

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 [...] first define [...] local double sente position.

"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

[RobertJasiek]

what is "late endgame"?

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

My temperature is CGT temperature.

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.

"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]

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

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.

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.

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.

I clearly don't understand something.

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.

[Gérard TAILLE]

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 [...] first define [...] local double sente position.

"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

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?

[RobertJasiek]

For most of the other theorems, such as related to gote and sente options, there is no requirement of no dominated and no reversible plays. That is because the proofs do not need such presuppositions. The double sente theorem needs the no reversible presupposition because it enables the proof for a more general values range also covering, as you have noticed, sekis and the like.

Mostly the theory of Bill Spight and me avoids CGT low level things, such as infinitesimals or preliminary tree simplifications. Usually, our theory is more accessible. Only sometimes we need CGT techniques.

[RobertJasiek]

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]

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

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.

Let me elaborate a bit more on this. At first glance, the theorem looks as if it was written without work because I have designed it as close to go players' thinking as possible. To achieve this, Bill had to work hard on the first theorem on local endgames with gote and sente options and then I did mathematical research on them for months writing and proving more fundamental theorems or preliminary propositions.

This starts, in particular, with the local move values MGOTE and MSENTE and the alternating sum ∆T of the environment's move values.

For every move sequence, one gets a term similar to M + C + ∆T, in which move values and a count are added. This required me to prove fundamental theorems for the validity of such a sum of move values and counts.

One compares such terms of every two move sequences while considering one player maximising and the opponent minimising. The purpose is to retrieve value comparisons in theorems, such as T ? MGOTE or 2∆T ? MSENTE for some appropriate relation '?'.

Other theorems establish equivalence of a) principles using move values and b) resulting counts of move sequences. (For other kinds of local endgames, there is also equivalence to using gains.)

Ω is an alternating sum for the tails of move sequences. ∆T|F is the alternating sum of the move values in the environment and the local follow-up move value F. T1 is the second-largest move value of the environment. The following term occurs in several propositions and proofs: ∆T1|F - ∆T1. Now, it was very useful for me to prove ∆T1|F - ∆T1 = Ω if the temperature is high. In a citation of the related proof, it is easier to see how work was done when writing out such in detail

"∆T1|F - ∆T1 = (T1 - T2 +...- TL + F - TL+1 + TL+2 -...) - (T1 - T2 +...- TL + TL+1 - TL+2 +...)"

then transforming it to Ω. This is for even L and a similar representation occurs for odd L.

Only as one of the last steps, the resulting counts C1, C2 and C3 are identified with the terms of move values for the move sequences. Before, several pages of detailed mathematical work on all the move values must be done.

While the theorem looks like copy and paste of an informal method, I established it by proper, detailed mathematical proving. That's why it is a theorem and not informal.


EDIT: high temperature

[dany]

What the theorem does is

"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]

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.

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,

And what are the other sequences besides C1, C2, C3? I see only one sequence family C4 - if the creator starts in the environment and the preventer replies in the environment.

[RobertJasiek]

And what are the other sequences besides C1, C2, C3? I see only one sequence family C4 - if the creator starts in the environment and the preventer replies in the environment.

Yours is another sequence family indeed and Bill found its treatment in the proofs.

We as players and my maths rely on taking simple gotes without follow-ups in order of decreasing-or-constant move values so this rules out most sequences indeed. However, even so it may not be obvious that the remaining follow-up with move value F shall abide by the same assumption. In my proofs, I treat this meticulously so that, eventually, the sente option only needs one sequence with its one resulting count instead of two sequences with two parameters for counts.

[Gérard TAILLE]

assuming simply T = MGOTE, I am asking for only one example for which:
1) one player has a gote and a sente option
2) F < T
That way I would know that this theorem makes sense.

Only showing T but not the remaining part of the environment, my example in [22], [23] has a remaining basic endgame ko so only |T - MGOTE| = 1/6:

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

Click Here To Show Diagram Code

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

Hint: T - F = 1/2 > 0, that is, the temperature is high, as we desire. You can easily modify this example from 2021 to emphasise a start either locally or in the environment. I do not recall exactly how long I needed to construct this example but I had invented this example shape class a few years earlier so I think it took me at most one hour to tweak the values of T and MGOTE close enough maintaining a high temperature. Therefore, you should have been able to find such an example yourself. For gote and sente options, I worked mostly with this shape class so, unlike for quite a few of other theorems, I do not already have exact equality, here T = MGOTE. If we should be lucky, shrinking the "42" example would do the trick - otherwise put it on a huge board with a suitable T to allow for an almost ideal environment with drop 1/2 or 1.

Let me come back to a previous question. Assuming simply T = MGOTE, I was asking for only one example for which:
1) one player has a gote and a sente option
2) F < T

You answered with a position in which |T - MGOTE| = 1/6. Robert, sure this position is interesting but this position has nothing to do with my question because I assumed T = MGOTE.

Seeing you did not find any other position (or tree) I conclude that we cannot have simultaneously
1) T = MGOTE
2) One player has a gote and a sente option
3) F < T
That means that your theorem cannot be applied if T = MGOTE