Robert just two questions to the go player (not the theorician)

Click Here To Show Diagram Code

[go]$$W
$$ ----------------------------
$$ . . . . . . 2 1 3 7 . . . |
$$ . . . . . . 8 4 6 5 O d . |
$$ . . . . . . . X O c X O . |
$$ . . X , . X . X X X . O . |
$$ . . . . . . . . . . O . . |
$$ . . . . . . . . . . O . . |
$$ . . . . . . . . . . . . . |
$$ . . . . . . . . . . . . . |
$$ . . . . . . . . . . O . . |
$$ . . . , . . . . . , . . . |[/go]

Q1 : after white 1 do you agree it is natural to assume that black cannot expect more than the result above (followed later by the exchange black "c" white "d")?

Click Here To Show Diagram Code

[go]$$W
$$ ----------------------------
$$ . . . . . . . 6 2 5 . . . |
$$ . . . . . . . 4 3 1 O d . |
$$ . . . . . . . X O c X O . |
$$ . . X , . X . X X X . O . |
$$ . . . . . . . . . . O . . |
$$ . . . . . . . . . . O . . |
$$ . . . . . . . . . . . . . |
$$ . . . . . . . . . . . . . |
$$ . . . . . . . . . . O . . |
$$ . . . , . . . . . , . . . |[/go]

Q2: after white 1 do you agree it is natural to assume that white cannot expect more than the result above, (followed later by the exchange black "c" white "d")?

Yes. (I would not call anything "natural" though.)



Uh, no. I might agree on such after the exchange 1 - 2. However, Black 2 might be a tenuki, of course.

Initial Position

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[go]$$B initial position, Black to move
$$ ---------------------------
$$ | O . X X . . . . X . X X .
$$ | 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]

Main Sequences

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[go]$$B gote option, H = -12
$$ ---------------------------
$$ | O C B B C C 2 3 X . X X .
$$ | O O C O B C O 1 O X X X .
$$ | . O O O C O O X O X . X .
$$ | . 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]

Click Here To Show Diagram Code

[go]$$B sente option, S = -19
$$ ---------------------------
$$ | O C B B C 4 1 3 X 5 X X .
$$ | O O C O B 6 O 2 O X X X .
$$ | . O O O C O O B O X . X .
$$ | . O . O O O C B O X . X .
$$ | . O X X X O B B O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$W White starts, R = -23 1/3
$$ ---------------------------
$$ | O C B B C C C 1 X . X X .
$$ | O O C O B C O C O X X X .
$$ | . O O O C O O B O X . X .
$$ | . O . O O O C B O X . X .
$$ | . O X X X O B B O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

Gote option's gote move value MGOTE = (H - R) / 2 = (-12 - (-23 1/3)) / 2 = (11 1/3) / 2 = 5 2/3.

Sente option's tentative sente move value MSENTE = S - R = -19 - (-23 1/3) = 4 1/3.


Follow-up

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[go]$$B start of the sente sequence
$$ ---------------------------
$$ | O . X X . . 1 3 X . X X .
$$ | O O . O X . O 2 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]

Click Here To Show Diagram Code

[go]$$B intermediate follow-up position
$$ ---------------------------
$$ | O . X X . . X X X . X X .
$$ | O O . O X . O 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]

Click Here To Show Diagram Code

[go]$$B Black continues locally, -9
$$ ---------------------------
$$ | O . X X . 1 X X X . X X .
$$ | O O . O X . O O O X X X .
$$ | . O O O . O O B O X . X .
$$ | . O . O O O C B O X . X .
$$ | . O X X X O B B O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$W White continues locally, -19
$$ ---------------------------
$$ | O C B B C 1 X X X 2 X X .
$$ | O O C O B 3 O O O X X X .
$$ | . O O O C O O B O X . X .
$$ | . O . O O O C B O X . X .
$$ | . O X X X O B B O X . X .
$$ | . O X X . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

Sente option's follow-up move value F = (-9 - (-19)) / 2 = 5.


Temperature Region

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[go]$$B Temperature T = 5 1/2
$$ ---------------------------
$$ | O . X X . . . . X . X X .
$$ | O O . O X . O . O X X X .
$$ | . O O O . O O X O X . X .
$$ | . O C O O O . X O X . X .
$$ | . O B B B O X X O X . X .
$$ | . O B B . O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

Assumptions

The assumptions are fulfilled: we have

H ≥ S > R <=> -12 ≥ -19 > -23 1/3 (meaningful options: the gote option might be considered because its result is at least the result of the sente option; Black does not pass because the result of playing is larger than the result of letting White play first locally; similarly, White does not pass),

F < T <=> 5 < 5 1/2 (high temperature),

MSENTE < F <=> 4 1/3 < 5 (sente option due to its increasing move values).

Therefore, the tentative sente move value of the sente option is its sente move value.


Theorem

At high temperature, the theorem for a local endgame with gote and sente options is:

"If F < T, usually start
- in the environment if MGOTE ≤ T,
- locally if MGOTE ≥ T (the creator chooses the gote option)."


Application of the Theorem

The example has a local endgame with gote and sente options, and the contextual and explicit assumptions are fulfilled so the theorem applies.

We have MGOTE ≥ T <=> 5 2/3 ≥ 5 1/2.

Black as the creator starts.

Applying the theorem means choosing the case "locally if MGOTE ≥ T (the creator chooses the gote option)".

Hence, usually Black starts locally choosing the gote option.

Click Here To Show Diagram Code

[go]$$B Black's correct starting sequence
$$ ---------------------------
$$ | O . X X . . 2 3 X . X X .
$$ | O O . O X . O 1 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]

Reading and Counting

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[go]$$B Black starts locally, count -23, Black continues in the environment
$$ ---------------------------
$$ | O C B B C C 2 3 X . X X .
$$ | O O C O B C O 1 O X X X .
$$ | . O O O C O O X O X . X .
$$ | . O C O O O . X O X . X .
$$ | . O B B B O X X O X . X .
$$ | . O B B 4 O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$B Black starts in the environment, count -23 1/3, Black continues in the environment
$$ ---------------------------
$$ | O C B B C C C 2 X . X X .
$$ | O O C O B C O C O X X X .
$$ | . O O O C O O B O X . X .
$$ | . O . O O O C B O X . X .
$$ | . O X X X O B B O X . X .
$$ | . O X X 1 O O O O X . X .
$$ | O O O O X X X X X X X X .
$$ | . . . . . . . . . . . . .[/go]

By the method of reading and counting, Black achieves the larger, better result by starting locally.


Judgement on the Example

The example has these remarkable characteristics:

- The theorem suggests Black's local start. The method of reading and counting suggests Black's local start. Both methods agree. This is no coincidence but related, in particular, to theorems 128 and 129 in [22] about comparing counts for a local endgame with gote and sente options.

- The values F and T are close.

- The values MSENTE and F are close.

- The values MGOTE and T are close.

For these reasons, the example is particularly well designed.


Inventors

Bill Spight has first

- identified the type of local endgame with gote and sente options,

- studied its MGOTE,

- studied its MSENTE,

- proved a theorem for a local endgame with gote and sente options at low temperature during the late endgame,

- derived a proof of a theorem for a local endgame with gote and sente options at low temperature during the early endgame,

- identified the assumption H ≥ S > R (for Black's start),

- compared F and T to identify low temperature,

- identified the assumption MSENTE < F for the sente options itself being a local sente.

He has been too scared to study high temperature by theorems though.

Robert Jasiek has first

- distinguished tentative from confirmed values,

- proved theorems for a local endgame with gote and sente options at high temperature during the late endgame,

- derived the proof of the theorem for a local endgame with gote and sente options at high temperature during the early endgame.

Therefore, everybody here should have been able to study the example according to Bill's study here about seven years ago except for the theorem, whose application has been possible for everybody at least since I have stated it here. Application of the theorem is the easy part. The time-consuming part is before and the difficulty is not getting confused with what sequences and value comparisons to use or avoid.


EDITs: proofs for late / early endgame; minor corrections.

Yes Robert but it was not my question. My question was
Q2: after white 1 do you agree it is natural to assume that WHITE CANNOT EXPECT more than the result above, (followed later by the exchange black "c" white "d")?
I am not asking for the best black sequence. It may well happen that BLACK MAY EXPECT more but it is not the question.

Clarified Q2: I disagree.

If you disagree that means that one the white moves might be a mistake. Which white move causes your disagreement in this white sente sequence?

I am not able to find a go example but at least I can propose the following symetrical tree of depth 2. I do not know if a go position can exist with this idea of tree but go can sometimes be a very surprising game.

With your theory both black b and white d are sente options aren't they?

I expected you to make this mistake...

You overlook tenuki!

If this should be double sente, the root would inherit Black's sente count -19 and White's sente count -16 2/3 but there can only be one count at the root.

For the earlier message https://www.lifein19x19.com/viewtopic.p ... 44#p280544

I have spent roughly these times:

- a few seconds for applying the theorem,

- a few minutes for imagining the move sequences, calculating the values and verifying the assumptions,

- almost two hours for writing the message.

The method of reading and counting applied to its two sequences of starting with the gote option or in the environment is also fast but cannot exclude the sente option and its third sequence, unless typically global reading is considerably deeper to quiet global followers. The theorem is also required to exclude the sente option before application of the method of reading and counting can be fast.

Click Here To Show Diagram Code

[go]$$W
$$ ----------------------------
$$ . . . . . . . 6 2 5 . . . |
$$ . . . . . . . 4 3 1 O d . |
$$ . . . . . . . X O c X O . |
$$ . . X , . X . X X X . O . |
$$ . . . . . . . . . . O . . |
$$ . . . . . . . . . . O . . |
$$ . . . . . . . . . . . . . |
$$ . . . . . . . . . . . . . |
$$ . . . . . . . . . . O . . |
$$ . . . , . . . . . , . . . |[/go]

My question was:
Q2: after white 1 do you agree it is natural to assume that WHITE CANNOT EXPECT more than the result above, (followed later by the exchange black "c" white "d")?
If you disagree that means that you think one of the white moves ( , or d) above is a mistake. Which white move is a mistake and why?

Concerning the tree I proposed in https://www.lifein19x19.com/viewtopic.p ... 48#p280548:

What do you mean Robert? I have never claimed the root of this tree was a double sente for two reasons:
1) Neither you nor me have defined what a double sente position is
2) In my post I did not consider the root position itself but only the black and white options. My purpose was only to know if it could exist a root position with both a white sente option and a black sente option.

In addition what do you mean by "the root would inherit Black's sente count -19". Coming back to the very simple tree I draw in https://www.lifein19x19.com/viewtopic.p ... 32#p280532? Having understood you consider black b as a sente option does that mean that the root would inherit this black sente option?

yes



no

Wrong.

My definition, e.g., in [22] is:

"M_GOTE > M_B,SENTE, M_W,SENTE"

where these are the tentative values of a local endgame with both players' follow-ups. Needless to say, _B uses Black's sente sequence and _W uses White's sente sequence, where sente sequence is an alternating sequence of even length. Adapting this to a local endgame with gote and sente options is straigtforward. Note that then the M_GOTE in the definition does not belong to the alternative gote options but still belongs to the sente options, that is, it is M_GOTE_OF_SENTE_OPTIONS.

Note that the two comparisons in the definition can be substituted by any of its equivalent comparisons.

However, strictly, the comparisons above are not enough. We also have to consider the possibility that either player prefers to pass. This is expressed in the formal additional assumption that the initial position P is without reversible plays.

In your example, Bw < Wb <=> -19 < -16 2/3. The proof considers the subcases Bw <= P (Black passes and Black's play is reversible) and P <= Wb (analogous), that is, we compare to the position P.



The same that is always meant when an initial position inherits the count of a particular, here Black's, sente follower.



The root does not inherit an option it has but it might inherit the count of its sente follower. Since you have suggested (double) sente, if indeed it should be one, the root must inherit the count of its sente follower. IOW, the count of the root must be the count of its sente follower. If and only if the root is a local (hopefully not double) sente (or ambiguous) indeed.


EDIT: reversible.

LOL you disagree but you do not want to explain why you disagree! Why do you take such negative attitude?

Oops, if "M_GOTE > M_B,SENTE, M_W,SENTE" then, providing no other options exist, the position looks like a seki.
Do you know a position which is not a seki and for which we have "M_GOTE > M_B,SENTE, M_W,SENTE"?

Because I have already explained...

For the simplest tree case, there is our (Francisco Criado, I, Bill Spight) theorem in [22] of non-existence of local double sente! So the only question is if something more complicated exists, possibly involving ko. However, the history of go and the history of go research have not produced any promising shape or CGT tree candidate yet. The closest attempts are (my) doubly ambiguous shapes with equalities in the value comparisons or my gote that every traditionalist would have perceived as double sente but whose values are just borderline attempts. If you disregard the web archives and literature, reinvent them (easy enough and a good exercise to eventually find your dream "counter"-example, if any exists)!

Click Here To Show Diagram Code

[go]$$W
$$ ----------------------------
$$ . . . . . . . 6 2 5 . . . |
$$ . . . . . . . 4 3 1 O d . |
$$ . . . . . . . X O c X O . |
$$ . . X , . X . X X X . O . |
$$ . . . . . . . . . . O . . |
$$ . . . . . . . . . . O . . |
$$ . . . . . . . . . . . . . |
$$ . . . . . . . . . . . . . |
$$ . . . . . . . . . . O . . |
$$ . . . , . . . . . , . . . |[/go]

You only said "You overlook tenuki!" which is not quite clear for me.
Does that means that it is better for white to replace or by a tenuki?
Remember that it does not matter for me if black does not play the best moves because my question concerns only white moves.

No, the white moves depend on the black moves!