I am starting this topic as a branch off of the topic, Principles from Basic Endgame Trees (Daniel Hu), https://www.lifein19x19.com/viewtopic.php?t=18166 , to avoid cluttering the discussion up.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Maybe this question fits here:

When to best consider a

a) combinatiorial game,
b) combinatiorial number,
c) settled position or
d) integer?

When to best refer to the
1) number avoidance theorem or
2) integer avoidance theorem?

Do you understand one point of territory?

That is the subtitle of O Meien's excellent book on the endgame in Japanese, Yose, Absolute Counting.

Early on, O Meien gives the following examples of one point of territory, using different diagrams.



The marked point in the top right corner is, OC, one point of Black territory. So is the marked stone in the bottom right corner, But it is one point of territory on average. White to play can leave 0 points in the corner with one move, while Black to play can leave 2 Black points in the corner with one move. There is 1 point on the board plus 1 point for the captured stone. A captured stone counts for 1 point of territory, as well. The average value is (0 + 2)/2 = 1 point.

Much later O Meien shows the following diagram, which is pretty much de rigeur for yose books these days.



There are three copies of the basic ko that often occurs at the end of play. Ko threats do not matter. Each player can guarantee one point for Black, one captured stone, even if the other player plays first. This position is worth 1 point of territory for Black. (Each ko is worth on average ⅓ point, OC.)

However, unless I have overlooked something, O Meien omits an example that Shimamura showed way back in 1954. And it is significant.



All stones are considered to be absolutely safe. As with the ko example, each player can guarantee one point for Black, no matter who plays first. Each marked point is worth ½ point, on average.

This is significant because it is a different kind of average from the one where White can play to 0 and Black can play to 2 points. Each corridor is in itself worth ½ point of territory, on average. (For territory scoring we ignore the dame, OC.) This kind of average is the basis for the Method of Multiples. (See https://senseis.xmp.net/?MethodOfMultiples.)

Some reflection will indicate that both averages must be equal when there are no ko complications. The Method of Multiples indicates that the average is intrinsic to the position, not accidental or contingent.

I am unaware of any yose book in the 20th century that shows Shimamura's diagram, I don't know why. It is just as significant as the 3 ko diagram that everybody shows.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

When there are no ko complications, the number avoidance theorem justifies Chilled Go, but nobody plays chilled go. The integer avoidance theorem, which is an example of the number avoidance theorem, justifies territory scoring as chilled area scoring when there are no ko complications and you are not playing by Japanese or Korean rules.

Both justify ending play by agreement with no passes, but all common rule sets these days effectively end play by some sequence of passes. If your audience is go players, why bother with either one?

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Bill - You said that you didn't know of any yose book published in the 20th century that shows Shimamura's diagram. The diagram occurs in two books by Kano that I just looked at, did you mean just the diagram or maybe the diagram with correct explanation?

The books are:

Nihon Kiin Middle Kyu Series Volume 8 - "Yose - Tesuji and Calculations" by Kano 1964 221 pages
- see page 41

"Yose Dictionary" by Kano 1974 348 pages - see page 18 (Shimamura's diagram modified)
- this is in the introductory chapter on deiri calculation and miai calculation

My limited knowledge of yose does not permit me to say if Kano's explanations are correct or not......

If you are interested I can scan in the relevant pages.

Take Care - John

Many thanks, John.

I found the diagram in Kano's Yose Dictionary. I am not sure why I did not recall it. He doesn't mention that the two corridors are miai, as Shimamura does. Kano also shows miai first line hane-and-connects on the next page, but he uses it as an example of getting the last play, which is bizarre.

Kano makes three statements about the diagram. The first one, which is expressed a little strangely, I think, calculates both A and B as worth ½ point of territory by miai counting. Then he says that the two positions are worth 1 point of territory, which is the significant point, I think. Then he says that by deiri counting A and B are worth 1 point. Kano stresses the difference between miai and deiri counting, which may be why I forgot about this diagram. But he did show it and evaluate it correctly, which is good.

Thanks again.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Suppose we have Black's alternating sequence along the positions A, B, C, D,... at the ideal environment's temperature t and want to characterise exactly the first two moves as being worth playing successively (the third move is less valuable so alternate local play should not immediately continue with it).

Let B_t(A) be the result of playing A in the ideal environment at temperature t if Black starts.

Let W_t(B) be the result of playing B in the ideal environment at temperature t if White starts.

How can we justify B_t(A) = W_t(B) - t?

Why complicate things? Position A has a mean value of m(A). That's the result of playing to A.



Position B is a follower of A, not the original game, G. Say that from G White plays to position Z. Then the result of that play is m(Z).



It's not true, is it?

If, at temperature t, Black plays to A and White replies to B, the result is m(B). If White plays to Z from G and then Black replies in the environment, the result is m(Z) + t. If m(B) < m(Z) + t, there is something wrong.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

For studying the Method of Making a Hypothesis, the complication may be necessary in an attempt to prove that "Black's alternating sequence is worth playing successively for the first k plays" <=> t <= G1, G2,..., Gk for its gains and the local endgame in maybe a rich environment.

"That's the result of playing to A." Not quite. I have meant move 1 from position A to position B, move 2 from position B to position C,...

"Say that from G White plays to position Z." My current hope is that the statement above can be proved without considering White's start.



The assumption was "the first two moves [of Black's alternating sequence are] worth playing successively" and I don't know yet why we would need to also consider White's start for this.

Well, if I understand your notation now, what should be true is this:

B_t(A) = W_t(B)

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Yes, I suspect this, too.

Black starts at A, the players alternate locally as long as at least as valuable as in the environment and then some player X continues in the environment.

This should be the same as White starts at B, the players alternate locally as long as at least as valuable as in the environment and then player X continues in the environment.

Does this mean that proving t <= G1, G2,..., Gk is trivial? Similarly for White's such alternating sequence. Next, we derive the initial count and move value.

A couple of years ago, you suggested that proving the correct initial values would require thermography. Now, somebody in private communication suggests that a rich environment and considering stops would do. Above, I wonder if it is all just trivial. Rocket science or trivial?!?

Since you use different notation from combinatorial game theory, I assume that Gi is not a game. I suppose you mean its miai value. I would not say that calculating the miai value of a game is, in general, trivial. But you have gotten the order of calculation backwards. First you calculate the mean values (count), then you find the move values.



If so, I did not mean to suggest that. After all, go players found correct mean values without thermography for at least a century, if not centuries. Ko is another story. Mean values work for certain ko positions, but are questionable, in general. I developed my own ko theory during the 1980s, but it was intractable for many ko positions. Berlekamp's komaster theory was superior in that regard, and I followed that. The theory is thermographic, OC.



In 1998 I redefined thermography in terms of a ideal (rich) environment. I suppose that is what is behind that suggestion.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Let me put it this way. If it's trivial, it's trivial. Take the game, {25|17}. From that it's easy to find the deiri value, 25 - 17 = 8. It's easy to find the miai value, 8/2 = 4. From that we may calculate the mean value. 25 - 4 = 21.

I say to calculate the mean value first. (25+17)/2 = 21. From that it's easy to calculate the miai value. 25 - 21 = 4. This calculation is slightly inefficient. But it gets the concepts right. Mean values are basic. A move from a position with a mean value of 21 to a local score of 25 gains 4 points. It's when the calculations are non-trivial that the concepts really matter.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Earlier, I declared that Gi are the gains (of the plays of Black's alternating sequence)!



Now that you know that I speak of gains, please reconsider:)

The most importantly, see the method of making a hypothesis in [16] in
https://www.lifein19x19.com/forum/viewt ... =17&t=8765

OK, let's call the original position G, with mean value m(G). Given correct play in an ideal environment with temperature, t, Black to play moves to A, then White moves to B, then Black plays in the environment instead of moving to C. The move to A gains, on average, m(A) - m(G), the move to B gains, on average, m(A) - m(B), and the move to C gains, on average, m(C) - m(B). It should also be the case that m(B) ≥ m(G), something that simply talking about gains instead of mean values does not make clear. That yields the following.

m(A) - m(G) ≥ m(A) - m(B) ≥ t ≥ m(C) - m(B)






I did, and still do, believe that thermography provides a conceptually superior way to think about these matters, as everything rests upon the concept of mean value. That is not to say that thermography is necessary to prove mean values, or any other values.

Kano is a good example of a pro 9 dan who does not quite get the concepts. For instance, in his Yose Dictionary he correctly states that the following corner is an example of miai. (By convention the outside region is Black territory.)

Click Here To Show Diagram Code

[go]$$ Strict miai
$$ ----------------
$$ | . O . . . . . .
$$ | . . . O X X . .
$$ | . O O O O X X .
$$ | . O X X X . . .
$$ | . X . . . . . .
$$ | . . X . . . . .
$$ | . . . . . . . .
$$ | . . X . . . . .
$$ | . . . . . . . .[/go]

So far, so good. But then he shows the following sequence and states that is the last play of the game.

Click Here To Show Diagram Code

[go]$$ Strict miai
$$ ----------------
$$ | . O . 6 4 5 . .
$$ | . . . O X X . .
$$ | 2 O O O O X X .
$$ | 1 O X X X . . .
$$ | 3 X . . . . . .
$$ | . . X . . . . .
$$ | . . . . . . . .
$$ | . . X . . . . .
$$ | . . . . . . . .[/go]

That's bizarre. The whole point is that and are miai. Sure, you can consider to be the last play, but doing so ignores . In theory the players could agree to the score without playing the miai out. The correct way to think about this is that White got the last play somewhere else, and then Black played the miai.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Now, you are on track:)

Why does talking about mean values already imply m(B) ≥ m(G)?

For longer traversal, m(A) - m(G) ≥ m(A) - m(B) need not be fulfilled. It is sufficient for each gain to be at least t.

Kano's example is not exactly miai because, even without ko threat play, playing at C19 can sometimes be correct and did win FJ Dickhut a German Championship game.

Yes, Kano also assumes that Black cannot win the ko in the corner.



You have to ask? If m(G) > m(B), Black plays to C.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

If you only use counts, everything is clear. However, if you use mean values, I have to ask because I am not firm with the CGT theory of mean values.

Well, mean values are basic. Berlekamp came up with the term, count, to cover both mean values of finite combinatorial games and mast values of kos and superkos. If you stick to finite combinatorial games, go players were calculating mean values well over a century ago.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Also to be clear. If we are considering a game, G, in an ideal environment, G may well consist of more than one local positions (games). A play in any of these positions is local to G. In particular, it may be correct to play suboptimally in one position in order to get the last play in another before a temperature drop.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.