Life In 19x19

RobertJasiek wrote:

$$B Example 3 for theorem 68
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$$|X X X X O O O O O O .
$$|. O X X O X X O X O O
$$|X O X X O X X O X X O
$$|X O X X O X X O X X O
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$$|X . O O O X X O . O .
$$|a O O X O X X O X O .
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$$|X O . O O X X X X O .
$$|X X X X X X . . X O .
$$|. . . . . . . . X O .

Click Here To Show Diagram Code

[go]$$B Example 3 for theorem 68
$$----------------------
$$|X X X X O O O O O O .
$$|. O X X O X X O X O O
$$|X O X X O X X O X X O
$$|X O X X O X X O X X O
$$|X O O . O X X O X O O
$$|X . O O O X X O . O .
$$|a O O X O X X O X O .
$$|X O X X O . O O X O .
$$|X O . O O X X X X O .
$$|X X X X X X . . X O .
$$|. . . . . . . . X O .[/go]

"Example 3: We have M_SENTE = 10, F = 10.5, T = 12, T_1 = 6, T_2 = 3 and T > F <=> 12 > 10.5. Due to the even number 0 of moves in the environment larger than F and excluding T, a plus sign starts Ω = F - 2(T_1 - T_2) = 10.5 - 2(6 - 3) = 4.5. According to theorem 68, the black creator starts locally as 2T - M_SENTE - F ≤ Ω <=> 2*12 - 10 - 10.5 ≤ 4.5 <=> 3.5 ≤ 4.5." [22]

Mandatory local start despite high temperature! You can also imagine the local endgame during the early endgame and draw its tree from B. I am not suprised that you could not find such an example easily; the unusual values made the difficulty of finding some of intermediate level. It is by far not the rarest, most difficult kind of endgame examples though; for the real difficulties (type a) late endgame with gote and sente options and mandatory start in the environment at low temperature; type b) the only correct choice gives tedomari to the opponent), I needed three to four days per first example.

I agree with you that a black move at "a" can be analysed as M_SENTE = 10 with F = 10.5
Now you built an environment T, T_1, T_2 = 12, 6, 3 and, though M_SENTE < T you conclude that a local black move is better than a black move in the environment which is again true.
Two comments:
1) the environment T, T_1, T_2 = 12, 6, 3 looks not an ideal environment according to your definition in your post viewtopic.php?p=277676#p277676
2) the environment you chose is very special. You can even see that you can suppress T_1 and T_2 and keep only the gote T = 12 to reach the same result. By using an ideal environment as defined in CGT or thermography the conclusion is that, in your example, a move in an environment with T = 12 is IN GENERAL better than a local M_SENTE = 10 move.

If you say that my result is wrong then you also have to say that CGT or thermography results are wrong for the same reason. I consider a move in an environment at temperature T= 12 is, in an ideal environment (meaning IN GENERAL?), better than a local M_SENTE = 10 but OC it might be incorrect for various specific environments.

Knowing that in practice a real environment is almost never ideal (even with your definition) what are you aiming for when you analyse a local position in the specific ideal environments you defined?

Hehe, this environment is not an ideal environment indeed.

For the late endgame, any environment of simple gotes is admitted and accounted in the alternating sum. For the early endgame, the ideal environment is my model for the approximative theorems and similar theorems. It is sometimes useful to consider an ensemble.

Now you are aware that you must not make too general statements as if everything would behave like simple gotes. However, you might presume an (or my) ideal environment, clarify the other presuppositions (C = 0, E settled, no kos now or later and whatever else may be necessary) on your generic tree, formulate a conjecture and try to prove it establishing a theorem.

RobertJasiek wrote:Hehe, this environment is not an ideal environment indeed.

For the late endgame, any environment of simple gotes is admitted and accounted in the alternating sum. For the early endgame, the ideal environment is my model for the approximative theorems and similar theorems. It is sometimes useful to consider an ensemble.

Now you are aware that you must not make too general statements as if everything would behave like simple gotes. However, you might presume an (or my) ideal environment, clarify the other presuppositions (C = 0, E settled, no kos now or later and whatever else may be necessary) on your generic tree, formulate a conjecture and try to prove it establishing a theorem.

OK I understand your example was wrong as a counter example for my statement and I do not see other counter example.
Let me remind you my statement:

If vb is the miai value of a position B (using miai value and an ideal environment as used in thermography) then:
When temperature of the environment is greater than vb then it is correct for black to play in the environement (it may happen that a local black move could also be correct but playing in the environment is always correct).
When temperature of the environment is lesser than vb then the only correct move for black is to play locally (playing in the environment is not correct)
When temperature of the environment is equal to vb the situation is ambiguous and it is correct to play either in the environment or locally.
Note : the situation for white is exactly the same.

'an ideal environment as used in thermography'

Thermography has not used ideal evironments, AFAIK. Thermography has used rich environments. Study for these theoretical, abstract constructs must be CGT-formal.

Before I dig again into examples, please clarify the environments you want to use!

RobertJasiek wrote:'an ideal environment as used in thermography'

Thermography has not used ideal evironments, AFAIK. Thermography has used rich environments. Study for these theoretical, abstract constructs must be CGT-formal.

Before I dig again into examples, please clarify the environments you want to use!

Rich environments is fine for me with the idea that this environment can be considered as rich as you want.
Ideal CGT environment is also fine for me but I prefer to use rich environments, as rich as I need.

For rich environments, I cannot give examples. Usually, the study is beyond me, although I could prove a bit about definitions or relations of move values and counts.

For my ideal environments, I think I can dig out counter-examples, unless my memory plays me a trick WRT conditions.

BTW I made some progress in the definition of my own theory.
In order for you to have a better understanding of my view here is an attempt of new definitions. You will see that for me an ideal environment is really based on sufficiently rich environments.
In my mind the move value defined here is identical to the miai value used in thermography. Unfortunately I did not prove this but I did not find any counter example.
OC I have still a lot of work on my to do list to go further.

Pure gote position
let's call g(k) a basic gote position a position associated to the following tree made of only two branches : k is said to be the size of the pure gote position

Rich environment definition:
A rich environment Rich(t, N) depends on two parameters:
- t is said to be the temperature of the environment
- N represents how rich is the environment. Bigger is N and richer is the environment
Rich(t, N) is made of N basic gote positions with the sizes evenly distributed between 0 and t.
More precisely Rich(t, N) is made of the N basic gote positions : {g(t/N), g(2t/N), g(3t/N), ... g(Nt/N)}
The two main advantages of this rich environment are:
1) Each time you play in this environment the resulting position is still a rich environment but at a lower temperature
2) By choosing a big N you can assure that this temperature will drop as slowly as you want
The goal of this rich environment is not to be a real environment but rather a kind of average of all possible environments.

In order to analyze any local position P, the idea is now to start a game from an initial position made of the local position and a rich environment:
initial position of the game = {P, Rich(t, N)}
Starting with a rich environment with a high temperature the best move is surely in the environment but as the game progress it will happen that at a given temperature you will have to play in the local position to reach the best score.
Considering the game {P, Rich(t, N)} then, by definition, the move value of the local position P is the value m such that
1) if t > m then you CAN play in the environment to reach the best score, providing the environment is rich enough
2) it t < m then you CANNOT play in the environment to reach the best score, providing the environment is rich enough

That way the move value is expected to show at which temperature you must play locally and not in the environment.
More formally the definition of the move value is the following:

Considering a game starting with position {P, Rich(t, N)} with black to play, m is the move value of P if:
1) ∀t > m, ∃ N1 such that N > N1 => the best score of the game CAN be reached by playing first in the rich environment
2) ∀t < m, ∃ N2 such that N > N2 => the best score of the game CANNOT be reached by playing first in the rich environment

Ideal environment:

Ideal environment = 70, 60, 50, 40, 30, 20, 10.
Simple gote with move value M = 53 and Black's follow-up move value F = 30.
The temperature is still high on move 3.
White starts.

I) On move 3, White making the follow-up unavailable is a mistake:

-70 + 60 - 53 + 50 - 40 + 30 - 20 + 10 = -33.

II) On move 3, White making the follow-up available is correct:

-70 + 60 - 50 + 53 - 40 + 30 - 30 + 20 - 10 = -37.

White takes the smaller move value 50 in the environment instead of taking the larger move value 53 of the local gote.


Rich environment:

I guess, such anomalies do not occur. However, I am not sure. I think that the rich environment must be sufficiently dense and the temperature large enough. The former is achieved trivially by letting the granularity converge to infinity. The latter might be the problem, but you presume a high temperature. The enriched score of a local position is the same so becomes the count regardless of the starting player for a large enough temperature by

"Proposition 9 [equal scores in case of large enough temperatures]
B_T(P) = W_T(P) for all T » 0. This value is a constant for large enough T:
If T, S » 0, then B_T(P) = B_S(P) = W_T(P) = W_S(P)." [24]

Note that "no kos" is a presupposition! Citation from the proof: "Black's start in the environment to P + T + E_T-D dominates his local start to P_B + E_T" [24] Hence, I guess you are right (if there are no kos) but this must be thought through more carefully.

Reference: https://www.lifein19x19.com/forum/viewt ... =17&t=8765

In Bill Spight's or my study with some ideal environment, we compare the temperature to the follow-up move value(s) to identify a high temperature. In a rich environment, a high, sufficiently large temperature is given by defining the local count. These two approaches are rather different! Rich environments work for abstract study in their model but the arbitrarily dense environments of sufficiently large temperature are unrealistic and cannot be represented on the board.

Now, I have read messages 36 - 38 of kvasir and Gerard.

When I read "B - va = A", I thought "for gote", only to read two lines later "Which are true when we talk about the gote (unforced play) case". Hehe, forcing the reader to think for himself before stating the missing presupposition:(

kvasir, if leaves carry counts, earlier nodes also carry counts. If you want to assign move values / miai values to earlier nodes, your tree annotation must make this clear. E.g., write v: 3 instead of just 3. (I prefer M.)

"Instead of scafolds you may also, for such simple corridor, calculate recursively the count, the miai value and perhaps the gain of each move."

For a tree "corridor", this works as you show by distinguishing simple gotes from simple sentes because the longest alternating sequences have two plays. Therefore, my more general method of making a hypothesis on possibly greater lengths of alternating sequences is not needed. For general trees, the method of making a hypothesis works in practice but there are pathological CGT tree examples for which it does not work correctly. So in theory thermography is needed. I do not know wether tree corridors are simple enough to always work correctly without thermography. A counter-example is not known so a theorem should be be proved.

Gérard TAILLE wrote: Considering the game {P, Rich(t, N)} then, by definition, the move value of the local position P is the value m such that
1) if t > m then you CAN play in the environment to reach the best score, providing the environment is rich enough
2) it t < m then you CANNOT play in the environment to reach the best score, providing the environment is rich enough

[...]
More formally the definition of the move value is the following:

Considering a game starting with position {P, Rich(t, N)} with black to play, m is the move value of P if:
1) ∀t > m, ∃ N1 such that N > N1 => the best score of the game CAN be reached by playing first in the rich environment
2) ∀t < m, ∃ N2 such that N > N2 => the best score of the game CANNOT be reached by playing first in the rich environment

It seems there are different equivalent definitions. Unlike Bill, I have not been a fan of Conway's via infinite numbers of multiples. I forgot which Berlekamp used. Siegel's relies on rich environments, I find easier to understand and is rewritten by me from us expert go players' perspective:

"Definitions 9 [count and move value]
For a position P and large enough number ∞, the count is C_P := B_∞(P) = W_∞(P) and the move value M_P is the smallest value of T for which B_T(P) = W_T(P) = C_P." [24]

RobertJasiek wrote:In Bill Spight's or my study with some ideal environment, we compare the temperature to the follow-up move value(s) to identify a high temperature. In a rich environment, a high, sufficiently large temperature is given by defining the local count. These two approaches are rather different! Rich environments work for abstract study in their model but the arbitrarily dense environments of sufficiently large temperature are unrealistic and cannot be represented on the board.

In your theory, do I have to understand that you use ONLY realistic environments?

Late endgame:

Any environment of simple gotes without follow-ups is used in the theory. As a model environment, this is realistic and can be represented on the board.

Early endgame:

An ideal environment with any drop is used in the theory. As an approximation of real environments, this is realistic and can be represented on the board by choosing, e.g., 1/2 or 1 as the drop.

Simplification:

Of course, these environments do not have kos or follow-ups. In a (usual) model, such is necssary or complexity would be NP-hard or worse. Only for special general cases, there is more sophisticated theory, such as certain positions with several local endgames with simple follow-ups.

RobertJasiek wrote:Late endgame:

Any environment of simple gotes without follow-ups is used in the theory. As a model environment, this is realistic and can be represented on the board.

Early endgame:

An ideal environment with any drop is used in the theory. As an approximation of real environments, this is realistic and can be represented on the board by choosing, e.g., 1/2 or 1 as the drop.

Simplification:

Of course, these environments do not have kos or follow-ups. In a (usual) model, such is necssary or complexity would be NP-hard or worse. Only for special general cases, there is more sophisticated theory, such as certain positions with several local endgames with simple follow-ups.

With such realistic environment, because the drop cannot be lesser than 1/2 I guess you can harldly expect to have an accurate move value. OC you can argue that in practice you do not need an accurate move value and I agree with you. Now I understand why you use the wording "good approximation" in some of your theorems.
With my definition, because I use irrealistic rich environment that can be as rich as necessary then my move value is alwyas accurate without any approximation. For the same reason move value for ko can be calculate accurately (a result like for example 5 2/3 is easy to prove).

BTW the drop 1/2 seems fine in the context of territory scoring. What small value can you use in the context of area scoring for your realistic ideal environments?

Rich environment is useful to define things in a particular way and enable sentestrat. My theory redefines count and move value differently. Ideal environment is used for often (much) better strategic advice than sentestrat because the late endgame can be exact (in standard positions) or the approximation is tight during the early endgame. In sentestrat, the error depends on the temperature. In my approximation, the error depends on the drop, which is much smaller than the temperature during the early endgame.

I do not use ideal environment to define count and move value! I need not. I define via tentative counts and tentative move values - gote or sente values being larger, equal or smaller. Local evaluation independent of environments! Environments for value definitions are overkill for practical application. Counter-examples (without kos) to the method of making a hypothesis are so hard to find that none is known on a go board so far.

Area scoring: any drop, but exceptions would occur more easily in the 2 points and below range of move values.