Click Here To Show Diagram Code

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

You are right Bill, because a local ko is essential to the difference game, difference games are not appropriate to this comparison.

Let's look for another example you will find here : https://lifein19x19.com/viewtopic.php?p=197731#p197731



You conclude a little farther that the hanging connection is best ... but you mentionned also yourself that we have to take care of the ko threats created.
Finally we understand that the choice between hanging connection or descent is very difficult in practice because you have to anticipate the future potential ko fights.
BTW do you know if some attempts as been made in thermography to give some value to ko threats created : 0.1 point per ko threat, or 1 point or t/4 points or whatever else?
Do you know if some statistics on real pro games exist to give a value to a ko threat created ?
Assume you (or a pro if you prefer), in the beginning of the middle game, must choose between a move gaining 13 points and a move gaining only 12 points but creating some very good ko threats for the future and its uncertainty. What is your intuition for the best choice?

Well, for a specific ko position and threat you can calculate its thermographic value.

As for the question of the descent vs. the hanging connection, unless there is a special tactical value to the descent, the pros avoid it. The real question is between the hanging connection and the solid connection. The pros most often prefer the solid connection. Probably because of ko threats, I expect.

Way back when, before I knew about thermography, I estimated the average value of a ko threat at around 0.1 point, as most threats are not decisive, but may affect the temperature.

The evidence from today's top bots is unclear. Sometimes they don't seem to care about them, but usually they do, by a very small amount. Katago might provide some point sizes.



No.



My guess is that a 1 point difference in the gain, if the pros are aware of it, is too big for them to care about ko threats for unknown ko possibilities.

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

Visualize whirled peas.

Everything with love. Stay safe.

More on my old method.

Now that it seems that the largest gote matters to the question of playing a global sente, let's add it to the game explicitly. We have

{2b|0||-a} + {h|-h}, b, h > g0

We do not need to have b > h.

Black plays first. The question is whether to play the sente or to play the gote.

1) Black plays sente, White plays to 0, Black plays to h

s = h - g0/2

2) Black plays sente, White plays to -h, Black plays to 2b

s = 2b - h - g0/2

3) Black plays to h, White plays to -a

s = h - a + g0/2

White chooses between 1) and 2), so for Black to play the sente, both 1) and 2) should be at least as good as 3).

h - g0/2 ≥ h - a + g0/2 , that is,

a ≥ g0 , and

2b - h - g0/2 ≥ h - a + g0/2 , that is,

a + 2b ≥ 2h + g0

----

Now, those are comparisons you don't see every day.

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

Visualize whirled peas.

Everything with love. Stay safe.

Interesting Bill and we have to understand why such old method could not get success.

Assume in your game you encounter the position P = {2b|0||-a} somewhere on the board. The environment E of this position is all the board minus the position P. It is the real environment and you have all the information concerning this environment. Because you have all the information you know in theory if you have to play in the position rather than in the environment.

What kind of help can we expect from a theory? OC we would like to know when it is time to play in the position rather than in the environment but at the same time we do not want to know in detail the environment. IOW, we accept to analyse in detail the position P but we are reluctant to take into account various environments.

The genius of thermography is to have understood that point : all you know about the environment is what is called "temperature" and you try and analyse in great detail the concerned position in this rather unknown environment.

Looking at your old method we can see three major drawbacks which are solved by thermography.
First of all you add to the position P the area {h|-h}, and, instead of analysing a position P in the environment you analyse the position P1 = P + {h|-h} in the environment. I do not say it is not interesting but that way the result are more complex (the results depends on (h,g0) against only t) and that is not what a go player may expect from a theory.
Secondly this old method do not give a clear picture of the result: by concluding a ≥ g0 , and a + 2b ≥ 2h + g0 you show in which cases black "has to" play the sente but you do not show when black "can" play the sente. Thermography with colored mast give us this information
Thirdly, for a human, a picture with a left wall and a right wall is far more easier and pleasant to read than equations like a ≥ g0 , and a + 2b ≥ 2h + g0.

More later to go further

The old method was intractable when it came to approach kos, 10,000 year kos, and the like. Berlekamp's komaster theory handled those.



If you have all the information about the real environment you can, given enough time, solve the whole game tree. As I have said, if you can read everything out, you do not need theory.



Make that almost every environment. Even the old method represented the environment as composed of simple gote.



Before 1998, when I redefined thermography, it was not based upon an environment at all.



Actually, the old method is correct to analyze the game, P + (H = {h|-h}) and make the choice between playing in P and playing in H, instead of playing in P and playing in the environment. In this it anticipated thermography.



Cart before the horse. I added colored masts to thermography to include that information, based upon my old method.



I definitely agree.

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

Visualize whirled peas.

Everything with love. Stay safe.

Critique of my old method

Traditional Japanese go theory classified positions into four classes, double gote, sente, reverse sente, and double sente. But it did not define these terms. The old method implicitly defined them globally, at least for relatively simple positions, and did not worry about more complicated ones.

Thus, given that all variables are greater than or equal to 0, {h|-h} was a double gote, as was {2a|0||-d|-d-2b} when a ≤ g0 and b ≤ g1. {2b|0||-a} was a sente if b > g0, but not if b ≤ g0. {2a|0||-d|-d-2b} was a double sente if d > 0, and if a > g0 and b > g1.

One possible problem with my old method was that it did not give the average territorial values of the positions. But it was not designed to do so. It was designed to choose which play to make. For the average territorial values (counts, as Berlekamp dubbed them), it relied upon the traditional theory. If a play was double gote, the count was the average of the counts of its two followers, as we say today. If a play was sente, the count was the same as the count after the sente play and reply, which was enshrined in the go proverb, Sente gains nothing. The count of a double sente was not calculated.

It may have given pause that the average territorial value of the game, {2b|0|-a} + {h|-h} was 0 if b > h and (b-a)/2 otherwise, but it did not affect the correctness of the results about the choice of plays. Any anomalous territorial values were the result of traditional theory, which did not define its terms, not of my method.

More later.

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

Visualize whirled peas.

Everything with love. Stay safe.

What is combinatorial game theory (CGT)?

CGT proper is a theory under which numbers and games form a group under addition. That is, numbers are games, and games can be added and subtracted. Numbers, OC, have numerical values. Other games have average values.

Combinatorial games were studied before CGT was born in the 1979s. That certain games have average values and how to calculate them was established by the mean value theorem in the mid 20th century. I presented the method of calculating them on the SL page, the Method of Multiples, https://senseis.xmp.net/?MethodOfMultiples . The page includes a section on kos. Kos are not combinatorial games, but we gave the method a shot. It can tell us something about regular kos. Three kos can, in a sense, add to a number.

Thermography, among other things, is a simpler method of finding the average values of games. It makes finding them tractable. When I read about thermography in the 1970s, I failed to appreciate that fact. I had already, in fact, discovered that {2b|0||-a} is a local sente when b > a, with no reference to an environment. Some of my friends encouraged me to publish that fact, but I thought that it was obvious. Mea culpa. Even today, I am unaware of any Japanese, Chinese, or Korean text that defines local sente and gote. I had my own method of finding average territorial values.

More later.

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

Visualize whirled peas.

Everything with love. Stay safe.

Yes Bill I see two basic questions.
Let's assume we are studying a local game G. For that purpose we imagine a global game H made of the local game G and an environment E (H = G + E).
Question 1) what is the best move for the global game H? Is it a move in the local game G (in that case which move?) or is it a tenuki (a move in the environment E)?
Question 2) what is the average territorial value of the local game G

Let's take as example the local game G = {9||4|0} + {3|-3}
The point is that you cannot answer the two above questions without having a detail view of the environment E.
For question 1 obviously you need information concerning E.
For question 2 the problem is to anticipate the mean value of {4|0} and without any assumption on the environment you only know that this mean value is between 4 and 0.

The thermography approach is quite interesting : we define an ideal environment depending of only one parameter called temperature, with good properties modeling more or less the behavior of a vast number of real environments.
If you want to study a local game G then you add to G an ideal environment E to build a global game H and you play the global game using the special properties of the ideal environment.
That way, taking the example G = {9||4|0} + {3|-3} you may find that the average territorial value of the local game G is 5½ the temperature of the local game may be 3½, the best black move if t ≤ 1 is in {3|-3}, the best black move if 1 ≤ t ≤ 3½ is in {9||4|0} and the best black move if 3½ ≤ t is tenuki.

But now is really the point: all these results being made in a special environment (the ideal environment) none of them are reliable for a real game: stricly speaking you cannot accept to say that the territorial value of the local game G is 5½ and you cannot accept to say that the best black move if 1 ≤ t ≤ 3½ is in {9||4|0}.
How can we formulate the results? We have to give a "warning" like, "on average", "in general" or whatever you want.
Curiosly however my feeling is that we generally accept easily to say "on average" territorial value of the local game G is 5½ but it happens more difficult to accept to say "in general" the best black move if 1 ≤ t ≤ 3½ in {9||4|0}. With this last sentence the more often reaction is conter by showing an environment in which the best move is another one. Consequently the notion of "domination" was defined with a large panel a possible environments (called non-ko environment) but in practice when two moves seem quite near, none of them dominates the other and we learn nothing.

More later.

I disagree because

- I find equations easier to read than graphs of mappings,

- graphs of mappings rely on equations and we must use the equations anyway to justify correctness of the graphs of mappings,

- after every move, a new graph of mappings occurs,

- when applied while playing a game, calculating equations is simpler than imagining and mentally constructing graphs of mappings.



This is not thermography. Even my non-thermographic ideal environment has a granularity, such as 2, 1 or 1/2. Thermography also relies on such a second paramater: the arbitrarily small granularity so that a RICH environment is formed.



1970s?



Now that our English texts define these, they can translate them to Asian languages.



You emphasise "old" but it is very useful as long as we are aware that kos etc. are outside the theory.

That is a very strong claim for the local game, since it contains no environment. Both traditional go theory and thermography do find an average territorial value for that local go game.



Speak for yourself.



OC, that was not the case for original thermography. Nor it is in my redefinement of thermography. The idea is that there exists, for any particular game, a universal enriched environment with the property that, at the temperature of every subgame of that game, a play in that environment alone gains one half of that temperature. That is a very restrictive requirement for the environment, which the vast number of real environments do not meet, hardly any of them, as a rule. Otherwise my redefinement would not yield the same thermographs as the original version, with no environment.



You get the same result as the original formulation of thermography. The average territorial value of the game is 5½ by thermography.



You can define the territorial value of a go game which depends upon its environment. But that does not invalidate the traditional and thermographic ways of defining that value. Although the traditional way has the problem of leaving important terms undefined. The traditional way, however, does not explicitly mention any moves in the environment.



There used to be a guy on rec.games.go who denied the traditional territorial value of 5½ for the local game in this note, because go scores are integers. Statements that that value was only an average value did not sway his opinion.



You are shifting the discussion away from that of the average territorial value, which you dispute.



That's irrelevant. Besides which, you do not disprove a general statement by finding an exception.



Domination is not part of thermography.

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

Visualize whirled peas.

Everything with love. Stay safe.

Yes. Combinatorial games, as defined in On Numbers and Games (ONAG), had been studied at least as far back as the 19th century, in the case of Nim, and even, although informally, in the case of go positions, and average values were added together. However, it was only with the definition of both numbers and games in ONAG that it was proven that they formed a group under addition. Combinatorial games can, themselves, be added together, not just their average values.



Well, I found that Berlekamp's komaster theory was much more tractable for ko positions.

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

Visualize whirled peas.

Everything with love. Stay safe.

You seem to not agree that we need some information on the environement in order to answer my two questions. What do you mean?
We all know that, depending of the environment, the best move to play can OC change. I said nothing else by saying that to discover the best move you must have information on the environmment. I do not understand what really hurt you.

BTW in your sentence "Both traditional go theory and thermography do find an average territorial value for that local go game" the key word is "average". Is it an average over a set of environments or is it an average over something else?

Combinatorial games as defined by CGT, including your game, G, have been proven to have average values, which in territorial go, mean average territorial values. The proof does not involve any environment for the games. Your statement says that that proof is false.

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

Visualize whirled peas.

Everything with love. Stay safe.

Let's try to proceed slowly because it seems there is some misunderstanding.

Consider only my first question:
Question 1) what is the best move for the global game H? Is it a move in the local game G (in that case which move?) or is it a tenuki (a move in the environment E)?

My feeling is that, when we are talking about a "best move", we have to add in which kind of environment the considered move is the best. It could only with an ideal environment in a specific range of temperature, it could for all simple environment made of only simple {u|-u} gote points, it could be on any non-ko environment or ...
If you say that it does not depend of the environment, does that mean that, when your are talking about a best move, then this move remains the best in all environments? OC it is impossible because a local move cannot be best of the temperature of the environment is very high.

This question may be answered with von Neumann game theory, in which the final score after correct play is the value of the game.

The average value of the game is not part of von Neumann game theory. If the game is a combinatorial game, then its average value may be calculated without regard to any other game, such as an environment, by the method of multiples. Either the average value is produced by some finite number of multiples, or it is approached in the limit as the number of multiples approaches infinity. You seem to dispute that idea.

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

Visualize whirled peas.

Everything with love. Stay safe.

OK I proposed to handle question 1 but you prefer to handle question 2.

The problem is then to evaluate a position without knowing the environment. Because you mentionned on many occasions Berlekamp's komaster theory I keep in mind that ko are not excluded.

Click Here To Show Diagram Code

[go]$$
$$ ---------------------------
$$ | . a X X O O O b O . . . -
$$ | X X . X X X X X O . . . -
$$ | O X X O O O O O O . . . -
$$ | O O O O . . . . . . . . -
$$ | . . . . . . . . . . . . -
$$ | . . . . . . . . . . . . -
$$[/go]

Click Here To Show Diagram Code

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

Comparing these to diagrams any go player with white will without any hesitation prefer the first diagram because white has potentially a ko threat at "a".
But if I am correct the method of multiples gives the same value to the two positions.
Maybe nobody knows if the difference between the two diagrams is 0.001 or 0.01 or 0.1 or 0.2 or 0.5 or whatever you want but the differnece is > 0.
That does not mean that I do not like the value given by the method of multiples. On contrary I consider it is the best estimation in the context of our current knowledge of go theory. I am convinced we can invent a new count for positions, taking into account ko threats, but it is for future work ...
BTW if these two positions are part of the same game and it is white to play white will obviously choose to play at b rather than c.

I do not understand how you present the results of your analyse when using two parameters (temperature + granularity)
Taking my example: G = {9||4|0} + {3|-3}
With only the temperature t the results are:
1) average territorial value of the local game G : 5½
2) temperature of the local game may be 3½
3) best black move if t ≤ 1 is in {3|-3}
4) the best black move if 1 ≤ t ≤ 3½ is in {9||4|0}
5) the best black move if 3½ ≤ t is tenuki.
I suspect your result are completly identical for the point 1) and 2) but what about the results for the best moves in your envrironments with two parameters?

I edited the diagrams to keep the number of stones the same. Given that the player knows nothing about the environment, then we agree.



Right again.



Make that ≥ 0. Once enough is known about the environment.



Bueno.

All of the go textbooks I know of will produce the same average territorial value for both diagrams, and, I expect, so would have Hayashi Genbi.

And for any ko ensemble including the first position, thermography allows us to compute a mast value, even if it is not, strictly speaking, an average value.

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

Visualize whirled peas.

Everything with love. Stay safe.

Yes Bill all of the go textbooks we know produce the same average territorial value for both diagrams but how many books try to handle ko threats.

Let me add explicitly a very small ko:

Click Here To Show Diagram Code

[go]$$
$$ ---------------------------
$$ | . a X X O O O b O . . . -
$$ | X X . X X X X X O . . . -
$$ | X O X X O O O O O . . . -
$$ | O O O O . . . . . . . . -
$$ | . O . . O . . . . . . . -
$$ | O X . . . . . . . . . . -
$$ | X X . . . . . . . . . . -
$$ | . . . . . . . . . . . . -
$$ | . . . . . . . . . . . . -
$$[/go]

What is the average territorial value of this diagram?
We may ignore the ko threat and say that the territorial value is 6⅓.
But after white "b" we may also take into account the ko threat and decide for the remaining ko white black is komaster. In that case are we allowed to change the territorial value and how ?