I try to have a better understanding on double sente moves but it is not easy.
I open here a new item because it seems the item thermography is not really a good answer to my question.

Let's modelise a sente move in the simpliest way: let's call b the black gote threat, w the white gote threat and n the double sente basic value. I need now a base (the 0 value) for calculating the score. Let's take as a base the position after white plays double sente and black blocks immediately:
if black takes the double sente point and then black takes her ko threat, the score is : n + b
if whites takes the double sente point and then white takes her ko threat, the score is : -w
if black takes the double sente and white blocks, the score is : n
if white takes the double sente and black blocks, the score is : 0.

Let's write this double sente area ds(b, n, w);

In addition because I consider it is really a double sente point I assume also that b and w are both above the temperature t of the environment.

Now the problem is the following:
Taking several ds areas and an environment at temperature t what is the best method to find on which ds point I must play first?

We can make a first simplification. Because I assume all b and w values are above temperature t, the ds(b, n, w) will be completly resolved before playing in the environement => the problem is exactly the same if we replace the environment by a simple t gote.

What is the point?

Without any method, in order to find the best move to play, you may imagine all the possible sequences and you will of course find the best move. Surely it works. This method works perfectly with only two ds(b, n, w); with three ds(b, n, w) it looks already difficult and above four ds(b, n, w) it looks quite impossible (unless you are a computer OC).

In any case we have to find a method to choose the best move or maybe the probable (?) best move.

Computer algorithms tells us that in order to gain time for finding the best move it is a very good idea to estimate first each move by an appropate heuristic. Then you can explore the tree of possiblities beginning by the best move according to the heuristic defined and continuing by using an alpha-beta algorithm to prune large branches of the tree of possibilities.

Now you understand my question. What is the best heuristic to estimate a ds(b, n, w) ?

Assume black to play (the estimation may be differnent for black and white doesn't it?)
At first sight you can have at least four ideas:
1) estimate(db(b,n,w) = n
2) estimate(db(b,n,w) = n + (b+w)/2
3) estimate(db(b,n,w) = n + b
4) estimate(db(b,n,w) = n + b + εw

My preference goes to the fourth one with small value for ε but it is only a feeling without any justification

I am pretty sure such problem has been tackled in the past. Do we know some results?
Who has an idea to try and progress?

Yes. You cannot evaluate double sente that gain anything as double sente.

Prototypical double sente: Seki.

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Oops OC I can evaluate double sente Bill.
Let's take for example the following evaluation:
estimate(db(b,n,w)) = n + b
and let's simplify the problem by avoiding exploring the tree of possibilties.

Surely this estimation will guess the correct move in a lot of cases but sometimes this estimation will fail to give me the best move but anyway the guessing move may not be far from the real best move and (rarely?) the guessing move will be quite bad.
In any case this estimation is one possible evaluation and on average I fell it is not too bad.
Now the question is : is this evaluation, on average, the best one? If not how to improve it?
That is the point: for a theoritical point of view it must exist an evaluation which is, on average, the best one!
Surely this evaluation will sometimes fail to give the best move and you might conclude this evaluation does not exist but that is not the point.

Thermographically double sente form two vertical lines. Where do those lines intersect?

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

Visualize whirled peas.

Everything with love. Stay safe.

You find the answer of how to evaluate double sente moves here: https://www.lifein19x19.com/viewtopic.p ... 72#p260572

In an informal sense, double sente has a good pragmatic meaning. In a strict sense, simply speaking, local double sente does not exist, as Bill Spight and I have proved mathematically. I will publish the basic proofs (case analysis including some proof by contradiction for the most difficult case) later, but more research for a deeper understanding won't hurt.

A local endgame that 1) is without kos and without double ambiguity and 2) we naively perceive as local double sente tends to be a local gote if calculated carefully (although rarely we might find it to be one player's local sente). Therefore, the pragmatic evaluation is as if having a local gote, see the link above.

Remember my first post Bill, I claim thermography is not a good answer to my question and for that reason I opennned a new topic on this forum.
Yes thermography with its ideal environment made of gote points with good caracteristics put on double sente a value which is INFINITY.
That does not mean we cannot build a new tool (beside the thermograhy) which would be able to compare double sente moves. For sure we know in advance this new tool will not be perfect but thermography is not perfect.

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$$ | . X O O O O O |
$$ | . X X X O . . |
$$ | . . . X O O O |
$$ | . . . X X X X |
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When we analysed this position with thermography we found the best move for black is always (whatever the temperature) the monkey jump. But we also perfectly know that in the real life the keima move could be strictly better that the monkey jump.
Anyway thermography is a very useful tool which tells you that on average (I mean when taking millions of real environments) the monkey jump is really far better than the keima.

When searching a new tool for comparing double sente moves it is exactly the same. The goal is only to be able to compare two double sente moves in order to say on average you should play this double sente before that other one.
Sometimes it is obvious
For example you have surely:
db (13, 4, 16) > db (13, 4, 14) or
db (14, 4, 16) > db (13, 4, 16) or
db (13, 4, 16) > db (13, 3, 16) or also
db (13, 4, 16) > db (13, 3, 17) etc
but you will also find double sente incomparable just like monkey jump and keima in my previous example.
That does not harm if you can find a way to compare them on average (here again I mean when taking millions of real environments)

BTW I made some progress.
Taking a set of double sente moves and adding an "standard" environment at temperature t, made of simple gote areas we can proof that the player who has to move first will in any case, after having solved all double sente moves, take first the environment. As a consequence we can always in the analyse ignore this environment (or assume t=0).
This result is quite interesting indeed.
When you say double sente move value is INFINITY comparing to simple gote move you are right OC but you can formulate this in the other way: the value of a simple gote move is 0 comparing to the value of a double sente move.

As you see in this new tool built to compare double sente moves these moves will have finite values and all the simple gote move will have the value 0.

The Ogawa-Davies book on the endgame pointed out that problem in the 1970s, with no knowledge of thermography at all. It is thermography that makes sense of double sente. A position or combination of positions may be double sente at a certain range of temperature where both sides of the thermograph are vertical. (Edit: I.e., where each player normally plays with sente.) OC, that does not include the range at which we find the territorial count, if the double sente is gainful.

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

Visualize whirled peas.

Everything with love. Stay safe.

You are comparing moves to local endgames. Instead, you should compare local endgames (in which either player has moves) to local endgames (in which either player has moves).

For a set of local endgames with one player's simple follow-up, it is known that there is no general solution. We only have general solutions for subsets. Therefore, I expect the more complicated local endgames with both players' simple follow-ups to be at least as complicated, i.e., that there is no general solution for correct move orders, in particular not for several perceived double sentes with arbitrary values.

Therefore, please reformulate the problem you want to study!

OK let's try.

In almost all games, somewhere at the end of middle game or the beginning of the yose, any go player have experimented a very frustrating phase of the game in witch one player takes five or more double sente points for a total gain of 15 points or more, before beginning playing the "simple" gote points.
What I am looking for is "simply" a new tool allowing me to play this phase in the best possible way. Of course I know that a perfect tool does not exist but I am pretty sure we can find a good estimate fonction f(b, n, w) of a double sente move db(b, n, w) which compare the double sente moves between them and give the best chance to play the sente moves in the correct order.

To modelise these double sente moves my intention is to use the following tree for each double sente area:


with the following assumption:
taking the local score on each leaf, here D,E,F and G, I assume E-F quite small (typically under say 8) and D-E and F-G quite big (typically above say 10). The idea is to use a model as close as possible to what real games are facing on average. The idea is the same as the ideal environment in thermography but for the double sente moves.
After discussion this model will be adapted accordingly OC.

I already have some candidates for this estimate fonction:
f(b, n, w) = n
f(b, n, w) = n+b
f(b, n, w) = n+b+w
f(b, n, w) = n+b+εw
f(b, n, w) = n+b/2
etc. etc.
and I will soon be able to compare these fonctions on the model I have in mind.
Some more later. I have to explore a promising new idea to progress

While you have every right to study such a class of abstract positions, we should agree on terminology. You study a set of local endgames. For each such local endgame, we know (because we have proven it) that none is a local double sente.

I call such a "local endgame with simple follow-ups of both players". Each might be a local gote, Black's local sente or White's local sente. Of this type, you want to study a subset with significant "threats". In the global context, we then have a "perceived double sente", or "global double sente", i.e., that might be double sente in some typical global contexts.

Concerning your mentioned practical experience, I disagree that there would be ca. five global double sentes during the transition from middle game to early endgame - IMX, it can be zero to circa half a dozen. Furthermore, it is often wrong to let the player with the initiative play in all global double sente endgames in sente. Instead, a frequently correct strategy is that of mutual reduction! In practice, many mutual reductions increase complexity of available follow-ups. Therefore, before we can hope to understand well the general case in arbitrary positions, we might first study the abstract, simplified case of a set of / few local endgames with simple follow-ups of both players. First without any environment, then with some temperature T>0.

Hint: already a set of two local endgames with simple follow-ups of both players and without environment is demanding to study. (I know because we needed quite some effort for the much simpler study of two local endgames with one player's follow-up!)

I do not (yet) see value in your function f(b, n, w). You construct it from the assumption "if white takes the double sente and black blocks, the score is : 0" but this means that you only study a strange subset of local endgames with simple follow-ups of both players.

Furthermore, you need to clarify: "let's call b the black gote threat". What do you mean? The count? The move value? "Threat" is ambiguous; do you refer to the first move or the second successive move of the same player, the resulting count after the former or latter, or the move value of which? Etc. for your other variables.

A good example of where the belief in (intrinsic) double sente is detrimental. If they really exist, then the answer is plain. When your opponent plays the first double sente, then, instead of answering it, play one of the other double sente.

See this post with an example from a traditional book that was not bamboozled by double sente. https://www.lifein19x19.com/viewtopic.p ... 35#p194535

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

Visualize whirled peas.

Everything with love. Stay safe.

OC I try to take into account all what you say.
Following Robert suggestion I looked at only two local endgames.

Taking the tree I defined earlier with four leaves and local scores D, E, F, G
I defined
b = D-E
n = F-E
w = G-F
and I search an estimate value for this area (I daren't use the double sente wording!) with a function f(b, n, w).
In addition the idea is to consider areas for which n is rather small and b and w rather big.

For the time being I tested several functions against couples of local endgames according to the kind of tree above.
For that I start from the following examples (it is very very easy for me to change the overall model).
I take all the areas (b, n, w) we can build with n ∈ {1, 2, 3, 4, 5, 6, 7, 8} and b,w ∈ {8, 9, 10, 11, 12, 13, 14}. You immediatley calculate that the number of possible areas is 8 x 7 x 7 = 392 areas.
From that you can build 76 636 (= 392 x 391 / 2) couple of areas.
Each of these couples is quite easy to analyse in order to know which you have to choose for playing first.
After this analyse I find 67 998 couples for which it is relevant to choose the correct area to play first (for the other couples it is indifferent to begin by one or the other area of the couple).

With this sample of 67 998 area couples I tested various estimation function in order to mesure the pourcentage of good guess reached by this function.
Here are my first results:

f(b, n, w) = n => pourcentageOK = 81,47%
f(b, n, w) = n + b => pourcentageOK = 81,94%
f(b, n, w) = n + b/2 => pourcentageOK = 87,17%
f(b, n, w) = n + b + w => pourcentageOK = 87,93%
f(b, n, w) = n + 0.5b + 0.4w => pourcentageOK = 94,54%

Any idea to improve the model? the sample of area couples? the function itself?
I can easily test other configurations if you are interested in of course.

Bill: You can judge it much better than I can, so can I direct you to game 1703-01-11a in the GoGoD database. It seems to me both an example and a counter-example of what you say.

When Hayashi Genetsu played 128 he evidently expected Black to answer around there (as he did very soon after - Black 135), and equally clearly it a was reverse sente move in that he didn't want Black (Dochi) to around 128 himself.

But Dochi - only 13! - was alert and followed your advice. He played a different double sente: Black 129. But White had to answer that one: he couldn't sensibly go on a mutual damage rampage, even though a double sente in centre still existed. That is, he couldn't follow your advice! In short, 128 should first have been 129. But then Dochi would have had to answer in the corner - he couldn't divert to the double sente in the centre. Another counter-example?

But this was also a case of what you said in your other thread about pros being significantly better than even high-dan amateurs (or in this case a 4-dan pro) at the endgame. Both Sakata and Segoe picked up on this mistake. (For anyone else looking up the game, a White move at S4 is the killer threat.)

I don't follow more than a few percent of this discussion, but insofar as I do it seems clear to me that errors in thinking are occurring because some people are mixing up sente and the initiative. In the above game Dochi can ignore 128 because he has the initiative even though Hayashi has sente. I think Robert is making the same point but his language is too opaque for me to be sure. I'm more confident that you have made that sort of point before, but you usually scuttle down the rabbit hole of numbers before I can catch your bobtail. One word you use quite a bit leaves me queasy: infinity. My attempt to get a grasp on what is going on (lots of double sentes hovering around) is simply to say these are moves 'left in abeyance' and the timing of when to play them is determined by who has the initiative. But since the characteristic of having the initiative is that you get to decide how play proceeds, you may decide not to play any double sentes. E.g. if you are behind and need to make complications, you may prefer to 'leave them in abeyance' and keep them open for aji. It is not a simple question of size, especially because double sentes, by their nature, are often big enough to make complications matter.

BTW there is no special significance in the game above. I just happened to play through it last night.

John, the informal phrase "double sente" suggests that whoever plays first in a local endgame can do so in sente. We all know that this is wrong as a general statement. Next, we become aware that such might occur at a suitable time.

The term "local double sente" suggests that, when it is correct for either player to make a first local play, the opponent's correct reply would always be a local play. However, simply speaking, local double sente does not exist.

The term "global double sente" means that, when it is correct for either player to make a first local play, the opponent's correct reply would be a local play because the reply is the most valuable move in the global position.

For example, the first move in an alleged double sente endgame might have the move value 19 and the reply the move value 17. In almost all practically occurring positions, we would have a global double sente. However, a rare global position might have a tenuki with the move value 18 so that the first local play is ignored by the opponent playing elsewhere. The local endgame assessed correctly is a "local gote". Without any large tenuki on the board, the local gote endgame is played as a global double sente endgame.

OK. Given a game, H = {d|e||f|g}, with d > e > f > g, we have a simple test for sente and gote. Let x = (d+e+f+g)/4. If x > e, then the game is a Black sente. If f > x, then the game is a White sente. If e > x > f, then the game is a gote. If d-g >> e-f, the game is unlikely to be gote. That would be threading the needle.



OK. Let there also be another game, J = {p|q||r|s}, with p > q > r > s. And let there be an ideal environment with temperature t. Since H and J are considered to be double sente, let (d-e), (f-g), (p-q), and (r-s) all be greater than 2t. Thus, each will be played before a play in the environment, and the same goes for H and J.

It doesn't matter which side is to play, the lesson is basically the same. So let Black play first.

1) Let Black play in H first.

1a) Let White reply in H. Then the result after an even number of plays will be

e + q.

1b) Let White reply in J. Then we are left with {d|e} + {r|s}. Depending on which play is larger, our result will be

d + s, or
e + r.

2) Let Black play in J first.

2a) Let White reply in J. The result will be

e + q.

2b) Let White reply in H. Then the result will be either

p + g, or
q + f.

The environment does not play a role in this decision, as the reader may verify. Also, the result will be the same, whichever position Black plays first in, if White replies in it. So we treat the double sente as gote, to decide between them.

The question then becomes that of finding

max(max(d+s,e+r), max(p+g,q+f))

Black plays in H first if
(
d + s ≥ p + g, i.e., if
d - g ≥ p - s,

and

d + s ≥ q + f, i.e.,
d - f ≥ q - s
)
(Back around 50 years ago, when I solved this question, I called d - f the forward value for Black of playing in H. (I was still in the swing value camp back then.) And I called q - s the backward value for Black of playing in J.)

or if
(
e - g ≥ p - r

and

e - f ≥ p - q
)

OC, e - f and p - q were traditionally called the value of double sente for each play. Note that choosing which to play based upon that value alone is not correct. However, you could start with that comparison. Without loss of generality, let e - f ≥ p - q . Then if the backward value for H is greater than or equal to the forward value for J, play in H.

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

Visualize whirled peas.

Everything with love. Stay safe.

Gerard, your case counting reminds me of one of my sins. As a mathematically interested pupil, I joined a week's excursion to university. One of our homeworks was a puzzle of dozens of tiles to be proven or disproven to fit into a square filling it. The puzzle was just small enough for my evening proof by enlisting each of some thousand cases:) The professor was not amused:) Of course, we were supposed to do an abstract proof:):) (I did better with my other homework.)

Thanks, John, I'll check the game out this weekend.
As for countre-examples (British spelling ) I said that was so if intrinsic double sente really exist; but they don't.



Dochi was a genius, wasn't he?



Well, some pros, anyway.

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

Visualize whirled peas.

Everything with love. Stay safe.

I agree 100% with you Bill. I used the same calculation taking (b,n,w) instead of (d,e,f,g) which is exactly equivalent because you can always choose arbitrary the value of one of your four d,e,f,g values.
And this calculation allowed me to know which amongs two sente moves I have to play first. Then I used this correct result to compare to the result got by the estimation function.

The point is the following: if you have only two double sente areas on the board you can read all possibilities as you did and you will find for sure the correct order. Fine. But if you have 3, 4 or more double sente moves on the board it becomes extremely difficult to read all possibilities and my view is that it could very useful to have a tool guessing the correct order with a quite high probablity to be the best one. After having viewed the result of this sequence you may, with your experience, look for a better sequence but you will gain a lot of time by beginning with this good (not best) guessing. BTW if you know that the tool is quite a good one and if you manage to find a better one your are almost sure it is not necessary to search another better sequence (unless you are a pro OC).

In this and your Thermography thread you have suggested that having more than one global double sente on the board at the same time is common. Well, it is not uncommon to have a double sente arise on the board, but it is usually answered immediately, and if not, soon, after a Zwischenzug exchange. It is not common for it to be left unanswered for another one that is not the result of said Zwischenzug to arise. What does sometimes happen is a shared blindspot, so that a double sente is left unrecognized on the board for some time. That even happens to pros, as the bots now tell us. Then another on may arise and you have two. It is even rarer to have two shared blind spots. As a rule, the time to handle the kind of position you are suggesting is earlier in the play. (Patient: "Doctor, it hurts when I do this." Doctor: "Don't do that.") )

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

Visualize whirled peas.

Everything with love. Stay safe.

OK Bill you mean that the position you mentionned by the link https://www.lifein19x19.com/viewtopic.p ... 35#p194535 is quite unusual?