We guess, but are they just shape dictionaries (possibly with values per shape) or do they also contain lots of go theory beyond our Western knowledge?



For greater clarity: "once +V for the player and once -V for the opponent calculated from the player's perspective". I.e., this amounts to V - V = 0.

Using a probability semantics, the probability of a gote is set to 0.5, and the probability of a sente is set to 1 - epsilon. There is certainly room for fuzziness, but it has not been developed, AFAIK. I have seen positions where whether the play is sente or gote is a close question. E. g., if a play is sente the reverse sente gains 1.75 points, but if it is a gote a play gains 1.78 points. In that case the probability of playing the sente does not seem like 1 - epsilon.

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

Visualize whirled peas.

Everything with love. Stay safe.

My own preference would be to say that the sente gains 4 points and the reverse sente gains 1 point. However, there are good reasons to favor the 1 point value and just say that it is a 1 point sente. As a rule, the play is only urgent as the value of other plays approaches 1 point. The 4 point value does not mean very much.

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

Visualize whirled peas.

Everything with love. Stay safe.

Except perhaps with some ko positions, my evaluations of local positions and plays will give the same results as O Meien's, barring arithmetic errors.



My criticism of Kano is about his presenting this position

Click Here To Show Diagram Code

[go]$$ Double sente???
$$ --------------------
$$ | . . . . . . . O X .
$$ | . . . X O . . O X .
$$ | X X . X O . O O X .
$$ | . X . X X O . O X ,
$$ | X X X X O O O O X .
$$ | O O O O X X X X X .
$$ | . . . O O O . . . .
$$ | . . . . . . . . . .[/go]

as a double sente when it is, in fact, a 7 point sente for Black,

and for presenting this position

Click Here To Show Diagram Code

[go]$$B Two point double sente???
$$ -----------------------
$$ | . . . . . . . . . O .
$$ | . . . . . . . . . O .
$$ | X X X X X . O O O O .
$$ | . . . , . . . . . , .[/go]

as a double sente (assuming the existing stones to be alive and safe) when it is, in fact, a gote that gains around 3 2/3 points. (It may possibly be a White sente if I have made an error with the White follow-up after White plays the kosumi.)

Kano probably had a ghost writer, but he put his stamp on both of these examples.



It sounds like, as with the gote with the huge replies that gain 19 points and 17 points, it would be very likely that either player could make the play with sente. IMO, that is not worth quibbling about.



It sounds like these are also plays where the follow-ups gain more than 15 points. If so, not worth quibbling about.



It was my study of pro games that first made me question the idea of double sente, because the pros so often left what the textbooks call double sente unplayed or unanswered. The idea of double sente, understood correctly, is not a problem. The problem comes with identifying certain local positions and plays as double sente. Why do the books persist in doing so? Especially as the pros do not play that way? I can only guess. (One reason is that the pros do not follow the textbooks. But then, they are not expected to. ) O Meien's book has broken that mold, fortunately.



Oh, I identified the 7 point sente and the 20 point gote in 1975, some 19 years before I heard of combinatorial game theory. All it takes is the traditional go evaluation. I did not bother calculating the other example, as it was obviously not a double sente.



John, ask yourself this. If Bill is wrong, why doesn't O Meien include double sente in his book, like every other writer on yose? Double sente being such a useful cog and all. He does not need to, does he? As you point out, practical, global double sente arise in nearly every game, often more than once; they are played and answered, and nobody blinks an eye. They are not the problem. It is the textbooks that are the problem.



The 7 point sente position is so far from a practical double sente it's not funny. If you presented it to O Meien and asked him if it was a 7 point sente for Black, I am reasonably certain that he would agree, barring an error on my part. He would not say, Oh no, it is really a double sente. Why should it be given priority over other sente where the reverse sente gains around 7 points?

And if you presented the gote that gains 20 points, I doubt if he would disagree. He might well say, why bother with calculations, when it will almost always be double sente?

And if you presented the "two point double sente" position and asked if it was double sente, I hope that he would just laugh. Or he might just look at you funny, like Jujo did to me when we were kibitzing a game and I pointed out that one of the players had not responded to a "double sente".

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

Visualize whirled peas.

Everything with love. Stay safe.

Bill

Thank you for coming back to this, but I still feel we have a complete disconnect. Accordingly, to save precious time for both of us, I will make this my last post on the topic by presenting the evidence for other people to draw concusions. But as regards your points, I would say that your noticing that double sente plays were played at a different time from what you expected can (and should?) be easily explained by the caveats made by all the oriental writers (including Kano) as regards aji and timing.

First, what Kano says in his 1974 book about the “big” double sente.



He then goes on to single sente, but if I may highlight a couple of things in the above: (1) he is using “I” and so is not just signing off on a ghostwriter’s script; (2) he is clearly differentiating categories of double sente, and despite that he sticks to the term double sente; (3) he is using the word “rights” even before O Meien latched on to it.

Now from Kano’s section on TWO-POINT MOVES:



I still do not see anything objectionable there.

Moving on to O Meien, he is redefining things for his own purpose. He points out that when people say a move is worth X points it’s not clear whether they means by deiri counting or something else. So, for example, he defines gote X points to means specifically what others refer to as double gote. Similary, while he avoids double sente, he does imply it by his constant use of “rights” (Quote: If we were to give a definition of “right” here, it would be: if the next move is bigger than the move just played, that move has been your right.)

It is all rather like when RJ uses connect-1 and connect-2 that doesn’t stop the rest of us continuing to say connect if we prefer.

As regards what most of us would consider under the heading of double sente, O gives the following position:



White has just played A, and O says this was the only move because he has to prevent a Black play at B. He adds that this move at B would be worth less than 20 points in de-iri terms, but it is much bigger in reality because it creates very bad aji for White. Now O had to think about whether to block on the lower side. It was obviously a difficult decision because he spends 7 pages explaining why he ended up playing at C, so he didn’t regard it as double sente but we can easily see why any amateur would – but that’s not my main point. He may not use the term double sente but he does explicity say that a White jump into Black’s lower side territory would be “only a 7 point sente” (though he later modifies this to a 5 point sente”. But it’s not a sente because he’s prepared to ignore it, as he goes on to show. But he doesn’t advocate not calling it sente for that reason, no more than I think we should avoid calling Kano’s kind of positon a double sente. It’s accepted, it’s practical, and it’s not confusing. What is confusing are cases like the above game, and they are confusing not becauase of terminology or proverbs, but because go is difficult.

What it all boils down to for me is that you say tomayto and Kano says tomahto and you are saying is he wrong. I say it is just a preference. O Meien may seem to be calling it ketchup, but it’s a different product, and that doesn’t make Kano’s usage wrong either, even if the ketchup turns out be tastier.

I use neither, but your point is that some writers create advanced terminology and its applications. I disagree that the "rest of us" would continue to use old, simpler terms - instead, still a significant percentage rejects immediate application of such advanced terminology. Now, the question is who benefits more: those using the more detailed, modern theory or those using the simplistic, old theory. In the case of double sente, I am not sure yet.

As may be. After all, I was a 4 kyu. However, timing covers a lot of territory. Not to reply to a properly played sente may be correct, as a matter of timing, but it is unusual, normally something to note and explain. Nowhere did the game commentaries note that a player ignored a double sente or give a reason for playing elsewhere. That was part of what made it puzzling.



The smallest threat comes to 34 points deiri, which is larger than 30 points. Quite right.



I suppose that right is what the Go Players Almanac calls privilege. The concept, if not the term, long antedates Kano, because it is a defining characteristic of (kata) sente. The player whose sente it is has the privilege of playing it, which amounts to a very high likelihood (certainty) that he will get to do so.

Kano almost gets there. He recognizes a problem with double sente, one that requires coming up with a new term (hitsuzensei, certainty) to explain why one player is much more likely to make the play than the other. I think that he was caught up in the idea that the double hane-and-connect where each player has a follow-up is double sente. But in this case the follow-up for Black is a gote worth 34 points deiri while the follow-up for White is either a gote worth 6 points deiri or a 3 point sente (Edit: depending upon Black's play). There is just no comparison. Talk about certainty! If it were not for the double hane-and-connect, would Kano have called this a double sente?

Calling it a double sente is, IMO, confusing because it obscures the very meaningful asymmetry of this position. If you calculate that it is a 7 point sente, you see that, except in very unusual circumstances, Black would have to sit around while the temperature of the rest of the board dropped to a mere 3 points before White could play the hane and Black would reply. Consider the proverb about not ceding double sente to the opponent. White hardly has a choice in the matter, does he?

Be that as it may, recognizing that there is a problem with double sente was an advance. In 1974 every writer on yose was calling such positions double sente.



That is not reason enough for White to reply. Black's threat is obviously small. There is no good reason to think that is sente, much less double sente. The only reason to think that this is double sente is because "the double kosumi on the second line is double sente". That is how the concept of double sente leads people astray.




There is nothing inevitable about it. Yes, White's threat is larger than Black's was in the previous diagram, and on a lot of boards Black will reply, but is hardly inevitable. (At some point I calculated White's threat as gaining a little more than 3 points. It is a monkey jump, but without much of a follow-up. I could have misread the position or miscalculated, so that this is actually a White sente. But a double sente? Puleaze!)



It's bad enough to call this a 2 point double sente, but then to imply that it is among the largest plays is extremely misleading. Do not cede this kosumi to the opponent? Really?

OK, here is a little test. I admit that I am taking some risk, because I have not read everything out. If the position is normal enough, then the biggest play is also best. I have not checked that this position is normal enough. I hope that it is. Anyway, the largest play outside the Kano corner is a middling endgame play. If this kosumi really is a double sente, among the largest of plays, then it should be the right play. (Edit: Cast iron rule, right? ) Try playing it out and see if it is.

Click Here To Show Diagram Code

[go]$$ Either player to play. Is "a" the best play?
$$ -------------------------
$$ | . . . . . . . . . O O O |
$$ | . . . . . a . . . O O O |
$$ | X X X X X . O O O O . O |
$$ | . . . , X . O O . , . O |
$$ | X X X X X X . X O O O O |
$$ | X O O O O . O O O X X O |
$$ | X X X X X X X . X X X O |
$$ | X O O . O O O O O O O O |
$$ | X X X X X X . X X X O O |
$$ | O O O O O O O O O O O O |
$$ -------------------------[/go]

Edit: Territory scoring. No komi.

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

Visualize whirled peas.

Everything with love. Stay safe.

Bill was a bit short with the detailed value caclulations of one of his initial diagrams, so let me try to work out the details:

Click Here To Show Diagram Code

[go]$$W Dia. 1: Initial position
$$ -------------------
$$ . . O . . O X . . .
$$ . . O X X O X X . .
$$ . . O O X O O X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

The outer strings are assumed to be unconditionally alive. To be determined: territory count. We need to consider Black versus White moving first and the counts of the follow-up positions before we can determine the count of this position.

Click Here To Show Diagram Code

[go]$$W Dia. 2: Locale
$$ -------------------
$$ . . O C C W X . . .
$$ . . O X X W X X . .
$$ . . O O X W W X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

Black or White moves first. Let us start with Black moving first.

Click Here To Show Diagram Code

[go]$$B Dia. 3: Var. 1: Black starts
$$ -------------------
$$ . . O 1 . O X . . .
$$ . . O X X O X X . .
$$ . . O O X O O X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$W Dia. 4: Var. 1: Result
$$ -------------------
$$ . . O X . O X . . .
$$ . . O X X O X X . .
$$ . . O O X O O X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$W Dia. 5: Var. 1: Count = 9
$$ -------------------
$$ . . O X C W X . . .
$$ . . O X X W X X . .
$$ . . O O X W W X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

Now let us study White's start:

Click Here To Show Diagram Code

[go]$$W Dia. 6: Var. 2+3: White starts
$$ -------------------
$$ . . O 1 . O X . . .
$$ . . O X X O X X . .
$$ . . O O X O O X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$W Dia. 7: Var. 2+3: Intermediate position
$$ -------------------
$$ . . O O . O X . . .
$$ . . O X X O X X . .
$$ . . O O X O O X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

The count of this position is still unclear. We need to study Black versus White moving next.

Click Here To Show Diagram Code

[go]$$W Dia. 8: Var. 2: Black continues
$$ -------------------
$$ . . O O 2 O X . . .
$$ . . O X X O X X . .
$$ . . O O X O O X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$W Dia. 9: Var. 2: Result, prisoner difference = 4
$$ -------------------
$$ . . O O X . X . . .
$$ . . O X X . X X . .
$$ . . O O X . . X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$W Dia. 10: Var. 2: Prisoner difference = 4, count = 8
$$ -------------------
$$ . . O O X C X . . .
$$ . . O X X C X X . .
$$ . . O O X C C X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$B Dia. 11: Var. 3: White continues
$$ -------------------
$$ . . O O 2 O X . . .
$$ . . O X X O X X . .
$$ . . O O X O O X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$W Dia. 12: Var. 3: Result
$$ -------------------
$$ . . O O O O X . . .
$$ . . O X X O X X . .
$$ . . O O X O O X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$W Dia. 13: Var. 3: Count = 0
$$ -------------------
$$ . . O O O O X . . .
$$ . . O X X O X X . .
$$ . . O O X O O X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$W Dia. 14 = Dia. 7: Count = 4
$$ -------------------
$$ . . O O . O X . . .
$$ . . O X X O X X . .
$$ . . O O X O O X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

The count of Dia. 7 is calculated from the counts of the black follower B = 8 in Dia. 10 and the white follower W = 0 in Dia. 13 by C = (B + W) / 2 = (8 + 0) / 2 = 8 / 2 = 4. We divide by the tally 2 because the black follower is created by 1 excess black play and the white follower is created by 1 excess white play. We add B and W before dividing by 2 because the count of Dia. 7 shall be the average of the counts of the followers.

Click Here To Show Diagram Code

[go]$$W Dia. 15 = Dia. 1: Count = 6.5
$$ -------------------
$$ . . O . . O X . . .
$$ . . O X X O X X . .
$$ . . O O X O O X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

The count of Dia. 1 is calculated from the counts of the black follower B = 9 in Dia. 5 and the white follower W = 4 in Dia. 7 by C = (B + W) / 2 = (9 + 4) / 2 = 13 / 2 = 6.5. We divide by the tally 2 because the black follower is created by 1 excess black play and the white follower is created by 1 excess white play. We add B and W before dividing by 2 because the count of Dia. 1 shall be the average of the counts of the followers.

***

Now that we have the counts of every position, we can also determine the per move values of either player's move in one of the positions.

Firstly, let us determine the miai value of a move in Dia. 1 from the counts of the black follower B = 9 in Dia. 5 and the white follower W = 4 in Dia. 7 by M = (B - W) / 2 = (9 - 4) / 2 = 5 / 2 = 2.5. We divide by the tally 2 because the black follower is created by 1 excess black play and the white follower is created by 1 excess white play. We form the difference of B and W before dividing by 2 because, with the miai value of a move, we want to express how far the count of the Dia. 1 position is away either from the black follower in Dia. 5 or (same value distance) from the white follower in Dia. 7 (the value distance is the same because we have set the value of the Dia. 1 position, i.e. its count, in the average, middle value position in between the values of the followers).

From Dia. 1 with its count 6.5 the moving Black gains the move value +2.5 to create Dia. 5 with the count 6.5 + 2.5 = 9. From Dia. 1 with its count 6.5 the moving White gains the move value +2.5, which equals Black losing the move value -2.5, to create Dia. 7 with the count 6.5 - 2.5 = 4. These are just recalculations of already known values.

***

Interlude: From Dia. 5, there is no move on the board in the locale. Either player's pass would gain 0 points and keep the position's count the same.

***

Secondly, let us determine the miai value of a move in Dia. 7 from the counts of the black follower B = 8 in Dia. 10 and the white follower W = 0 in Dia. 13 by M = (B - W) / 2 = (8 - 0) / 2 = 8 / 2 = 4. We divide by the tally 2 because the black follower is created by 1 excess black play and the white follower is created by 1 excess white play. We form the difference of B and W before dividing by 2 because, with the miai value of a move, we want to express how far the count of the Dia. 7 position is away either from the black follower in Dia. 10 or (same value distance) from the white follower in Dia. 13 (the value distance is the same because we have set the value of the Dia. 7 position, i.e. its count, in the average, middle value position in between the values of the followers).

From Dia. 7 with its count 4 the moving Black gains the move value +4 to create Dia. 10 with the count 4 + 4 = 8. From Dia. 7 with its count 4 the moving White gains the move value +4, which equals Black losing the move value -4, to create Dia. 13 with the count 4 - 4 = 0. These are just recalculations of already known values.

***

Even if a count to be added or subtracted is 0, one must always perform the correct arithmetic operation: addition versus subtraction. While an error with 0 can be overlooked easily, an error with any value unequal 0 creates further wrong values.

***

Now let us speak Bill again:

My previous calculations have been done as if everything was gote. However, the gote value of White's initial play is 2.5 while the gote value of Black's answer play is 4. From a value perspective, a local position is called a 'sente' if the value of the answer is greater than the value of the initial move. We have a sente here. All the previous calculations have mainly served the purpose of identifying the sente nature for White of the initial position.

The count of a sente position is calculated differently from the calculation of a gote position: the sente sequence starting with the move of the sente player is imagined and creates the related follower (the thereby created position) in Dia. 10. Its gote count we know and may use: it is 8. The initial position "inherits" (well, it is the opposite direction, but the word fits) this count of the sente follower. Therefore, the sente count of Dia. 1 is 8.

Now, how do we determine the value of a move in the initial sente position in Dia. 1? It is also inherited. However, we inherit it from the value of a move in Dia. 7, of which we know the move value because this position, if it were an initial position, is a gote position. Since the value of a move in Dia. 7 is 4, Dia. 1 inherits this move value 4.

Only now it is possible to say that White 1 gains 4 for White (i.e. loses 4 for Black) and Black 2 gains 4 for Black. The net effect (from Black's perspective) is -4 + 4 = 0.

If White were allowed two successive local moves, each would have the black value -4, so -4 * 2 = -8 is the total value of both white plays. Since the initial position's sente count is 8, the effect on the count is 8 - 8 = 0, and this is the count of the position in Dia. 13 created by two successive white plays.

Is everything correct or have I not understood something?

Remember that there is yet another value: the value of Black's reverse sente play in the initial position. This value is calculated differently.

The discontinuity in the calculated value of a position or a move, on crossing a "sente" threshold (followup worth more than initial move), seems like a pretty good indication that this is just one approximation scheme, with nothing mathematically definitive about it.

No, these values are not just approximations. For instance,

Click Here To Show Diagram Code

[go]1.25 point position
$$ _ X X X X O _
$$ _ X . . . O _
$$ _ X X X X O _[/go]

The territory value of this corridor is 1.25 points. That is not just an estimate. To see that, let us evaluate 4 copies of that position by play.

Click Here To Show Diagram Code

[go]$$B 4 copies, Black first
$$ _ X X X X O _ X X X X O _
$$ _ X . . 1 O _ X . 5 2 O _
$$ _ X X X X O _ X X X X O _
$$ _ X . . 3 O _ X . 6 4 O _
$$ _ X X X X O _ X X X X O _[/go]

If Black plays first, she gets 5 points.

Click Here To Show Diagram Code

[go]$$W 4 copies, White first
$$ _ X X X X O _ X X X X O _
$$ _ X . 5 1 O _ X . . 2 O _
$$ _ X X X X O _ X X X X O _
$$ _ X . 6 3 O _ X . . 4 O _
$$ _ X X X X O _ X X X X O _[/go]

If White plays first, Black still gets 5 points.

The 4 copies are miai, with a total value of 5 points, and each copy is worth 1.25 points.

With mere estimates, the error increases with the number of copies. With these values, the error stays the same or decreases.

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

Visualize whirled peas.

Everything with love. Stay safe.

Now let's look at a sente position.

Click Here To Show Diagram Code

[go]4 point position
$$ _ X X X X X O _
$$ _ X O O . . O _
$$ _ X X X X X O _[/go]

The territory value of this corridor is 4 points. That is not just an estimate. To see that, let us evaluate 4 copies of that position by play.

Click Here To Show Diagram Code

[go]$$W 4 copies, White first
$$ _ X X X X X O _ X X X X X O _
$$ _ X O O 2 1 O _ X O O 6 5 O _
$$ _ X X X X X O _ X X X X X O _
$$ _ X O O 4 3 O _ X O O 8 7 O _
$$ _ X X X X X O _ X X X X X O _[/go]

If White plays first, Black gets 16 points.

Click Here To Show Diagram Code

[go]$$B 4 copies, Black first
$$ _ X X X X X O _ X X X X X O _
$$ _ X O O . 1 O _ X O O 5 4 O _
$$ _ X X X X X O _ X X X X X O _
$$ _ X O O 3 2 O _ X O O 7 6 O _
$$ _ X X X X X O _ X X X X X O _[/go]

If Black plays first, she gets 17 points.

The average is 16.5 points, 4.125 points per copy. Multiple copies of sente never become miai, but each new copy adds 4 points to the totals. That justifies the average value of 4 points for this position. No other numerical value makes sense.

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

Visualize whirled peas.

Everything with love. Stay safe.

Choosing 4 multiples is arbitrary. How about choosing n multiples? If V is the average value per copy and the starting player is Black, then lim (n->oo) V = 4. However, I find this justification unnecessarily complicated. Shouldn't we better explain carefully how just 1 copy works?

Actually, there is no discontinuity in the value of the position upon crossing the threshold to become sente. The value changes smoothly as the threat increases without ever jumping - it merely stops increasing further once the move become sente. See example game trees below - moving downward and to the left represents a black play, moving downward and to the right represents a white play, leaves are labeled with the net score of that position from white's perspective.

A is double-gote, value at A = 0.5, temperature at A = 0.5


A is double-gote, value at A = 0.75, temperature at A = 0.75


A is right on the border of being sente for white, value at A = 1.0, temperature at A = 1.0, white's followup threat is worth 1.0.


A is sente for white, value at A = 1.0, temperature at A = 1.0, but white's move is forcing as long as elsewhere moves are worth less than 1.5.


A is sente for white, value at A = 1.0, temperature at A = 1.0, but white's move is forcing as long as elsewhere moves are worth less than 2.0.


In general for x >= 1, the value and temperature at A is min(1,(x+1)/4). The move is sente when the "1" side of the min is the constraining side, and the move is gote when the "(x+1)/4" side of the min is the constraining side. This makes sense - the position's value should worsen gradually for black as white's followup threat increases, but only so much. It can't be worse for black than 1 because he can commit to always responding whenever white plays and thereby always achieve a value <= 1 here, and unless ko is involved, doing so costs black nothing on the rest of the board.


Another sign that these values are not arbitrary is that they are *exact* under Token Go / Environmental Go (http://senseis.xmp.net/?TokenGo) in the limit as you have a large number of tokens that are infinitesimally spaced apart in value and there is no ko. This variant is simply normal go except there is also a pile of tokens that are worth different numbers of points, and on your turn rather than playing a move or passing, you may take one of the tokens. In this limit under this variant, the optimal move is always to play the move with the largest possible urgency or temperature as calculated in this way, counting also each token as a play with temperature or urgency equal to the value of the token. And the result of the game under optimal play is precisely the sum of the values of the positions calculated in this way, plus half of the largest move for whoever has sente at the start.

Considering that the effect of the tokens is merely to remove the value of tedomari, and that seemingly any reasonable method of assigning (real-numbered) values to positions in a completely local way will also have trouble with tedomari and ko, this should give one some strong suspicions that this counting method is in some sense the "unique best" such method (up to scaling by constants).

Working on lightvector's trees. He specifies "leaves are labeled with the net score of that position from white's perspective". I.e., contrary to ordinary convention, positive counts favour White in the analyses below. Expressed in ordinary convention, the leaves would have negative numbers and counts be calculated accordingly.

Given tree:



Gote counts C (first numbers after colons):



Gote move values M (second numbers):



MA = 0.5 > MB = 0

Therefore A is gote.

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Given tree:



Gote counts:



Gote move values:



MA = 0.75 > MB = 0.5

Therefore A is gote.

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Given tree:



Gote counts:



Gote move values:



MA = 1 = MB = 1

Therefore A is ambiguous sente for White and gote for either player. lightvector says: "A is right on the border of being sente for white." Since this is also a sente for White, alternatively the values CA and MA can be determined in the manner of sente value calculation:

Kept gote move values:



Inherited count CA with I being the source of inheritance:



Inherited move value MA:



Note that we have got the same values as by the gote move value calculation.

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Given tree:



Gote counts:



Gote move values:



MA = 1.25 < MB = 1.5

Therefore the assumption of A being gote has been wrong; A is sente for White. We need to proceed with the sente calculation for A:

Kept gote move values:



Inherited count CA:



Inherited move value MA:



lightvector says: "[...]temperature at A = 1.0, but white's move is forcing as long as elsewhere moves are worth less than 1.5." Here, I am lost in my attempt to understand his calculation of the (local) temperature at A, i.e., the move value at A, which I abbreviate MA. I think that MA = 1.5 because two successive plays by White, each gaining 1.5 for him, result in 4 because 1 + 2*1.5 = 4. What do I or lightvector not understand here?

LightVector, thanks for the clarification and demonstration that there is no sente discontinuity. Given the internal definition of sente, I see your counting method makes sense and is well behaved.

However, it is still true that the value of a position changes abruptly, depending on whether it is deemed sente or gote. So if I want to inject non-local information about values of moves elsewhere on the board, I need to know more than just the value and temperature of the local position and move under consideration; I also need to remember the value of the followup move under the gote assumption.

Just to make sure I got it all, here is one more example, slightly more extreme than the cases shown above, labeled with position and move values.

What you have found is the temperature of the environment (how much the largest gote in it gains) at which White can play the sente. In that case, it will form a miai with the largest gote, so Black will not have to reply, but it is still correct play. However, Black cannot afford to make the local play (reverse sente), because it gains only 1 point. When the size of the largest gote equals the local temperature, both players must be indifferent between making the local play and playing in the environment.

For non-kos you can define the local temperature as the smallest temperature of the environment at which both players are indifferent between playing locally and playing in the environment.

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

Visualize whirled peas.

Everything with love. Stay safe.

So the local temperature IS defined as the smallest ambient temperature (the temperature of the enviroment) at which both players are indifferent between playing locally and playing in the environment? If so, why is it defined like this? Is this local temperature the same as the local miai value of a local move? The miai value is supposed to be a particular, fixed number while temperatures can vary. It seems that there is a range of "local" temperatures at which both players are indifferent between playing locally and playing in the environment. This ranges varies from the smallest value, which appears to be called the "local temparature", to the greatest value. Is the greatest such value always (except for ko considerations) equal to the gote move value at the intermediate position's tree node on the tree's currently considered sente path?

And then I would have more questions about the nature of the environment, but I postpone them for now:) Already the questions above are enough I do not understand at the moment.

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In my next attempt at understanding value calculation of sente, I visit the sente example on this webpage http://senseis.xmp.net/?MiaiCountingWithTrees and learn a new way of determining the local temperature: pruning those parts of the tree dominated by sente sequences. I do not know if this method is correct. However, since it looks like another interesting candidate method, let me try to apply it. Please clarify if this method is correct and why.

Now I use the usual convention of positive values favouring Black.

Given tree:



Gote counts:



Gote move values:

Note that move values are expressed as positive numbers. If Black plays a move, he gains this value and the count changes accordingly when moving from a position to its child position. If White plays a move, he gains the same value, but this gain for him is Black's loss so that Black's gain is the negative of this value.



MA = 1.25 < MB = 1.5

Therefore the assumption of A being gote has been wrong; A is sente for White. We need to proceed with the sente calculation for A:

Reset given tree:



Pruning the tree:




Inherited count CA:



Note that B does not have any sente count at all.

Sente move value:



This calculation (not an inheritance) is a bit tricky. Of the position A, we have the 0 leaf as the black follower and the -1 leaf as the white follower. From A to the black follower, the number of black excess moves is b = 1. From A to the white follower, the number of white excess moves is w = 0 (*). The miai value of a move in A is defined as swing divided by tally. The swing is the difference of the followers' values: 0 - (-1) = 1. The tally is the sum b+w of the numbers of excess moves to the followers: 1 + 0 = 1. Hence, we get the sente miai move value at A = swing / tally = 1 / 1 = 1.

Provided this is the right kind of calculation, this caluclation is correct and the miai value equals the local temperature, I would now understand why lightvector says: "[...] temperature at A = 1.0".

Summary of the pruning method:

1) prune the tree along the sente sequence except for its start and end
2) inherit the count from the sente follower
3) calculate the sente miai move value as swing / tally along the sente sequence

(*) Note for those who do not get it: When a tree bends rightwards, it is White's move; when a tree bends leftwards, it is Black's move. On the sente way from A via B to the -1 leaf, White makes the first move, then Black makes the second move. Each player makes one move, so the number of White's excess moves (expressed from Black's perspective) is -1 + 1 = 0.

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lightvector also says: "white's move is forcing as long as elsewhere moves are worth less than 1.5." To understand this at least from its local game tree perspective, we need to consider both the gote move values and the sente move values:



White would play in A because he considers his move with its sente move value 1 forcing. After White's move, we are at B. There, Black must decide if he should accept White's move as a sente or if Black should play a gote move elsewhere. In the latter case, Black's reference is the gote move value 1.5 at B. If Black finds elsewhere a move with at least the gote value 1.5, he can (if greater than 1.5, should) play. Else, if all gote moves elsewhere have a gote value smaller than 1.5, then Black must accept White's local move from A to B as forcing, reply and move to the leaf -1.

Have I now understood correctly what lightvector has been saying and what Bill seems to be referring to when mentioning the ambient temperature?


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Earlier viewtopic.php?p=178175#p178175 I wrote: "how do we determine the value of a move in the initial sente position in Dia. 1? It is also inherited. However, we inherit it from the value of a move in Dia. 7, of which we know the move value because this position, if it were an initial position, is a gote position. Since the value of a move in Dia. 7 is 4, Dia. 1 inherits this move value 4."

Although Bill Spight and lightvector "liked" that post, I am still unsure about the correct calculation of the local temperature and local sente miai move value. The citation relies on inheriting the gote move value, but isn't the pruning method discussed above correct?

Now assuming the pruning method is correct, I correct (?) the calculation of the sente miai move value for the example in viewtopic.php?p=178128#p178128

Click Here To Show Diagram Code

[go]$$W Initial position
$$ -------------------
$$ . . O . . O X . . .
$$ . . O X X O X X . .
$$ . . O O X O O X . .
$$ . . . . X X X X . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .[/go]

Given tree:



Pruning the tree:



Inherited count CA:



Sente move value:



Swing = 9 - 8 = 1.
Tally = 1 + (-1 + 1) = 1 + 0 = 1.
Sente move value = swing / tally = 1 / 1 = 1.

Oops. This may be the reverse sente move value, but it is certainly not what Bill would call the sente move value.

So why is the method of inheriting the move value correct in some examples and is the pruning method correct in other examples? What else do I not understand here? There is more: I do not have the slightest idea yet when and why min or max comes into play. I have a suspicion about an explanation for Bill's (a+b+c+d)/4 tems, but I am not sure yet what exactly I need to ask, except that it migh be related to my confusion about which is the right method.

I defined the count and local temperature in terms of an ideal environment in 1998. Doing so produced the same count and miai value as traditional go evaluation, and it produced the same count and local temperature as the earlier definitions of combinatorial game theory. In addition, doing so allowed me to find the count and temperature of multiple kos and superkos, which earlier methods could not evaluate in general.

Informally, we are interested in when to make the local play. Experience has shown that an ideal environment of simple gote is in general sufficiently like the real environment of other plays on a go board that knowing when to play locally in an ideal environment usually tells us when to play locally in a real game. Even if we lose nothing by playing locally, we do not have to make a play until the opponent threatens to make a local play. That threat is real when the temperature of the environment drops below a certain value. That is why we take the minimum value where both players are indifferent between a local play and a play in the environment as the local temperature. It is the crucial value.



Positions in which there is a range of temperatures at which both players are indifferent to playing locally or in the environment are in a decided minority. The most frequent examples, I think, are double ko threats, miai in which either player can play with sente.



I am not sure that I understand you, but for there to be a range of local temperatures, at the high end both players must be able to play correctly with sente. So the reply must be gote or reverse sente.



To apply this method you have to know that the play or sequence is sente. Often you can tell by looking, but telling by looking is how people came to believe that there are local double sente that gain points.



OK, you have identified this position as a White sente. Good.



That is O Meien's approach. Not that he prunes trees, but he does not assign a value to sente plays or sequences.




Well, it is important to understand that he and I are positing an ideal environment. In an ideal environment the hottest play is always correct. That is generally true in go, but not always. Berlekamp showed how, for any go position, it is always possible to construct a finite ideal environment. For the position under discussion, {0 || -1 | -4}, such an environment would be the simple gote positions, {0.5 | -0.5}, {1 | -1}, {1.5 | -1.5}, {2 | -2}. This environment has a temperature of 2, and each player prefers to play in the environment. After the top play has been taken, the temperature has dropped to 1.5. Now White is indifferent between playing locally or in the environment, but Black prefers to play in the environment. Assuming that the 1.5 point gote has been taken, the temperature has dropped to 1. Now each player is indifferent between playing locally or playing in the environment. Assuming that the top, 1 point gote has been taken, the temperature of the environment has dropped to 0.5. Now Black is indifferent between the local play and the play in the environment, gaining 0.5 either way, but White prefers to make the local play. So the local temperature is above 0.5. It is, in fact, 1.



It is the traditional "sente value" and the local temperature. I prefer to emphasize the asymmetry of sente, to show that the sente move is normally answered. The pruning method "forgets" the value of the reply and the value of the sente play. You do not need them to find the count of the position or the local temperature.



Both are correct. But the pruning method depends upon correctly identifying sente and does not retain what may be important information under some circumstances.



Outside of go, minimax play is used to determine the value of a position, with a certain player to play. Go evaluation does not depend upon who has the move. However, I discovered how to use minimax play in an ideal environment at a certain temperature to evaluate positions. That explains why the equation has minimax values with a temperature term. Best play in general depends upon temperature. Equality means that it does not matter who plays first. (One side of the equation shows the minimax value when Black plays first and the other side shows the minimax value when White plays first.)

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

Visualize whirled peas.

Everything with love. Stay safe.

The close relation to CGT (and, in some sense, to traditional evaluation) and the ko generalisation are very good reasons and, it seems, more relevant than relying on only traditional evaluation.

The following is important enough to be presented well for ordinary players (not now, but maybe in 2015?):
- Your definition of the count and local temperature in terms of an ideal environment.
- Your explanation why it produced the same count and local temperature as the earlier CGT definitions.
- Your general evaluation of the count and temperature of multiple kos and superkos.
- Your minimax method in an ideal environment at a certain temperature to evaluate positions.
- Berlekamp's construction of a finite ideal environment for any go position.



Would you like to report your experience? Most of us have not studied play in ideal environments and we lack related experience. Since you have it, it must be very useful for us to learn from it.



This sounds a bit as if always the environment would reach an ambient temperature equalling the local temperature, however, this need not be the case:) It is more common in ideal, dense environments of arbitrarily many tokens of "all" values, I imagine.



Surprise. I imagined something like this: local temperature 1.5, gote moves elsewhere being worth 2.5, 2.0, 1.4, 1.0, 0.6. Then, for the range of temperatures smaller than 2.0 and at least 1.5, both players are indifferent to playing locally or in the environment. There is always such a small range of temperatures between the local temperature and the miai value of the next bigger gote in the environment. Therefore, I do not understand why you say that such positions were in a decided minority. What exactly do you mean?



Pruning is optional. Instead, one can also "simply" follow the sente path and "ignore" its intermediate siblings.



As I understand from what you are saying, this sente move value is a sente move value FOR WHITE, but not for Black.

Thus, a gote move value at a node is a move value FOR EITHER PLAYER while a sente move value at a node is a move value FOR ONLY THE PLAYER FOR WHOM THE MOVE IS SENTE.

Similarly, the reverse sente move value at a node is a move value for only the opponent, for whom the move is reverse sente.

Have I understood this correctly?

(Reminder: Above and below, I understand your value and preference of where to play arguments informally, but I still need to work out why indeed this is so in terms of actual value calculations.)





I think tradition considered both values (or related base values such as the leaves' counts) but did not always know exactly their sizes, meaning and relation, or confused everything and then resorted to considering only part of the value framework.



Many thanks.