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

Visualize whirled peas.

Everything with love. Stay safe.

I didn't mean exactly in that simple tree, but thought generally about the implication that if W moves to the same (or even worse) value than the EV of the original position, that line can be excluded (but maybe I misunderstood, and this is not a general principle?). So, maybe something similar to the original tedomari position, where W's sente resolves the position completely, but B's sente leaves a double gote followup.

While we are at it, let's show that A = D, below, where D is the same as A, but without the move to C.


I have added a ' to -BIG because it is not necessarily the negative of BIG. So let's subtract D from A and show that A - D = 0.


Plainly, the second player can mirror any play in -D, and guarantee a final score of 0, and any play in A except White's move to C. That's all we have to look at.

Let White play from A to C. Then Black can reply from C to 0. Then White can only play from -D to -B, and Black replies to 0. So A - D = 0. QED.

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

Visualize whirled peas.

Everything with love. Stay safe.

Well, as long as we are talking about averages, which is what I think you mean by EV, we have a lot of leeway. Like changing 0 to D ({1 | -1} in CGT notation}, which has the same average value, justified retaining C. There are plenty of situations where playing a losing sente is correct, that is, a sente that loses pts. by comparison with the average evaluation. But if all we have are average values, then the best we can normally do is to make the largest play, and if we have a choice, choose at random. But if we can make strict comparisons, we may be able to choose correctly between plays without reading the whole board out.

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

Visualize whirled peas.

Everything with love. Stay safe.

Bernd Gramlich, writing under his own name, posted a note on 2003/07/15 with this whole board position.



It contains a natural button. I think that I had pointed out that possibility earlier in the discussion.

Edit: Looking again, Bernd and I discussed this position on the 15th. But I am quoting him, so he posted the original a little earlier, I think.

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

Visualize whirled peas.

Everything with love. Stay safe.

I count the stones, I count the stones
(with no apologies to Barry Manilow )

A very recent post by John Fairbairn ( forum/viewtopic.php?f=15&t=14556&p=223379#p223379 ) in which he references Mizokami's ideas about counting the number of stones of each player in one half of the board or other and using that in making the decision of where to play. Sonoda also uses the relative stone count as a heuristic, also on a large scale. John is familiar with Sonoda, as well. It would be interesting if he compared the two stone count heuristics.

The explicit use of stone count as a heuristic is relatively new, I think. I have even noticed Michael Redmond mention stone count in passing a few times in his videos. I think that pros used it implicitly and unconsciously well before that. It is such a simple heuristic that it is almost embarrassing to talk about. Why, even beginners can count stones. Obviously, it needs to be supplemented by other heuristics and by reading.

I stumbled across stone count as a heuristic in the 1990s, when I was trying to classify go moves and positions in a very general way, so that it would be easy to program a computer to use the classification scheme to generate statistics. Initially I used the number of Black stones and the number of White stones. It quickly became apparent, even without using a computer, that most top level plays were clustered near the line where the number of Black stones and White stones were equal. I concluded that a better classification scheme was the sum and difference of Black and White stones. (I did not look at positions as large as half the go board, however. )

I derived two quite general heuristics using stone counts. Both can be used by beginners. Greatly to their advantage, I think. The first, and stronger heuristic is to play where each player has approximately the same number of stones. (Yes, this can be used to justify following the opponent around. ) The second heuristic is to play where there are fewer total stones. (Yes, this can be used to justify playing "where the stone makes the loudest sound." )

Now, neither heuristic mentions efficiency or urgency or vital points or the strength and weakness of groups, or anything else. If you make a play that kills a large group, it probably violates both. These heuristics are very basic, and probably more susceptible to exceptions than other heuristics, but I do think that it is a good idea to keep both the total number of stones and the difference between the number of Black and White stones explicitly in mind.

In the note I linked to above, John Fairbairn makes an interesting observation. (Emphasis mine.)

It seems to me that, as long as the two players are evenly matched, the relative number of stones in a region will reflect their relative influence. With the caveats that the stones of weaker players generally have weaker influence, and that the weaker the players, the greater the variation in efficiency and the greater the variation in the strength of stones.

Keeping in mind the weaknesses of stone count heuristics, I do think that they provide some good lessons for amateurs, and not just beginners. I was able to subdivide the strongest heuristic, the one about the stone difference, into ones about specific differences.

The most frequent types of plays were those that either added one stone to an even count, to go one stone ahead, or added one stone to make an even count, to catch up. The next most frequent types were those that went two stones ahead or added a stone to get one stone behind.

Getting three stones ahead risks overconcentration. And playing where you are badly outnumbered is not always a good idea, either.

IMO, the larger the region being considered, the less valid these heuristics are. Does anybody really think that the Chinese opening is overconcentrated? (Although playing too much on one side in the opening is questionable.)

The second heuristic, about the total number of stones, is weaker, but is also one that many amateurs don't believe. As a rule, they think, the more stones in an area, the more urgent it becomes, unless the stones are secure. If you are pincered, you have to do something, right? Actually, things are just the opposite. In general, the more stones in an area, the less urgent it becomes, unless stones are weak or heavy. Amateurs do not tenuki often enough, and, to judge by the AlphaGo vs. AlphaGo games, maybe pros don't either.

These heuristics are weak, so it may sound like I am making stronger claims that I actually am. But I do think that there is food for thought in them.

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

Visualize whirled peas.

Everything with love. Stay safe.

Bill: You are right but I'd like to point out that Mizokami says rather more about the topic than I've revealed here, and so with his guidance you can make the heuristic work much better. In other words, you still have to buy the book, chaps! The fascinating part is knowing how to halve the board - at least I didn't make the right choice in several cases.

There are several of these finger-countable heuristics floating around and I'm often surprised at how little known they seem to be. Redmond mentioned one in (I think) No. 8 of his AlphaGo commentaries: you can often count the game tolerably well by counting the number of groups. It's a very long time since I heard that! (London Go Centre, I think.) It was interesting that he also stressed that you very, very often can't count the game accurately. Readers here may recall that I've banged on quite often about pros not counting the game very much (at least in the way amateurs talk about it - x points of territory) and instead rely on heuristics such as counting the relative number of inefficient plays (works well for them because most of their plays are efficient, of course, but the idea is applicable by dan players, I think).

Another one that is not as well known as it should be is "five alive" (my name, for the principle that in a contact fight you can treat a group as safe if it has five liberties.). I came across this in Korean a long time ago, but apart from Bruce Wilcox I've not seen anyone else mention it. In fact, I've never even come across it in Japanese, but the Chinese seem to have several similar heuristics that have come down from classical people like Zhou Donghou, though my memory is a bit frail on these things now.

None of these heuristics are foolproof but they seem especially valuable for amateurs who mostly tend to play very quick games. The ones you've devised also seem like a first-rate addition to their armoury.

I went to a lecture at the US Go Congress this year in August, taught by On Sojin. I've been trying to take notes on things, since I forget so easily.

Here is an excerpt from my notes from his lecture:



Basically, as I recall, when he's evaluating a position, one of the first things he considers is the number of stones involved. Comparing the relative size of territory, influence, strength/weakness of stones, etc., can be evaluated with that in mind.

This is slightly different than the idea of counting stones to evaluate play on a large scale, like dividing the board in half. It was more to identify local benefit that a given player might have achieved (I think).

As a side note, which I thought was interesting, he didn't seem to care as much about sente as I would have imagined. He was of the mind that, if you get more than your opponent in a local area by taking gote, it's often good enough. I'm not sure if this aligns with AlphaGo, which seems to like taking sente aggressively, but it's still interesting to think about.

_________________
be immersed

Fuzzy go positions and comparisons

OC, go positions aren't fuzzy. What may be fuzzy are the game in the position and its evaluation.

Click Here To Show Diagram Code

[go]$$Bc 1 fuzzy point
$$ -------------------
$$ | . . X . . .
$$ | . X X X . .
$$ | O O O X X .
$$ | . . O O . .
$$ | . . . . . .[/go]

We evaluate this corner as 1 pt. for Black. (Assuming, per convention, that the Black and White stones are alive.) But, OC, the local score will either be 0 or 2, depending upon the play. One way to look at the value of 1 pt. is as an expected value. We don't know who will play in the corner, so we reckon the odds as 50:50 and calculate the value as 0/2 + 2/2 = 1.

So far, so good. We can calculate gote on a 50:50 basis, and calculate sente using a probability of 1 - ε, to indicate a probability very close to 1 that the sente will be played (with sente). This works for individual positions.

But it doesn't work for combinations.

Click Here To Show Diagram Code

[go]$$Bc 2 points (miai)
$$ -----------------------
$$ | . . X . . O O . . . .
$$ | . X X X . X O . O . .
$$ | O O O X X X X O . . .
$$ | . . O O O O X X O . .
$$ | . . O X X X X O O . .
$$ | . . O O O . X X O . .
$$ | . . . . . . . . . . .[/go]

In terms of probability, the expected value for Black in the corner is 2, which we can calculate as 0/4 + 2/2 + 4/4 = 2. But, OC, as go players we can consider the two regions in play as miai, with a value of 2 pts. Either player can guarantee at value of 2, no matter who plays first. Black can guarantee at least 2 pts. for Black and White can guarantee at most 2 pts. The exception occurs in a ko fight where one player plays twice in the ko while the other player plays twice in the corner.

Now, fuzzy numbers do not have to add up to a precise value. But combinatorial games can. And indeed, this miai is the sum of two combinatorial games. We may write the equation this way.

Finding the count for a position may be regarded as a process of defuzzification. (Not that we actually say that, but I just did. ) I am going on like this in part because I find it interesting, but in part because I believe that the only known theoretically valid way of evaluating go positions is the one that humans use, at least when we are able to do the calculations, using fuzziness, not probability.

Now, the mean value of A is 1. And, obviously, we can't say that A > 1, nor can we say that A < 1. We say that A is confused with 1. A is also confused with ½. The mean value of A is greater than ½, but A itself is not. But obviously A < 3 and A > -2, so we can make strict comparisons between fuzzy games and numbers.

Is A confused with 0? That is not exactly obvious, but according to CGT, it is. We phrase the question this way. Is A ≥ 0? It is if White plays first in A and Black can reply to a score of at least 0. Well, White plays from A to 0, but then Black has no reply.

What about this sente?


Well, we know that it is not greater than 0. Is it confused with 0? Is it less than 0? Answer to come soon.

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

Visualize whirled peas.

Everything with love. Stay safe.

Does sente really gain nothing?

What about this sente?


Well, we know that it is not greater than 0. Is it confused with 0? Is it less than 0? Well, if Black plays first and White can reply to a score of 0 or less then S ≤ 0. Plainly that is the case. S obviously is not equal to 0, so it is less than 0. And that means that playing sente in S gains something.

How much does it gain? What it gains is the difference between 0 and S, or -S.

Well, that's a big help.

But it shows that the gain from playing this sente is not a number, but a game. The count of S is 0, and that is the same as the final result of playing the sente. That's what the saying, Sente gains nothing, means. The count stays the same. (Remember that combinatorial game theory is only decades old. The saying came before that.) In ordinary parlance we say that what sente gains is that it takes away the reverse sente option from the opponent. But sente gains no points. Nada.

OK, now what about this game? Does sente gain anything?

That's easy. No, it doesn't. As we know, A = D, and the sente gains D - A = 0. Zero. Nada.

And since it actually gains nothing, we don't treat it as sente (except perhaps as a ko threat), and we keep going. Black plays from D to 1. This play is called a reverse. At D the play reverses direction, from right to left. There is a play in American football called a reverse, where the quarterback hands the ball off to a halfback going in the opposite direction. Same metaphor.

What about the game in the local position around E-15 in Kano's beauty?

Well, if this is the correct game tree, fully reduced, then the play does not reverse through D. And that means that A is not greater than or equal to D.

What is the difference between A and D? For convenience, let me show A - D, and not the other way around. As we know -D = D, so A - D = A + D.

Gotta run. That's next.

_________________
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 Kano's Beauty



This shows A - D, which is the same as A + D, which is Kano's Beauty. Let's quickly compare it to 0.

1) Black plays from A to B and White replies from B to D. D + D = 0.
2) Black plays from D to 1 and White replies from A to -1. 1 - 1 = 0.

So A - D ≤ 0.

3) White plays from A to C and Black replies from C to 0. Then White plays from D to -1.

So A - D < 0. I.e., A < D.

That's not so easy to understand, and, even though I showed it, I don't claim to. But the thing is, no matter who plays first in Kano's Beauty, White can get the last play. And that's not so in either A or D by itself.

Kano's Beauty simplifies to this.


I guess we can call this an ambiguous 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.

It is good to see that some repeat my ideas I had described earlier in much greater detail.

Why?



Why does it gain the difference between 0 and S?



Why?



Why?



In the Berlekamp/Wolfe sense? Not in the go term sense of creating the same position using a different sequence?



Please explain.

Because White can play from S to -1, a score which is less than 0. Black is unable to reply to a final score of at least 0 if White plays first. So it is not true that S ≥ 0.


S = 0. While White can move from S to V, which has a mean value of -1, Black can then move from V to 0. Each player can guarantee a final score of 0; Black can guarantee a score of at least 0 by replying if White plays to V, and White can guarantee a score of at most 0 by replying to if Black plays to V.


What else would it gain?


It's not. You left out the .


Take a look. It's a game. (Numbers are games, too, but not in ordinary parlance. )


Yes, in the CGT sense, of course.


Transposition is a well established term for that. Maybe your term will catch on. IMO, it is too early to say that it has.


It's visual. CGT calls the players Left and Right. In Mathematical Go Berlekamp and Wolfe make Black the Left player and White the Right player.

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

Visualize whirled peas.

Everything with love. Stay safe.

Can you please show the calculation that the profit is not a number but a game?

We start with S and end up with 0. The profit is 0 - S. 0 - S = -S, which is a game.

OC, the mean value of -S is 0, so the average gain is 0. Hence the saying, Sente gains nothing.

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

Visualize whirled peas.

Everything with love. Stay safe.

It seems that I need to find out on my own why the count of S is not a number. Maybe I find time for this in a few months.

The count of S is a number. It's 0. S is a game. So is -S.

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

Visualize whirled peas.

Everything with love. Stay safe.

The profit is the value change from the count of S to the count 0. Why do you instead say that the profit is the change from the game S to the count 0?

What do you want to call the difference between S and 0? Oh, and 0 is the score, not just the count. {1 | -1} has a count of 0.

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

Visualize whirled peas.

Everything with love. Stay safe.