Numerical evaluation theory of thickness

dhu163 · #1

There used to be a lot of discussion in L19 about the true meaning of thickness. There is an old heuristic of counting 3 points for every height of a thick wall (note that the thickness of the wall doesn't directly contribute to this equation, and instead is normally discounted for the value of an attack by the opponent, assuming the wall can be defended).

I will add a theorist's perspective. If a group G is completely alive in empty space, they regardless of how the opponent approaches, unless they occupy the empty positions neighbouring G, then G has at least 50% chance of occupying any (dame) point adjacent to G. There is also good chance of making territory locally too. Even if the opponent is strong, they still can't afford to play too loosely near to wall, or else they can be captured.

If we estimate, a la Spight influence functions, the influence of strength decays by a factor of two with every move played, then under stone counting, the wall has a value of 1 + 1/2 + 1/4 + ... = 2 for each height. I can't really justify that the possibility of making territory doesn't change this value, but atm, I can't even tell you if it should increase or decrease this value.

If the opponent has a strong group nearby, then they compete for value, perhaps by cancelling out value somewhere along the way so it becomes 1 + 1/2 + 1/4 + 1/4 = 2 (sente-gote tends to double the probability a boundary play occurs). This is no change at all. (note we haven't counted the value of the opponent's group)

If the opponent doesn't have a strong group nearby, then pushing the value of the wall up to 3 per height seems a good estimate. There is probably an argument for why it must be less than 4.

Now compared this to if G wasn't completely alive or could be cut. For every move that the opponent plays, it is more likely to be sente. If the group is killed, then though it still has lingering influence with threats to save it, this becomes a drastic reduction. For example, perhaps we should instead count influence as 0 + 0 + 1/4 if the capturer expects to answer threats two moves away from saving the group. Then for every move that the opponent plays nearby, if we assume the owner of G may ignore the threats (ignoring the sente reduction), we can add a value of (2-1/4)/(2^n) if the threat is n moves from capturing G to the opponent's moves.

Accounting for the sente reduction (the amount depends on the local temperature), then we should add less value. However, be careful as cuts can be double attacks if the player has another group H nearby that depends on G for support, increasing the value.

Overall, in summary, a good rule of thumb seems to be that having thickness can reduce the value even of strong moves by the opponent nearby by up to a factor of 2x. (and weak dead moves by as much as the global temperature). There were many weak assumptions in this derivation, but I think this is a good summary regardless. Playing near thickness is like playing in the centre. It is less valuable than the corner, but far from worthless.

RobertJasiek · #2

See https://www.lifein19x19.com/viewtopic.php?f=17&t=18734 for the good definitions!

kvasir · #3

Maybe influence functions go back all the way to Zobrist?

The three points per length of wall should be compared to counter examples. Such rules seem to fail because they are too mechanical and there isn't a criteria when to not "count" thickness. Personally I like the QARTS system because it works with the players understanding (or lack of) and not against it. What I mean is that QARTS subtracts 20 for each weak group that isn't able to make eyes and 10 for weak groups that can make a single eye, the problem for the player is then to find the eyes and group safety is what most of us should be thinking about before counting thickness. It is also straight forward to extend to cases where you make judgment to subtract anywhere between 0-20 points for a weak group.

I'm not sure if "the evaluation theory of thickness" is supposed to work in the opening, middlegame or the endgame? It can't be the case that thickness has the same value at every stage of the game. Probably there should on average be a gradual decline in the value of thickness from the opening to the endgame, and in the middle game it is probably more concreate in that you try to effect the thickness in the middle game. Every game is different though and there is the kind of whole board thickness that only really becomes useful in the endgame.

My own experience with counting three points for walls in the late middlegame / early endgame is that it doesn't work: 1. because it overvalues the thickness when there aren't weaknesses to exploit; 2. it is a biased estimate and doesn't get you the right answer when you are close to solving the endgame; 3. it doesn't seem to guide where to play. Maybe others have a more positive experience (and I probably put it in a more negative way than needed)?

RobertJasiek · #4

kvasir wrote:

20 for each weak group that isn't able to make eyes and 10 for weak groups that can make a single eye

Far too simplistic.

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I'm not sure if "the evaluation theory of thickness" is supposed to work in the opening, middlegame or the endgame?

Always, of course!

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Probably there should on average be a gradual decline in the value of thickness from the opening to the endgame

Groups representing thickness can survive or die so the connection and life aspects of thickness decrease or increase. For surviving thickness, its aspect of new territory potential can increase if a) the connection and life aspects of thickness increase or b) the opponent's stones in the environment become weaker to increase the new territory potential of thickness. Otherwise, for surviving thickness, its aspect of new territory potential decreases to eventually zero at the game end while it should be realised.

The values of thickness must be reevaluated after each move.

My model allows all that.

John Fairbairn · #5

I'm encouraged that at long last at least some people are making the distinction between thickness and influence. But there's still a long way to go if we want to get into synch with the Japanese pro usage of the words.

I think in particular that the first step is to make a case that a numerical evaluation of thickness can (or even should) be made. To say it is necessary for computer algorithms to work is no sort of case for humans.

In my now rather vast compendium of index of Go Wisdom concepts, thickness is one of the biggies. For example, in Kamakura there are about 40 instances for just 10 games, and that is without counting closely related topics such as thinness, walls, and influence. In not one of those instances, as far as I can recall (and likewise in all the other GW books), is there even a hint of a pro attaching a number to thickness.

It is true that a couple of Japanese pros have written books in which they appear to put a value on potential territory associated with thickness, but (a) the value is on the territory not the thickness and (b) they don't appear to use such numbers in their own games/commentaries. I infer these books are just sops to lazy amateurs, and maybe even ghost-written by amateurs. The much quoted 3 points per stone heuristic is something I associate with Bill Spight, though I think he told me once that he got it from someone else - certainly not a pro. The related heuristic of 6 points per stone in a moyo is something I heard from Korean amateurs. I've never seen it linked with a pro.

So, apart hearing why thickness should be counted, it appears we need also an explanation why amateurs cleave so much to counting thickness (and other things) whereas pros don't.

In real, pro-commentary life, the way thickness is talked about is rather about the way it adumbrates the game. It provides context. It determines strategies (including strategic mistakes). It's a gross form of signposting. It tells you what you can or should do next, or shouldn't do. And, along those lines, the one phrase that comes up most often in pro talk about thickness is "keep away from thickness - including your own". That's seems a lot more valuable than numbers of spurious accuracy.

RobertJasiek · #6

Reply to John: https://www.lifein19x19.com/viewtopic.p ... 32#p273032

kvasir · #7

RobertJasiek wrote:

kvasir wrote:

20 for each weak group that isn't able to make eyes and 10 for weak groups that can make a single eye

Far too simplistic.

I think it was meant to accompany a positional estimate and a rough count that rounds to 5-10 points in the opening and middle game, but note that it is not a numerical evaluation of thickness but of the burden of weak groups, it was an example. In the context of rough estimates it doesn't appear too simplistic. It seems reasonable to assume the weak groups are going to survive and estimate the effect of having to take care of them, that is if one desires to compare weak groups to points. Also -20 points, while being a round number is also similar to 2 handicaps and 0.5 komi (-19 points by KataGo) and is not necessarily arbitrary and could be adjusted or based on an actual estimate of the cost of suffering the attack.

My point was not that this is a fantastically accurate and sophisticated system, it was that it works with the player's understanding and not against it. If the player thinks he can handle the group without suffering then he need not subtract 20 points, and if more precise estimate is needed then the player can estimate how much he will suffer by envisioning the attack. In contrast, systems that tally up stones in walls seem to fail to accommodate the player and his thought process (the player checks weaknesses, groups safety, safe territory, potential, center control, and etc. -- importantly, using the ability to see a few moves ahead in the game) and instead work against the player by diverting his attention to something that is of little importance or at least is not accomplished with the the usual playing skills. I think a "numerical evaluation" of thickness is more useful if it builds on the players skills instead.

RobertJasiek wrote:

Quote:

Probably there should on average be a gradual decline in the value of thickness from the opening to the endgame

Groups representing thickness can survive or die so the connection and life aspects of thickness decrease or increase. For surviving thickness, its aspect of new territory potential can increase if a) the connection and life aspects of thickness increase or b) the opponent's stones in the environment become weaker to increase the new territory potential of thickness. Otherwise, for surviving thickness, its aspect of new territory potential decreases to eventually zero at the game end while it should be realised.

The values of thickness must be reevaluated after each move.

My model allows all that.

In general it is probably easy to come up with numerical estimates for certain balanced positions but I have doubts about unbalanced positions (i.e. thickness vs. weakness, thickness vs. moyo, thickness vs. point lead, and etc.). I don't see in the other thread that you have a model that gives a numerical estimate that is comparable to territory, except for potential territory, possibly you never proposed to compare thickness with territory? Heuristics that give information about if the position is balanced in regard to thickness, or anything else, are certainly useful.

RobertJasiek · #8

kvasir wrote:

systems that tally up stones in walls seem to fail to accommodate the player and his thought process

There are different systems for numbers of stones in walls. Some systems are bad while others are good. Height of a wall for finding an extension is a weak system. Counting all stones of a wall has some meaning for tewari but not for assessing thickness. Counting only the significant, outer influence stones is a good system for the application 'influence stone difference'.

Knowledge of sophisticated concepts or previously insufficient thought processes are never good excuses for ignoring the most basic concepts. One very basic concept is to count numbers of stones! Any player can do this! Any player ought to do this because it has important applications for efficiency, joseki evaluation, other balance evaluation, neutral stone difference and influence stone difference.

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if it builds on the players skills instead.

Each player has the skill to count stones!

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In general it is probably easy to come up with numerical estimates for certain balanced positions

No. It was hard because it is basic. Recognising the basic things is hard. I needed circa two decades before I could state the simplest numerical estimates for certain balanced and fighting positions. One such value is the neutral (or dead) stone difference of newly played stones. Very basic, very easy and very important but I had to develop more complicated concepts before I recognised this simple concept. Every professional applies it presumably subconsciously (unless making severe mistakes) but nobody taught it until I discovered it as an excplicit concept. Am I still the only one to teach it?

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but I have doubts about unbalanced positions (i.e. thickness vs. weakness, thickness vs. moyo, thickness vs. point lead, and etc.).

I have developed some theory (lots of principles) for that but these topics are advanced. An exhaustive, profound, coherent theory is still missing.

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I don't see in the other thread that you have a model that gives a numerical estimate that is comparable to territory,

For joseki evaluation, there is my model that relates stone difference, territory difference and influence stone difference to each other.

For other theory, I could only provide bits comparing territory to thickness or influence so far. However, there is also quite some theory by me that relates them by a) partial application of numbers (e.g., only the territory balance or only the influence stone difference) or b) without numbers but using rough conditions, such as a player dominating a region. You know where to find this theory of mine.

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possibly you never proposed to compare thickness with territory?

As before.

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Heuristics that give information about if the position is balanced in regard to thickness, or anything else, are certainly useful.

Principles and procedures for that are even better, see above.

dhu163 · #9

Just to note down the thrust of my ideas. No formulae or concrete evaluations yet (though I am working on it).

Weak points and weak groups theory

In order to use influence theory, weak groups need to be included. How does this work so that a human can understand and apply it? I think the principle is that against weak points/groups generally 2 moves or more in a row will gain a lot. However, with just one move, the opponent can respond and sente gains nothing, so not only does it not help, but it might be ajikeshi. How to understand the influence then?

If a weak group/point is large enough that attacking it is sente, then this generally implies we expect it to reinforce itself closer to itself (locally). In contrast a large wall might need 7 moves by the opponent to kill it, so 4 moves in, it might start moving out, but even then, it will probably move out faster that a group only 2 moves from being killed. At the same time, we expect the opponent also has some influence near a weak group/point by attacking it in sente.

However this sente can only be used up once if the response fixes the weakness, so the main decision is which direction to attack from and how valuable that is locally vs globally. This is not as simple as attack from the lower side if you want the lower side because as we know, you should sacrifice on the side you don't want. This is because your attack stone can be counter-attacked (and so on) and your follow up might make your opponent even stronger on the lower side by cutting you off. Hence it is valuable to support attacking moves as preparation in order to lower the value of your opponent's attacks, forcing them to play less valuable reductions.

Weak point evaluation/judgement is about determining the boundary between the player and the opponent's likely development and how much this shifts with one move.

Hence, the owner of the weak point should avoid playing "on the other side" of the opponent's weak point since it is so big for the opponent to cut through as a double attack. This is unless the first player's initiative is enough to compensate. i.e. the opponent has a weak point of their own vulnerable.

In a sense, potential territory also acts like a source of weak points because they are worth defending, but it is much flatter and spread out so that one move won't make that much difference. Only an attack on a weak group that gains say 5 forcing moves will create a large area, and Go strength is about managing what the opponent can get from such forcing moves before they are played.

A distinction should be made between local weak points (territory) and potential eyespace points (which are worth the same for strong groups, but worth as much as the size of the group when weak). For example, it is often bad to extend from a 4-4 joseki wall as normal since the corner still has weak points, so the extension tends to be overconcentrated and too concerned with eyespace. This can even hinder your own follow ups against the corner weak points since you don't want to give up the eyespace and territory the extension creates.

In fact, this sort of analysis can be applied to general common shapes in order to understand why common patterns of attack are the best. The main principles are mathematical in origin such as cut, connect (block, extend), but as we get into Go strength, we think about attack and defence of vital points and double attacks, leaning on the opponent's weak points, tenuki (don't want to lean on opponent to defend, but can defend if attacked).

Good shape is about defending the most important weak points (normally one by one with a bias towards those more important), preventing threats without damaging self. Sometimes even if two weak points are nearby and important, you are unable to defend one without making a new weak group which your opponent can lean on to get more at the other weak point. In this unstable situation, tenuki may be best.

dhu163 · **#10**

I have done some whole board (computer assisted) calculations using equations I've worked out for influence functions, and am shocked to find that it predicts the first move as gaining 13.6 +/- 10.

This is entirely mathematical (no tuning), computation time scaling approximately linearly with board size (361) (10 seconds, though perhaps more accurate with more time), so I am shocked at the accuracy. Perhaps Go is really simpler than you might think? A lot of the complexity comes from life and death in the middlegame, but somehow endgame style miai counting averages gives such an accurate result for the opening too ?????

I'm inclined to keep the details a secret for now. I should test it out for larger boards, and improve the error.

The key equations fit on one chalkboard line or around 15 lines of computer code. It's so simple I feel someone must have thought about this before but perhaps not had the resources to do the computation, or the confidence/inclination to publicly discuss it preAI?

Perhaps its just the beginning for this idea, but for me it naturally closes several of the few conversations I've had with Go researchers. To be honest, I have spent time on the mathematical side, with little understanding of the history of ideas for Go programs beyond some GnuGo reading. It isn't clear what the advantages/disadvantages of my system are compared with classical programs. I have some hope my program has already reached 15k level with much room for tuning and improvement.

edit: 10 mins later. Not including systematic errors (from not quite right equations), it gives 17.7 +/- 0.1 ish. at the moment, systematic error is +/- true gain of first move (I think this is close to true from miai counting bounds even when sente/gote exists). By miai counting bounds, this "should" be an overestimate for twice komi as W has the next move which I didn't take into account. I am preparing to take it into account, but this multiplies the equation code by 4 times (with lots of room for mistakes).

edit: 4hrs later. Can seem to make useful progress with my new equations. I have written some down but the computation spits out nonsense (e.g. most of the time I get B+345 or something. I think I am missing some kind of balance to them).

edit: 15mins later. Ah, I accidentally played 19 moves on a row rather than one move. That's why! Now it is working again. Hmm, the new value is 10.5 +/- 3 for the value of the first move. That is too low, but still not too bad given that it gives better move suggestions. It says tengen for the first move, and otherwise 2-10, preferring the sides to the corner, but at least its no longer nonsense!

edit: 15 mins later. My main criticism of my engine is that it thinks 1 eyed groups are 100% alive. Perhaps this is why it likes the centres, then sides, then its favorite corner move is 2-2, and it likes to play 2-2 on any other corner move. I think my new equations are slightly insufficient and I need a few extra lines of computations, though I think I have the key variables.

kvasir · **#11**

RobertJasiek wrote:

Knowledge of sophisticated concepts or previously insufficient thought processes are never good excuses for ignoring the most basic concepts. One very basic concept is to count numbers of stones! Any player can do this! Any player ought to do this because it has important applications for efficiency, joseki evaluation, other balance evaluation, neutral stone difference and influence stone difference.

It certainly does helps to count stones in the center, I have assumed this is what you mean by influence stones but I am not sure if this is what you mean. What I tried to say above is that simple influence estimates can be useful if they can tell you if the game is balanced but I doubt the usefulness of methods that claim to predict how outside stones convert into territory based on simply counting the stones. That is not to say that such methods are total nonsense in every situation.

When evaluating position with thickness it is important to judge the effect on the game. This can then help with making objective decisions about how to play. The alternative way of finding a target number for converting thickness into territory based on a heuristic can work in some situations but I can't agree that this is an objective way of playing. As an example, saying that one player has a wall of length 6 and now they must find 18 points of territory somewhere is not objective. On the other hand, it is certainly true (as you said) that some heuristics, principles and theories are better than others.

RobertJasiek wrote:

For other theory, I could only provide bits comparing territory to thickness or influence so far. However, there is also quite some theory by me that relates them by a) partial application of numbers (e.g., only the territory balance or only the influence stone difference) or b) without numbers but using rough conditions, such as a player dominating a region. You know where to find this theory of mine.

I don't know if you mean your books or something in this forum.

dhu163 wrote:

I have done some whole board (computer assisted) calculations using equations I've worked out for influence functions, and am shocked to find that it predicts the first move as gaining 13.6 +/- 10.

With all these theories floating around maybe someone can try their theories on this position from a pro game and explain how they help with playing. I marked the last move only to show who's turn it is.

Click Here To Show Diagram Code: [go]$$B $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . X . . . . . . X . . . O . . . . | $$ | . . . , . . . . . , . . . . . , X . . | $$ | . . . . . . . . . . . . . . . O X . . | $$ | . . . X . . . . . . . . . . . O . . . | $$ | . . . . . . . . . . . . . . . . . X . | $$ | . . . . . . . . . . . . . . . . O . . | $$ | . . . . . . . . . . . . . . O . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . X X . | $$ | . . . . . . . . . . . . . . . . . O . | $$ | . . . . . . . . . . . . . . X X . . . | $$ | . . O . . . . . . . . . . . O X O . . | $$ | . . . . . . . . . X X X . X O X O X . | $$ | . . . O . . . . O O O O X . X O O . . | $$ | . . . . . . . . . . . . O . X . . . . | $$ | . . . . . . . . . . . . . . 1 O . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ ---------------------------------------[/go]

RobertJasiek · **#12**

kvasir wrote:

It certainly does helps to count stones in the center, I have assumed this is what you mean by influence stones but I am not sure if this is what you mean.

It is immaterial whether center or not. What matters is the outside for the sake of future potential.

The "influence stone difference" (around a region) is the number of Black's outside stones with significant influence minus the number of White's outside stones with significant influence.

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What I tried to say above is that simple influence estimates can be useful if they can tell you if the game is balanced

Or unbalanced. (E.g., territory count 0, a clearly positive influence stone difference and Black to move suggest Black leads.)

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but I doubt the usefulness of methods that claim to predict how outside stones convert into territory based on simply counting the stones. That is not to say that such methods are total nonsense in every situation.

Sure, usually, a simple count of excess influence stones does not equate a simple amount of new excess territory.

However, there are special applications for which something similar is possible. E.g.:

1) Joseki evaluation. Although the influence stone difference does not predict the amount of future territory, it is related to the current local territory count and to the pure local stone difference.

2) When a player pushes to take, say, 2 extra points along the edge, we can say that the opponent's new influence stone should, from the opponent's perspective, make at least 2 new points later.

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I don't know if you mean your books or something in this forum.

See https://www.lifein19x19.com/viewtopic.p ... 38#p273238

dhu163 · **#13**

I'll work on it a bit more today, it is still very crude.

Without any changes from birth a few days ago, my engine at no komi says B+1.2 +/- 2 (but as W is to play, probably -7 to this so W+6 approximately).

I think it thinks that (15,18) not a real tesuji as it helps W, and (11,17) and (11,18) are too slow, so I also think W is ahead, but not but as much as it thinks.

If I remove those two black stones, I get W+11 +/-3, so it thinks those B stones only gain an average of 6 points, much less than the 14 points expected for a good move.

Suggestions for B to play: (2,5),(2,10),(3,9),(18,6),(8,11)
Suggestions for W to play: (3,9),(2,10),(2,5),(2,8),(4,5)
(numbering from top then left)

These are pretty bad suggestions in my opinion, but actually the top suggestion for W is probably one of the top moves.

For comparison, katago says with W to play, it is B+4 on the board, so I am off by 10 points. Katago suggests (14,13), (2,18), (5,12), (3,9), (14,8).

some notes on difficulties

several hours later: There are still problems, but I could cry. It predicts the 3-3 invasion as well as the small knight's move!!! On the other hand, computation is several times slower.

Also I removed a factor of two in the equations to make the output much more correct. But I don't understand why that is correct. I suspect it cancels the error that it thinks one eye is enough for life.

At the moment the komi prediction is B+8.34 +/- 0.11 after 1000 rounds. B+8.82 +/0.03 after 2000. Clearly my error estimate is not taking all error into account, but I'm not too bothered atm.

In my opinion the number one problem now is LD understanding, and not using the chain rule (my calculus is only first order).

However, rerunning on kvasir's position, it says W+22.16 +/- 0.07 after 2000 rounds, with move suggestions (2,12),(2,10),(3,2),(4,5),(7,5).
After 3000, W+22.20 +/- 0.03, (7,5),(6,2),(2,10),(4,12),(2,12). I am impressed it found the attachment (4,12).

What went wrong? I think it is underestimating the problems at weak points in general. It needs to anticipate B's possible attacks even if they are premature right now.

OOTH, I have an idea for patching the 1 eye problem.

(10 mins later)
At least it no longer suggests tengen. Now opening is B+8.45 +/- 0.06, (8,2),(7,2),(2,16),(11,11),(14,10).

I can attach a picture its thinking in Wgain.png (W is where W wants to play, B is where W doesn't).
The control estimate is control.png

Even though the engine is very weak, I would like to stress how much simpler to understand these equations are than neural networks, though it will still take math and go skill to interpret results.

Compared to neural networks, pure equations are zero learning.

I think its not bad at finding good shape points overall, but it often plays too deep or doesn't defend weak points.

RobertJasiek · **#14**

Click Here To Show Diagram Code: [go]$$B $$ --------------------------------------- $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . X . . . . . . X . . . O . . . . | $$ | . . . , . . . . . , . . . . . , X . . | $$ | . . . . . . . . . . . . . . . O X . . | $$ | . . . X . . . . . . . . . . . O . . . | $$ | . . . . . . . . . . . . . . . . . X . | $$ | . . . . . . . . . . . . . . . . O . . | $$ | . . . . . . . . . . . . . . O . . . . | $$ | . . . , . . . . . , . . . . . , . . . | $$ | . . . . . . . . . . . . . . . . X X . | $$ | . . . . . . . . . . . . . . . . . O . | $$ | . . . . . . . . . . . . . . X X . . . | $$ | . . O . . . . . . . . . . . O X O . . | $$ | . . . . . . . . . X X X . X O X O X . | $$ | . . . O . . . . O O O O X . X O O . . | $$ | . . . . . . . . . . . . O . X . . . . | $$ | . . . . . . . . . . . . . . X O 1 2 . | $$ | . . . . . . . . . . . . . . . . . 3 . | $$ ---------------------------------------[/go]

I lack time for a full analysis. To start with, what is the LD and territory status in the lower right?

kvasir · **#15**

It would be a mistake to not fight the ko but besides taking the ko stone white's shape appears alive or escaping to the outside.

Click Here To Show Diagram Code: [go]$$Wcm4 Is white alive without the ko? $$ , . . . . . , . . . | $$ . . . . . . . X X . | $$ . . . . . . . . O . | $$ . . . . . X X . . . | $$ . . . . . O X O . . | $$ X X X . X O X O X . | $$ O O O X . X O O . . | $$ . . . O . X . O . . | $$ . . . . . X O X O 1 | $$ . . . . . . X . X . | $$ ------------------[/go]

My mind is tainted by KataGo but I can suggest this black follow up, assuming no escape and no effect on the outside. Maybe it is useful for endgame purposes but I am not sure if it is because it is a very strong assumption, something that may never be realistic.

Click Here To Show Diagram Code: [go]$$B Eventually black may be able to play like this. $$ , . . . . . , . A . | $$ . . . . . A . X X . | $$ . . . . . . . . O . | $$ . . . . . X X . 3 . | $$ . . . . A O X O 1 . | $$ X X X . X O X O X . | $$ O O O X . X O O . . | $$ . . . O . X 7 8 6 . | $$ . . . . A X O . 2 5 | $$ . . . . . . . 4 . . | $$ ------------------[/go]

If black tries the same thing right now:

I was toying with a precise endgame evaluation of this group myself yesterday and chose this follow up for black.

dhu163 · **#16**

several more updates and in many ways it has mellowed, but it is certainly sharper around weak points and boundaries, though it still likes crude pushes and cuts.

For around 3000 rounds, 3 minutes, B+9.71 +/- 0.13 (note it is much less confident)

Top moves (6,7), (7,12), (15, 9), (12, 6), (16, 18)

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For opening board, 1000 rounds, B+2.96 +/- 0.01, top moves (16,14), (16, 15), (6, 18), (10,16), (15, 14)
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Now I think the main problem missing is that it does zero reading, and that its attention is too local (only on neighbours, so it can't easily even understand that corners are more valuable than the centre), and it misses double attacks, but I'm not sure its worth spending more time on this, though I am definitely happy with progress.

Also, it assumes independence, which can go wrong in semeai for weak groups. It doesn't see snapback.
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The key correction required seems to be equations that find two eyes for every point counted as territory.
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20220530 signing off for a while. The pictures are very pretty, but really not as justified as they could be. Getting balance for pure control estimates is too fiddly. I think flow variables for sente/gote/eyespace should be more useful. Perhaps I'll investigate.

It seems difficult to combine value in without actually doing the reading. I probably can't expect too many nice general equations, but perhaps only special cases.

Numerical evaluation theory of thickness

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