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 Post subject: Numerical evaluation theory of thickness #1 Posted: Fri May 13, 2022 3:47 pm
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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.

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 Post subject: Re: Numerical evaluation theory of thickness #2 Posted: Fri May 13, 2022 11:24 pm
 Judan

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

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 Post subject: Re: Numerical evaluation theory of thickness #3 Posted: Tue May 17, 2022 3:37 pm
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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)?

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 Post subject: Re: Numerical evaluation theory of thickness #4 Posted: Wed May 18, 2022 12:25 am
 Judan

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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.

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

Always, of course!

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.

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 Post subject: Re: Numerical evaluation theory of thickness #5 Posted: Wed May 18, 2022 2:13 am
 Oza

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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.

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 Post subject: Re: Numerical evaluation theory of thickness #6 Posted: Wed May 18, 2022 2:29 am
 Judan

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

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 Post subject: Re: Numerical evaluation theory of thickness #7 Posted: Wed May 18, 2022 9:48 am
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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.

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 Post subject: Re: Numerical evaluation theory of thickness #8 Posted: Wed May 18, 2022 10:55 am
 Judan

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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.

Quote:
if it builds on the players skills instead.

Each player has the skill to count stones!

Quote:
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?

Quote:
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.

Quote:
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.

Quote:
possibly you never proposed to compare thickness with territory?

As before.

Quote:
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.

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 Post subject: Re: Numerical evaluation theory of thickness #9 Posted: Sat May 21, 2022 10:13 am
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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.

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 Post subject: Re: Numerical evaluation theory of thickness #10 Posted: Tue May 24, 2022 7:35 am
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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.

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 Post subject: Re: Numerical evaluation theory of thickness #11 Posted: Sat May 28, 2022 5:37 am
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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.

`[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]`

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 Post subject: Re: Numerical evaluation theory of thickness #12 Posted: Sat May 28, 2022 6:15 am
 Judan

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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.

Quote:
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.)

Quote:
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.

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

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

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 Post subject: Re: Numerical evaluation theory of thickness #13 Posted: Sat May 28, 2022 6:27 am
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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
My recent updates have only led to slight nonsense, though curiously it does want to invade 3-3 in response to a 4-4 now. I tried to add "player to move" and consider some spreading of move influence, but perhaps my equations aren't quite right again.

or the balance doesn't work. The owner also wants to play 3-3 unfortunately. Actually now I think about it that does make sense.

Ah, in order to fix, I think I need to prevent exponential decrease if the size of moves is similar.
__
Hmm, I have implemented a sente/gote solution. At least it realises that if it wants to play 3-3, then other local moves are less useful.

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.

tried a 15k game. After over 200 moves, I resigned, it was behind around 90 points However, it was doing ok until late middlegame, when the opponent cut through into the centre territory. I understand better where it is weak now. My approximations of reading just one move ahead were a speed up, but seem to have made it too dumb.

 Attachments: control.png [ 3.82 KiB | Viewed 1524 times ] Wgain.png [ 4.67 KiB | Viewed 1524 times ]

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 Post subject: Re: Numerical evaluation theory of thickness #14 Posted: Sun May 29, 2022 10:41 am
 Judan

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`[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?

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 Post subject: Re: Numerical evaluation theory of thickness #15 Posted: Sun May 29, 2022 11:45 pm
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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.

`[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.

`[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:
`[go]\$\$B\$\$ , . . . . . , . . . |\$\$ . . . . . . . X X . |\$\$ . . . . . . . . O . |\$\$ . . . . . X X 2 . . |\$\$ . . . . . O X O 1 . |\$\$ X X X . X O X O X . |\$\$ O O O X . X O O . . |\$\$ . . . O . X 6 5 7 . |\$\$ . . . . . X O 3 4 . |\$\$ . . . . . 8 0 9 . . |\$\$ ------------------[/go]`

`[go]\$\$B\$\$ , . . . . . , . . . |\$\$ . . . . . . . X X . |\$\$ . . . . . . . . O . |\$\$ . . . . . X X O . . |\$\$ . . . . . O X O X . |\$\$ X X X . X O X O X . |\$\$ O O O X . X O O 1 . |\$\$ . . . O . X O X X . |\$\$ . . . . 2 X O X O . |\$\$ . . . . . O O X . . |\$\$ ------------------[/go]`

I was toying with a precise endgame evaluation of this group myself yesterday and chose this follow up for black.
`[go]\$\$B\$\$ , . . . . . , . . . |\$\$ . . . . . . . X X . |\$\$ . . . . . . . . O . |\$\$ . . . . . X X 6 . . |\$\$ . . . . . O X O . . |\$\$ X X X . X O X O X . |\$\$ O O O X . X O O . . |\$\$ . . . O . X 1 2 . . |\$\$ . . . . . X O 3 4 7 |\$\$ . . . . . . 5 . . . |\$\$ ------------------[/go]`

and are a follow ups (can tenuki instead). I might correct it like this

`[go]\$\$B\$\$ , . . . . . , . . . |\$\$ . . . . . . . X X . |\$\$ . . . . . . . . O . |\$\$ . . . . . X X . . . |\$\$ . . . . . O X O 6 . |\$\$ X X X . X O X O X . |\$\$ O O O X . X O O . . |\$\$ . . . O . X 1 2 . . |\$\$ . . . . . X O 3 4 . |\$\$ . . . . . . 5 . . . |\$\$ ------------------[/go]`

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 Post subject: Re: Numerical evaluation theory of thickness #16 Posted: Mon May 30, 2022 7:13 am
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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)

__
For opening board, 1000 rounds, B+2.96 +/- 0.01, top moves (16,14), (16, 15), (6, 18), (10,16), (15, 14)
__
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.
__
The key correction required seems to be equations that find two eyes for every point counted as territory.
__
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.

20220708 I find it difficult to think clearly about this problem. A tiny thought on how to make a balance equation: perhaps if a position is dangerous for your opponent (you have some threats), then you are more willing to take on danger yourself to claim the profit. Note that this is a choice though and you don't have to.

 Attachments: Wgain.png [ 4.61 KiB | Viewed 1429 times ] control.png [ 4.12 KiB | Viewed 1429 times ]

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 Post subject: Re: Numerical evaluation theory of thickness #17 Posted: Thu Jun 30, 2022 1:28 am
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New to this forum, I was reading this interesting thread and thought John hit the nail on the head. I assume it's allowed to bump month-old threads.
John Fairbairn wrote:
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.
Responding specifically to the bolder part: I think it's because professionals, unlike amateurs, understand the game well enough to know that reducing it to math, while theoretically possible, is simply too complicated for a human brain. Decades of failed attempts to create strong go AI by explicitly programming strategic concepts basically confirms it.

It comes down to a sort of Dunning-Kruger-like concept that I've noticed in other games as well. You would get new players asking oversimplified questions, and good players answering "it depends."
In Go, for example you could have someone asking "How many points is a 4-stone wall worth?" and obvious to anyone with a basic understanding of the game is that you can't answer it except with "it depends."

It seems amateurs, due to our limited understanding of the game, are understandably more likely to believe that it's reducible because they don't know the extent of this "it depends." They are more likely to think we can reduce the wall question to a few core concepts, run the numbers and get an answer, but the intricacies are lost on them and the answer is unlikely to be accurate. Pros would know this.

Don't get me wrong, futile though it may be, attempting this is still no doubt an interesting and fun pursuit.

And I suppose another reason we like to try to put it into numbers in particular is that numbers are unambiguous in their value. You ask a pro to explain the value of a thick group in one game, and then a similar thick group in another game, you're not going to know which of the two was worth more from their answer.

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 Post subject: Re: Numerical evaluation theory of thickness #18 Posted: Thu Jul 07, 2022 5:09 pm
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re: highlighted paragraph

I assume your point is that in order to play a good move, you need to focus on profiting from the value of weak points before your opponent does, so think less about thickness. However, if you want to judge the score of a position, you start needing to think about how thickness affects life and death around.

Or perhaps in other words, that isn't even the really difficult part of the game. A few averages will work OK to estimate thickness. But reading the boundary of life and death can take arbitrary amounts of time. Determining whether a move is 2,1,0,-1 moves away from threatening a large group or cutting point can change its value drastically.

I mostly agree with both your points, and yet I still believe there are ways to improve the mathematics towards much better strength without needing too much more computing power, even if it might be beyond me. Of course, we default to asking the great strength of AI which is very efficient (though its training was very costly), but I don't think it is the end yet.

20220713
I always found it difficult to understand P=NP, but in terms of Go, I think it is about how the smallest scale differences (tiny fractions of eyes) can blow up to make a difference to the status of a large group, and hence change the game. This means that not only is finding the optimal move difficult, but in fact that even finding a move within say K points of the best move on general positions is just as difficult. However, I think conservative strategies which minimise point loss are possible if you play consistently. You need to make sure you never have too weak a group. I suspect this can be done in polynomial time, and is what I am thinking about. The key difficulty is in managing life and death, but I think that investigation can be bounded by the space required to guarantee eyespace.

Perhaps that is the point of probability. Given unknowns that eventually become real things, we try to associate them beforehand to real (but less than integers) fractions of space time.

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 Post subject: Re: Numerical evaluation theory of thickness #19 Posted: Thu Jul 14, 2022 7:35 am
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I have realised there is a simple trick to improve the balance, though the crude means some unjustifiable assumptions. However, I think it has greatly improved strength, though it is much slower to compute (I think a priority needs to be assigned somehow to the computations to cut down unnecessary ones). Also, its first instinct to play around the 2nd line or where it isn't alive yet remains, though with more computation, it does seem to reconsider more. The maximum influence on a neighbour is bounded at 1, but this doesn't really appreciate the extra value of eyespace when weak, mostly by overestimating the value of weakly controlled areas.

The score estimate seems to vary more freely now with more computation (less precise but it shows it is thinking about something), and there are also checkerboard patterns for gain that I can explain but can't easily get rid of. (they seem to act like a sign of life that it is computing something, a bit like oscillations have a non-zero energy).

Another problem I don't much understand is that it outputs a gain of at most 2 per move normally when it should be 14. The checkerboard probably isn't helping, though the shielding is a nice feature. I think all these problems are linked to not checking for 2 eyes. And yet I'll just be content with the strength improvement for now. It still wasn't able to win at 15k, but as usual the centre shape was fine but it couldn't hold onto it.

I attach output after 1000 rounds, 200 seconds. B+7.34 (accounting for who to play)

lots of good top move suggestions were outputted, but at the end, it suggested nothing so good
W to play:
[(5, 2), (4, 7), (2, 17), (11, 10), (2, 7)]
B to play:
[(11, 5), (14, 7), (12, 5), (17, 14), (14, 6)]

I can't completely claim it does zero reading now as it does some fuzzy (fractional) addition of imagining moves that it wants to play on the board for both sides.

30mins later: some tweaks and gain is up to 8 in some situations (though 2 still seems normal). However, it now wants to play on the 1st line. This seems to be because it overemphasises the principle that your opponent's best point is your own and both seem very determined to play on the 1st line. It doesn't blow up, but it reaches a high stable point.

It lost again at 15k, but it was leading most of the game this time. But towards the end though that an eyeless stick in the opponent's area was alive and didn't both to defend its own area. My eyespace patch months ago was a bit too simple. But I may want to rehaul that whole part of the system.

I like that the system tries to start from first principles. However, I think the next step in strength requires concepts like group and eyespace. As a mathematician, it is nice to get a fuzzy proof for why such concepts are necessary.

 Attachments: Bgain.png [ 4.71 KiB | Viewed 515 times ] Wgain.png [ 4.72 KiB | Viewed 515 times ] control.png [ 4.05 KiB | Viewed 515 times ]

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 Post subject: Re: Numerical evaluation theory of thickness #20 Posted: Fri Jul 22, 2022 3:28 am
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My intuitive feeling for eyespace theory.

For this sort of more general theory, if eyespace can be understood within it, it must at a minimum be able to give predictions for multi-dimensional go. My feeling is that larger boards in 2d is not good enough, especially when komi appears to only depend on the nature of the corners (or the most valuable area of the board). I think the powers of two arguments might break down for example (at least before the endgame). However I have less doubt than I used to that a reasonable theory should exist even for the opening, but it might just take many components with functionals rather than functions say (I have been using a mix, but haven't found the right functionals yet).

In 1d go, upon placing a stone anywhere, it becomes uncapturable except by ko, so you can almost play whereever you want. Pass-alive becomes impossible unless you control the entire board. In 2d go, capturing secure eyespace is less easy because even in the corners, there are 2 dimensions of escape. Even if you respond by blocking, your opponent has a chance to escape via the other. But this makes the extra control of the corners even more important because the profit isn't so much that you make life yourself but that your opponent finds it even harder to make life inside a 4x4 square around the corner. Every cost like this must be paid for in Go though there may be double defenses (like 2 for the price of 1).

In 3d go, it is even more difficult to make eyespace or territory, so I suspect the whole board will intuitively be a bit more like playing on the first line where anything can die (but sometimes having the first move can mean that like the monkey jump you control much more than normal). Certainly the corners remain the most valuable, and I suspect playing closer to the 2nd line is a better way to start a new group (rather than 3/4th lines), but even this isn't trivial to prove with current theory. A group only needs 2 eyes, and with more dimensions, there are more options. It is difficult for me to predict komi at the moment. I will naively bet that komi increases slowly with dimensions.

Comparing to GnuGo style theory. In 1d, control of a stone once placed is maximised, but influence drops rapidly from 1 in the two adjacent intersections around (sufficient for 2 "eyes") to 0 just two intersections away.

In 2d the smallest number of moves it takes to make pass-alive is 6. Is this related to the similar value of komi? It sounds naive, but I think it might be (though the multiplier factor probably varies with dimensions too). If one stone can block the opponent from having enough space to make pass-alive, then they "control" that space in the sense that even if the opponent can escape, you can probably get value from responding.

Two moves are needed to surround a corner, then at least 3 for another nearby intersection. If on the edge, at most one is shared, so a total of 2+3-1=4 moves required to surround two eyes. If in the centre, 2 may be shared, 2+4-2=4. If two on the edge, 2 may be shared, 3+3-2=4. Finally, 2 extra moves are required to connect the 4-2 moves that aren't shared to one that is.

In nd, it seems that 2n moves are required to surround 2 eyes, after which, assuming centre/corner, n are shared. To connect the other n to the shared ones, at least n more stones are required (it is impossible in 1d as we can't connect (-1) to (1) without going via 0, but with more options we can rotate to connect say (0,-1) to (1,0) via (1,-1)). This gives 3n moves in general to make pass-alive. I'm not sure if this is optimal, but the order must be correct.

What if you can't use the corner and have to live in the centre? Then 3n moves are required to surround 2 diagonally placed eyes, with n shared. Then probably 2n moves are required to connect the 2n non-shared with the n shared. This is a total of 5n moves to make life in the centre.

In 3d, I think influence drops more rapidly because each stone blocks a smaller fraction of the escape routes of adjacent intersections (1/3). However since there are more dimensions, the number of intersections n moves away grows rapidly as n^3, so the sum total of influence may still be very high. This means that one wall can control a very large area. However, even making eyespace with such a wall isn't that easy. Probably this makes the game much tenser for longer, probably not great for your heart.

20220803 oops
this is just not correct. The numbers for minimal group size are a lower bound that is not achievable. But it does certainly remain linear. My new bounds are 8n-6 moves required in the centre and 4n-2 in the corner.

Meaning of temperature in go relative to thermodynamics

I don't think go makes any claims as to the nature of reality, but it does tell us about the prejudices of the structure of our model.

In physics 1/T is dS/dE, where S is entropy that must increase and E is the conserved energy. S is considered a measure of disorder and randomness. Energy flows from hot to cold because the same energy loss won't change the hot object much but will greatly increase disorder in a cold object. But everything is perspective if there are many dimensions.

In Go, S can be thought of as move number that increases (perhaps in the sense that the human is hotter than the cold maths of the board) and E as the conserved number of points (361) on the board. Note that in an alternating play game we can think of entropy going up for both players at the same rate as another conserved quantity.

And in a zero sum game, the total energy is the sum of the energies of each player. Where energy is expected reward.

Temperature increases with fighting, and cools over time, at least with strong players.

Chilling is removing a portion of "energy" by tax (accounting) in order to simplify. It is like ignoring the lowest mode, which works well at high energies.

Definitions philosophy, (e.g. eyespace)
From the games analysis perspective, why are certain concepts highlighted? I think we can think of Go like alphago

Board position -> (patterns, theory) -> Move, control evaluation

What patterns, theory becomes important? Probably the shortest routes from position to accurate moves/eval. As well as those which improve efficiency. For example, if the board position changes slightly, what key ideas are general enough to still be useful for predicting move/eval. This improves efficiency by not needing to recompute the same things, or allowing finer theory of increments when more accuracy is needed than the base approximation. (calculus)

Life of groups becomes clearly useful since control of an intersection at the end requires connection to a living group, and once a group is alive it is very rare for it to be advantageous to sacrifice it (except in ko), so there is a close association between life and definite control. And if you know the control function, you just have to maximise it to choose the best move.

The difficult part of theory is what happens when groups aren't alive yet. Like the critical exponents of critical point theory, we have theories (like powers of 2) that can bound the values of moves before a situation is settled, but sometimes these bounds can be very wide apart (as they should) when we don't know if a group is alive or dead, or what the cost it will have to pay is, or if can recuperate it by a sacrifice. Furthermore, while the key points of the theory generalise to higher dimensions, I am unable to tell you many of the qualitative consequences of higher dimensions.

What about other ideas such as light/heavy? These are harder to explain and unpick in terms of these details, but I think they can be. Many further ideas are qualifiers of a tree of principles to explain when they should not be used. For example take
"Play away from thickness."

This doesn't really tell you where you should play, but why? Is it not true that you should "play near to weak points?" Unfortunately this formulation is not quite true, but to explain why, you start needing to understand "sente gains nothing", "kikashi loses aji", "defending when your opponent isn't yet able to threaten you may be overconcentrated". But even if such moves aren't best, they tend to be better than playing near thickness, which is the simplest maxim, so it becomes good to teach it first.

I think even alphago will find it a bit difficult to hold the optimal concepts since random slippage can break them in a fragile net. But there are so many dimensions, they probably find ways to subtly stay alive in other forms.

Regarding the definition of 1 eye, detailed investigation shows this isn't really possible, at least not to the same rigour and completeness as the definition of having two eyes because two connected false eyes can lead to a living two headed dragon.

Back to the original question. What is the value of a wall? wordy thoughts. still no concrete calculation.
Our control estimates are correct when the value of the wall's expected attack gain balances the value of the reducer's profit. Being able to make 2 eyes is kind of a critical point, but the cost must be balanced, because the goal isn't to live, but to get points. Hence, we balance with the opponent's most valuable (double) attacks when evaluating.

Consider an infinitely long wall in 2d. What is the local influence? For a point adjacent to the wall, there is zero chance of eyespace adjacent to the opponent's wall, whereas the wall is essentially pass-alive, so any move it makes adjacent to the wall is instantly pass-alive. So it must connect to points one away from the wall to have a chance of becoming pass-alive itself.

Points two away from the wall are adjacent to potential but very unlikely eyespace. If there is support from neighbours, this means that the wall is blocked from attacking at those neighbours, but it may still attack indirectly. But such a point may not only survive but also support making eyespace.

A friendly to the wall always has the retreat option unless the reducer dares to go for the kill for eyespace. Basically, we expect that the reducer has less incentive to play close to the wall until a living group is nearby as there isn't much development nor eyespace beyond reducing.

In terms of what are the required variables?

For each point, what is the extra eyespace that it offers each side? and at what cost? We know that complete control over it has a swing of 2 points (chinese rules). If it can say offer 1/2 eye at the cost of only one more move, then you may only need a group with 1 3/4 eyes for it to be worth making a move there, at which point one should probably estimate 50-50 control of it. Though if that eye isn't very valuable, it may be even more likely that the wall attacks by pushing the reducer into making that eye with greater attack profit. Normally the reducer shouldn't depend on eyespace that close to the wall.

Intuitively a wall makes a channel where it is more difficult for the opponent to make eyespace, pressuring the points inside to escape to the opponent in the only ways they can, or let your follow ups claim them.

My imagination is that some gamma like function should exist to tell the control of each row of intersections from the wall.

I admit I seem to be floundering here.

I must note that semeai becomes vital for accuracy, and at the least this requires counting and comparing liberties, which my program doesn't take into account at all.

Intuition for using semeai results for points evaluation.?
I know this is a hole in my thinking, both because it is difficult and I didn't notice it.

Don't just assume the value of a group is only complete if it can reach two eyes. If it has more liberties, then every such can count. As usual we think in terms of the number of moves required to live, and if both sides depend on killing the other, then the dame moves around suddenly become very valuable.

My program wasn't thinking about sente/gote but averaging everything. In a semeai, for many moves in a row, they may be the largest moves on the board, and hence completely align with time/move number etc.

An analysis of a basic model for miai defenses that could be extended to double weak points, double defenses, double attacks, etc.
Not written into a paper because it is only one variant with several possibilities and I don't understand it well enough, just working through it now.
Most moves are double (or triple) attacks of some sort since their influence extends in different directions.

However, sometimes, that attack is powerful on both sides that can't connect except via your move (at which point such a move can be called a cutting stone).

Or the whole group is connected, but a move threatens two weak points simultaneously, and the double attack is more to get access inside the opponent's area. Then it is called a vital point. Normally a vital point move can't be saved directly without killing the opponent's group. But sometimes it is just a combination which means you have miai (or better) by threatening to connect enough on one side that you get enough forcing moves to connect the other side.

A basic model is -A-B-C- where you can play A or C solidly, but with control of both access points, you may be able to play B with miai of A and C. However, such adventures always have a debt to pay, which is that the opponent has the options of tenuki as well as either A or C in sente which could be big if they affect the liberties of your supporting groups. Writing letters for their summed values, we can call this position after you play B as

{{ABC|A-BC},{ABC|C-AB}|{A-BC|-ABC},{C-AB|-ABC}}
where you are Right and trying to minimise the sum (I think of this as Left's area)

The balance here is that normally the values of A,C are small compared to B (since B is deeper in Left's area), so Left's plays are sente, and it settles to A-BC or C-AB. We see this as Left's follow up threatens to gain min(2BC,2AC) while Right's follow up threatens to gain min(A,C) which is necessarily smaller.

However, when A,C threaten your outside, the fact the position is unsettled loses you something. Using miai counting, Left can move to max(A,C), while you can move to min(-BC,-AC). What this means is: suppose your opponent can threaten D from the threat of A,C and you defending gains E. This affects the local area by not changing the scores if you play first (since D only appears if Left plays two moves in a row since it is a sente option).

WLOG A>C. Assuming that Right playing A solves the problem (normally it won't if Left has follow ups responding to B).
In the original the count was A-BC. Now?
We have the position

So if D,E are bigger than the gain of BC, then Left can move to a count of A+(D-E)/2, while Right can move to -BA. If D>E>BC, and A+B/2>(D-E)/4, then nothing is sente and then Left's gain compared to -B is (D-E)/4 + A/2 + B/2 - A/2. The 4 probably comes from the fact that two moves are required (A,D) in order to complete the threat. There is no gain at A since it isn't sente. B/2 is lost because originally B worked as a move, but now it doesn't, but you still get some of its value as you can connect with C.

If A + B/2 < (D-E)/4, then A by Left is sente, and the count is A-E. As usual the defender against a double threat has an advantage (this lesson from game mechanics is actually true, at least in go)(logical as otherwise the defender has nothing worth defending), with a shift of (E-D)/2 in Right's favour. Left's first move gains A and B/2 as above as well as (D-E)/4 since D and E go from 2 moves away to 1 move away. The second move gains (D-E)/2.

What is this condition intuitively? Note that I have assumed that Right playing A removes such threats completely (which is unlikely). Normally forcing attack moves gain something in the sense that although sente gains nothing so that the result should be taken as the count, it does gain relative to expected local endgame if the groups were alive. Here the attack. We should compare to the situation if Right hadn't played B, and instead think of Left A as forcing on E rather than forcing on C. We can just ignore B. Then by assuming the sente result, the count should be A controlled by Left, E by Right, and B/C neutral. But B is only neutral rather than controlled by Left because Right has made a move there. Without such a move, Right would require first preparing with C, so the count would be A-E+B/2. Overall the value of Right playing at B has gone down from B to B/2 (assuming A=C). The condition says that normally we only compare A to (D-E)/4, but now the value of B must be included so that D needs to be even more threatening in order to assume A forces E.

If E>D>BC, then this means that you shouldn't play A because it isn't the best attacking direction and the local count isn't affected. Normally this means that the position around E will get settled until either D disappears (towards zero) or at least E gets smaller. At that point, A becomes a viable local move, and the local count needs to be able to predict the values of E,D at that point (far into the future), though simple bounds will often work well. Note that the error is bounded by BC, which although helpful, shows why go is so difficult. We can say the count of the board is within +/- 361 but that is obvious.

At equality, with D=E, Left can't gain anything from the threat.

If D>BC>E, then nothing changes. Right will respond as it is still sente.

I think this is useful for describing several beginner style mistakes in thinking, such as how to use aji and when. But I'll stop here for now.

Differential vs absolute
Tewari analysis compares different moves by comparing sequences that lead to the same result and asking if the additional moves are better or worse. This requires a way to evaluate the value of such moves, which often goes wrong, because it is so dependent on the surrounding position.

The keys are harmony for attacking weak groups/weak areas, as well as giving yourself more (valuable) options for making life even if you can't be killed, in order to prevent attack. I think I have some skill to explain why some moves must be better than other, more rigorous than normal theory. But how to formalise?

Groups, komi
strange that my program seem to get close to correct komi. I'm unsure if it was my imagination now I've added several tweaks. Also, komi does seem to greatly depend on the size of a living group. For each group, imagine the most compact way it can be forced to make life, or at least the local bump in size of moves before a large kill/invasion (if dead, then perhaps seki aji after a cut). Then this can be considered the "size" of the group, and the extra value of attacks (other than territory) takes cuts from this value.

I never seriously considered eyespace or tense fights. But perhaps I treated each intersection like having intrinsic value, and said that each intersection is valuable if its neighbours are and it can change the control of them. The influence value of each move really does seem to be proportional to its aliveness (so this tells how much value can be extracted from attacking it), and after it is played, the neighbouring area acts like it is split into independent situations, which can be recalculated, helping to estimate the count and gain of the move.

With my miai model, A acts like a double attack for Left, assumed to be completely alive when played, after which B and (D/E) are counted independently. When it is not completely alive, there is a dancing fight for eyespace.

I don't think I can push this further without numbers I am confident in.

Complexity measures
This is why its hard for me to leave Go despite my intentions. Ideas keep running. If I think about measuring complexity as:
T, the time it takes for an error bound minimisation algorithm to reduce error below B points in a solution to a problem.

Then, T scales approximately linearly (or a little slower) with board size (number of intersections) since the control on each intersection must be computed, and Go is mostly local, so each must be independently computed (approximately).

However, T is dominated by computing the most connected and hence most "valuable" (in the sense on move/mistake can affected pts most here), areas of the board. This is only difficult if there is a tension with both sides near equally strong/weak, and this is where the exponential issues occur as tiny details can affect the situation. Tsumego problems is the most complex concentration of complexity. As it isn't the only factor reducing complexity, perhaps amateurs are assigned to think about the other aspects. But pros focus on tsumego for the reason that it dominates at all large B.

So I ask, what aspects of tsumego affect this definition of complexity?

Just from today's 101 weiqi https://www.101weiqi.com/qday/2022/7/26/

1) a cutting point X between your groups A,B defended on all but one side Y loses a liberty (for nothing, i.e. in opponent sente) if the opponent throws in at X and you must take Y (i.e. both A,B must be saved especially if they concern the life and death of an opponent's group C), when you wouldn't have had to play there otherwise (the liberties/connections Y offers to A,B are worth much less than temperature).
2) making an eye can lose a liberty especially if such a move doesn't have liberties itself beyond the eye. similarly connecting can lose a liberty. playing on shared space loses a liberty. We see that W is in a bit of a bind here, but W isn't yet dead as the same applies to B. In combination, B first can kill (gote) rather than seki (in sente) by getting two moves in a row in the corner to capture another eye. If B only plays an atari without being able to capture, that simply loses B one liberty (for a price of B's one move, in gote, which is even worse). Over 26 points (since killing W will help the outside) but I can't say it is sente since W living threaten's B's upper right which is only 25 pts (chinese). Probably comparable to 26+25/2 = 38.5pts gote.
3) Not enough space in the corner, so the outside eye must be defended even with the cutting points. It is more about noticing that it isn't impossible.
4) Like 1, there is such a cutting point. In order to set up a double attack, sometimes you need to focus on the most important weak cutting stones to get the most forcing moves or otherwise kill. The simple move is insufficient as it doesn't get 2 forcing/killing moves where B has a weak point but only 1, leaving W two liberties where B is strong enough not kill W an eye.
5) The strongest move (smaller endgame is possible in sente) threatens the eyespace, and even uses W's weakest eyespace stones as tools for B's own escape.
6) The key is not to play heavy moves that connect your own stones but to keep pushing with "light" ataris that turn out the capture the entire opponent group since there are many forcing moves. Here B has something like a double attack if W tries to escape at the top, but if B escapes to the left/lower, B's other cutting stone also comes in useful. It is pretty clear that W can live with the next move as each part of the eyespace just needs one more move to secure. But less clear that B's move is the only option to kill. It seems that the point is that B's shape inside isn't good either so B needs the forcing moves from W's lack of eyespace to escape or else when W makes eyeshape, B is too far away to attack the final weak point.
7) The cut is vicious (meaning the follow up is very big), but doesn't work without support as the cut itself is threatened. So like a mouse, B can sneak in from the side to undermine W's boundaries stone. Like a double peep, it is too far for W to make a double attack, but it is close enough to support.
8) confusing as like 6, you mustn't try to connect your attacking stones, but rather use the forcing moves to make a double attack to get the 2-3 forcing moves to live. If you only play the nice shape points, you are too far from threatening the other side because your cutting stones remain weak. Instead you must first play to threaten the cut even though your stones are weak.

I can give a better definition of "strongest move" or "strongest resistance" (even if stubborn) now. Even if it only leads to ko, it keeps the temperature the highest, concerns the largest possible area.

20220810
with some more thought of how go value is divided into moves. I think balance equations that take into account eyespace really are the right idea. With the most valuable things, the costs also become valuable. i.e. can you get it for free by nullifying the value of opp retrieving the costs.

This is the physics of Go. Work out which points are valuable enough to be worth cutting and hence what sente a group has and if it is sufficient to live. It is sufficient to explain all points about overconcentration, efficiency, weak groups, up to proverbs about playing on boundary of weak group and moyo, last chance to invade. But the exact equations remain tricky if they exist even if they are clear informally in terms of case by case if else.

JF's points about pros talking in units thickness are particularly to do with potential fights and semeai, and hence the level of these costs locally.

A valuable area for you also attracts invasion, reduction, risk taking by the opponent.

__
if multiple chains depend on same eyespace for life, that is a group. Some may depend on other things, with other chances, and that must be discounted for. This sounds political or at least economic based.

however potential also clumps together. if such chains expect to always defend their eyespace, that may be a secure area with added value of potential from that strength. if the chains are independently alive, the potential decreases.

If A, B are adjacent to C and A is more valuable and weak, then C may be secured for reasons primarily due to A (while also gaining value from B), so that the B-C relationship overflows from C to B. The control at B cannot be deduced from C alone. And they are truly closely related. It needs to be understood that control of C connects to control and hence value of A and its potential to be attacked.

We can understand JF's not quite probes as to do with maximin on unsettled areas competing between close possibilities. But you need to force the possibility to maximise your own potential. Generally they are at a weak point fairly deep inside the opponent's area. And yet they may be the only weak point around, so leaning is important, but within the opponent's control as they have more stones. If you play too late, the weak point may have disappeared or otherwise become mitigated.

The key advantages of a better formal theory is more rigorously establishing that some moves are optimal. e.g. must play in corners, must cut. And how does it generalise to higher dimensions?

we say go is local, positive, but it is also symmetric. So that if opp can profit in more valuable area, then perhaps you should have competed for that area yourself, unless for some reason it is more easy to make eyespace the way you chose. i.e. if you played their direction, perhaps you play towards thickness and they still have more territorial moves, and so this way you still maximised.

When maximising point shifts for moves, a long weak group may have long lines that still allow life. Perhaps you threaten to make an eye locally or escape and the opp can take out that eye and let you escape and this repeats like a ladder. In such situations, deep reading of discrete if-else possibilities is the only way, but a support tool still needs to give good suggestions as to what is the local optimal shape. I think this is one sort of reason humans have been able to get fairly close to optimal play in say joseki. There often is just one key point bigger than rest and the fight is over that and it is just local reading for the semeai that question if such is possible or not or what the dependencies are, with reusable results.

But normally, say in endgame, alternating play happens and we expect big points of similar size to be exchanged. Though tsumego is important for optimising strength, a conservative player might not need it. When groups aren't on the boundary of life and death, everything is endgame, and we can just take averages rather than working out sente/gote.

My main question here is what is the probability of getting access to and controlling eyespace for a given large group (which dominates questions of value). Using control, we can estimate (via independence), the probability a given intersection becoming eyespace. But if each group needs 2 eyes, how to estimate such? Maths allows analysis in the limit of life/death, but what about recognising precursors to eyespace? This seems to be quite a bit more difficult than endgame theory.

In high dimensions eyespace is very valuable as it is so hard to obtain though there are more ways to get it. But there are also more ways to escape, so it isn't clear to me if a group can delay making eyespace for a long time. Influence may reach further with heightened threats on any particular intersection. Although a group needs to be larger to live (i.e. have access to enough space), there are more ways for any particular intersection to live. In 2d, a stone cuts say the intersection left of it from right and prevents them connecting, though if such is valuable, the opponent can still treat it as a clamp to join them up. In 3d, likewise, if an intersection needs 6 boundary stones to be captured, how strong are the boundary stones? What advantage does controlling that intersection have over them (divide and conquer)? Generally n boundary stones can connect with n-1 extra moves I think (graph theory, trees). There are a total of n cutting points. But the controller has the choice of which cut to choose based on value. i.e. balance the likelier control over weak intersections with longer term threats on a more difficult but plentiful area (calculus?).

My main question is what is komi in higher dimensions? Intuitively, it is to first order related to the size of smallest pass-alive group possible, and then to the smallest group the opponent can make nearby and so on. Note that diagonally adjacent points get further away as root of the dimension number. Locally the number of cutting points increasing linearly with dimensions. And the combination of possible cutting shape configurations is probably exponential. There are probably much more exciting tricks than the monkey jump near the edges.

I think connected shape becomes more important than immediate eyespace. and sacrifice is a more common strategy as capturing one stone is so difficult.

As go is symmetric, a sort of thing like opposition of kings in chess may occur to surround a group, until strong enough to leave a cut by hane. What it means to block or take a side may be to keep playing at the opponent's last move + (1,0,0) for example.

Is it really true that points in the centre are half as valuable as the corners simply as they have 4 access directions and the corner has 2? If so, perhaps the best first move can gain 4n-2 (size of smallest group) - aji + potential on sides? And yet playing the first move at tengen loses very little perhaps as influence extends in two more directions. Probably the only cost is that tengen itself isn't alive yet.

If the first move is alive by spending a move, then how to estimate influence? If opp's reductions can only live with group of size around 6n-4 on the n sides around it, then perhaps even it out on each side? If a corner needs to block n directions to take it, consider the endgame on each independently? If one move prevents opp living locally, if they can still escape through gaps, how much escapes? If one move of yours controls 6n-4 on the sides but leaves more gaps, then opp has at the least more centre influence there. It seems that we get some sort of spiral of alternating sectors. If the opponent approaches, the defender likely gains more influence in the side/centre around that, and the opponent is then biased towards reducing such and so on. These loops I still can't get my head around for a theory.

Ideally a theory would explain why complete graphs lead to seki under territory scoring and infinite loop under area scoring.

Based on above, my naive conjecture is that komi in n dimensions simply grows as 4n-2 + f(n) where f(n) is increasing but small. At the least, I think that it is very unlikely to be higher than quadratic growth. Perhaps f(n)(n/2)(4n-2) is another reasonable possible form.

the aji subtraction is moved into the powers. If 1 move can control corner, how to balance side potential? Recall if one move from territory, then split 1/2. If side is (6n-4)/(4n-2) approx moves from territory (note counting size often isn't as good as counting boundaries, depending on limits, but good approximation if the territory has many options), then the corner also takes some of side but not completely. Perhaps if m more moves needed for B, n for W, just use binomial ratio to split? sum of first m binomials of (m+n-1) go to W, the rest for B. bernoulli has standard deviation 1/2. So (m+n-1) has standard dev sqrt(m+n-1)/2. We need the errf integral up to (m-1)-(m+n-1)/2 over average inclusive, so add 1/2. So integrate up to (m-n)/2. If m>n, this is g(m,n)=errf((m-n)/sqrt(2(m+n-1))) (not 1/2 + g(m,n)/2 as we want the value difference), the value for W control

My 2nd guess is [(4n-2)+n(6n-4)g((10n-6)/(4n-2), (2n-2)/(4n-2))+?]/2

for n=2, this gives (6 + 8 * 2* 0.878+?)/2 = 10.024
I'm not sure about my count of aji. If I say that the first move doesn't have 1 influence on the outside but reduced due to aji, how much should I scale down? Probably according to the value of follow ups. If the side moves are the same size to corner still, then counting them as sente isn't unreasonable, after which it is more like g((6n-4)/(4n-2), (6n-4)/(4n-2))=0, which simplifies to
(6/2)=3

Somehow we want the answer to be (6+8)/2=7, but this seems like fiddling with equations that aren't justified. Somehow that would work in 2 dimensions, but I can't say I know what will happen in higher dimensions.

If such a pattern continues, perhaps [(4n-2)+n/2(6n-4)]/2 = (3n+1)(n-1)/2 isn't an unreasonable guess.
in n=3, this predicts k=10.

The understanding is that the centre doesn't need to be counted not because it isn't valuable but because the influence is shielded from the centre by the influence of the opponent's forcing moves. Some seeps through as those forcing moves aren't alive, but only after the corner adds a shimari does it really complete it.

By assuming side forcing moves are sente on the corner, this suggests that the standard corner moves aren't as good as you might think they are, because the commitment gives the opponent large moves on the sides. However, they main remain locally and globally best because if you say play on the side first, the opponent can play in the corner to counter that direction and may get a larger corner.

_________________
Give me triangles strong enough and I can measure the universe.

When Venus transits, we can align our clocks to one event. By measuring the angle to flat Earth at two places far apart on Earth, we can compute the distance to Venus and the Sun.

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