Numerical evaluation theory of thickness

[dhu163]

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.

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.

[mart900]

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.

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.

[dhu163]

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.

[dhu163]

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.

[dhu163]

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

{{ABC, AD|A-BC, A-E}|{C-BA|-ABC}}

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.

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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.

[dhu163]

What concepts are derivable from this sort of theory that isn't covered by sente/gote etc. ?

Well, shielding is certainly one. Aji too (though the theory isn't strong enough to estimate it well, only informally tell you principles). Weak points as a key source of value of moves that gets split up as more moves are played. Each move is like splitting the area around into more independent endgames. If you take a vital point of life, it may be difficult for the opponent to make eyes in each independent region, so you may be "winning" in each, and profit arises from this.

My better guesses for komi tune to the known k=7 for d=2. So suppose it takes d+1 moves to capture the corner.

Perhaps m+n=(6d-4)/(4d-2). And m-n= 1-2*(-d-1)

I think this m-n value is the closest I've come to JF's units of thickness/atsui

Then
k=[4d-2 + d(6d-4)errf( (1-2*(-d-1)/(4-2/(2d-1))]/2

Which predicts

d=1
k=1.5
note that this is unreliable as (4d-2),(6d-4) break down for d<2. However, the answer does seem reasonable as any stone you place on the board that has 2 liberties lives.
d=2
k=7
d=3
k=15.7
d=4
k=27.8

what goes wrong for the complete graph? well, the smallest group size, (4d-2) increases with degree and the number of neighbours, and it is larger than the whole board, so we can no longer talk about the "influence" such a group has on (6d-4) etc. because that is outside the board and so on.

This tends to something a bit more than

(3d^2+2d-2)/2

which has roots at (-1 +/- sqrt(7))/3 . not quite golden ratio.

Funny that d=0 gives k=-1 . There is only one intersection and play there is suicide.
__

next day: what?? please tell me this is a joke.
7th letter of the English alphabet: G, 15th: O.

[dhu163]

When points seem obvious, attention turns to how best to present and which words to use. Some ideas here are already an obvious improvement in how to think (at least for me), but much is fuzzy.

Today, I am pondering on two seemingly unrelated topics.

How does 2d affect the board concepts the opening/middlegame compared to well understood corridors in endgame theory?
How do physics/chemistry analogies help/apply/confuse?

e.g. Entropy, temperature etc

In physics, entropy maximisation, stationary action etc talk in terms of some optimisation. In Go, we have optimal play (lets say for score), and Go is discrete. Can this be thought of as some approximation to a problem of optimising one continuous function. I think it should be possible (in fact it is by definition in the game tree), but I haven't found a way merely as a function of the board graph, which is what I am focusing my research on. Perhaps its best to simply think of both players applying a "strength" of pull on the board. Equal strength should lead to equal score up to randomness. However, Go is not exactly stable (though not exactly unstable either except in semeai), and moves having positive value meaning that a single move usually doesn't affect the score that much. So on average, equal strength leads to equal score. Without even averaging, perfect play leads to perfect score deterministically as it is a combinatorial game (perfect information, finite, two player).

This is like a problem in non-equilibrium thermodynamics. There are many possible low temperature states that the game tends to over time (heat death), with completely different scores. Both B win and W win are stable possibilities (or else there is nothing to fight for). The board is the medium of interchange between the two players who are sources of energy like a star? The colder player draws the opponent into traps. When B plays a move, there is an interaction between B and the board that absorbs B's will (maximise their points) and moves oscillate between B and W. But which intersections maximise B's points and how does that change after a move? Are there waves spiralling outwards from a fight? Or could say that B has infinity temperature, W has -infinity temperature, and this is how they absorb the value of the board over time. Though this is probably not a good idea as the game is symmetric between B and W, and this tends to be misleading when compared to standard thermodynamics. Go is quite random in the sense that slightly different starting positions lead to vastly different outcomes on the board (and board positions are the only physical thing we have access to as observers), basically chaotic as it is so unstable (at least in perfect play), but principles are common to each timeline.

Temperature is a common term in Go endgame theory, but as I have studied some thermodynamics, I don't really want to use it here without understanding if it really matches the thermodynamics concept, or if it is actually a slightly different concept. Otherwise bad analogies will be made. In physics, temperature is defined as 1/T=dS/dE (entropy,energy)

Stephen Wolfram seems to say that energy is akin to computation activity. Physically, high energy tends to mean you don't know how the materials involved will spread and react (or where they came from, going backwards in time). And yet it also implies that it is stored and insulated from externals at the moment, not yet forming bonds and attachments that lower the energy. Or perhaps it is to due with more physical fluctuations than time fluctuations. It seems that by friction, hot systems tend to lose their energy to the environment until they reach a much lower local energy state. In the case of a ball rolling in a parabola, it falls to the bottom and remains there at the extreme low of energy local to this system. But by conservation of energy, this energy needs to go somewhere, and that may mean small (mechanical or even quantum) oscillations likely remain. Statistical physics uses temperature to balance the energy exchange of different (mostly) independent systems. High temperature systems are likely to give their energy to low temperature systems. If so much energy is stored invisibly (say in air heat, or the mass itself), this suggests that these are comparatively low temperature for their energy. Invisible suggests that our conscious brains don't interact with them directly. This may be because they are small and similar temperature to ourselves. Note that higher temperature systems change (entropy) less compared to lower temperature systems.

If moves are particles, their high or lower value (or energy change?) depends on the stability of the position and how many other unsettled intersections are nearby that it affects. Is the move itself stable enough claim the whole potential or only a portion?

Weak point theory in 2d

From the point of view of territory and eyespace maximally bonding to a secure group (pass-alive ideally), what about potential territory and insecure groups?

Given an insecure group, there are a sequence of possible directions to attack it. The opponent will attack from their best direction until the group dies or the moves aren't valuable enough. The best direction tends to be connected with other attacks into a potential moyo, or perhaps it is on the side/corner with better control of mutual eyespace (Territory before eyespace before potential is a reasonable rule of thumb). This is especially useful in securing (at least for now) both an attack and supporting weak attacking stones around if the defender must leaning from the centre on them as this helps the attacker reinforce their connection to side eyespace.

Likewise the defender will likely play on a specific set of these weak points themselves in response if they want to save their group. Aliveness and killing each take a finite number of moves, and at vital points, these often overlap. Normally, any attacking move is a valuable defensive option. Both are close to the heavier (larger, closer to boundary of life and death) weak group in question. Heavy is proportional to size of group plus potential (times appropriate reducing factors for aliveness to prevent double counting) and exponential in distance from life and death boundary. At the boundary it is highest. One move away is 1/2, two moves is 1/4, etc, as usual. However, not all chains of the group may be at the same point relative to the life and death boundary. Those closer to the outside might have sacrifice techniques that threaten to rescue them by leaning on the weakness of the attacker's seal.

Often, stretching a little to cramp the opponent in the short term, even if they get better shape than normally locally by spending two moves (in the long term) may profit, because those two moves might not be valuable direction for them (e.g. next to your living group even if they break your potential). For example, a 3-3 invasion sometimes continues with a one space jump instead of the knight's move, just to block the 4-4 wall's potential on that side. The corner is thinner, but taking more of the development potential also means that the wall is more attackable from the other side. This can make a difference if that side may be more valuable than the corner.

Stretching is the attacker's responsibility. Normally the side with eyespace has more influence locally. If you are attacking, you often have less eyespace since you have to come close to the opponent, and the opponent may pincer to sacrifice. Your eyespace and much potential territory comes from the potential kill or at least follow up attacks.

At the same time, if your attacking stone is too weak (which probably means the defender is too strong, so consider adding nearby support first), then the defender may threaten that attacking stone perhaps attaching or at least occupying or cutting it from the next follow up, or even pincering and counter-attacking. Then at the least, the attacker may have to respond and such a play may be gote, or not gain as much as simple estimations of value will predict. Don't overcommit to weak attacking stones (they are light) as even if they aren't easy to capture, one stone isn't many points, and their potential is already reduced by the defender's strength locally. This means delay adding a move to defend them. Ideally threaten to sacrifice them by profiting from other attacking moves first (using it as a threat without even losing a move reinforcing it), and only develop with them if the opponent remains weak or threaten to capture it on too large a scale. Remember even then that their main value comes not from making territory locally but from threatening the opponent's weak points. Even if the opponent doesn't have many weak points, playing higher to reduce the opponent's more solid potential may be better than a lower territory that remains flimsy.
A higher thinner move can make miai of a larger territory and just connecting. A tighter move is better to make secure territory/eyespace locally and prevent any incursions by the opponent, and is more appropriate for the defender of territory/eyespace.

They say don't play close to thickness, but where should you play? The above tries to explain that playing close to weak points (your own or the opponent's) is good but only if it works and is strong enough to support follow ups (even if the opponent responds). Bad shape arises when you play near weak points but in the wrong way, either inducing a defence, or spending too many moves defending (perhaps without probing the opponent's weak points and hence making the opponent's attack light). The worst extreme is playing in no man's land where both sides have strong stones and all areas are interchangeable. Hence each can be sacrificed with profit from tenuki or leaning on it with one's own local strength. So securing more intersections locally is less valuable. Sometimes there is a wall with few weaknesses (connected to a living group) that is threatening a large territory but you also have some stones there with some control of eyespace. This is not quite no man's land then, though you may still prefer to play lightly and consider sacrificing. An area which is may potentially be large B territory or W territory (without semeai, just both have large walls) is not that rare, and may arise from each taking territory on the other side and pushing the opponent towards such a less valuable area. This isn't quite no man's land, but is unlikely to grow very big. It is like: you take low, then I take high. You play high, then I invade.

Balance, miai, evaluation of bonds

Weak groups and potential territories often find themselves in balance with each other or close to in perfect play. The less clever explanation is that this is because highest temperature moves are played first, so that remaining moves are often close to similar in size if they are large. The clever explanation is that this is planned since having big moves in miai is good insurance for the second player. In perfect play, the first player shouldn't expect to get much profit except at similar cost. Biggest moves are those concerning weak groups or large territories. For example, if a group A is weak and doesn't have much potential (i.e. the opponent is strong nearby), then the value differences locally may be dominated by the attacks and corresponding defensive responses. If you can be certain that play will occur on such areas, then even if the local areas isn't so big (perhaps they are already close to miai), premptive control of the attacker's potential (often by 3-3 invasion or approach), or threaten a bigger moyo with the escape route. So in good play, the defender (who normally has more control of their group) should try and is able to balance the major attacking points by supporting their weak groups when choosing direction of play before they are 1 or 2 moves away from being forced to respond. We can call this "playing close to a potential fight", or just that the defender should try to make more options for themselves, even at the cost of territory. Of course the attacker tries to prevent this but can normally only choose the largest attacking direction in combinations with leaning on other weak groups of the defender. This is why they say playing Black is harder than playing White. The attacker maximises profit with a splitting attack on the defender's nearby two weak groups. (driving tesujis are the same point even if not a long term attack, but only one weak single cutting stones that were already heavy and quickly grow).

The least valuable region next to a wall such as the adjacent points is most likely not going to be competed for, and hence is most likely not going to be competed for as it isn't valuable for eyespace. So it can be thought of as part of the chain since its life and death is so closely linked, even towards the end of the game in dame filling. This is another way we actualise influence of strong shapes.

Similarly, if the opponent builds a moyo, it is often a good idea to build a moyo adjacent. This fight for the centre should make it more awkward for them to expand their potential (as you can just answer with a large territory with also threatens their boundary stones) and also to invade (as your strength or centre control from attacking will help support invasions of your own, or reductions, or give you a larger centre potential that works with reducing moves). In general, centre influence gives you more tuning "control" of how the game flows. This is in part because there is more balance between the value of the opponent's territory and their potential, so you have more choices of strategy which is better if you know how to use it, and if it compensates for your loss of territory. Normally the opponent's territory is much more valuable for you to fight for especially as it is a source of mutual eyespace. In part it is also that if they lose their territory/eyespace, then they may find it difficult to get compensation if you control the centre so they may remain weak without much potential (of course, your potential is likely reduced too).

A lives vs A dies vs A dies and the local area is the opponent's potential.

rather than territory. But perhaps there remains space to fight for locally. Then, the miai balance may be to play looser or higher to make A's potential territory miai with A's connection back to security.

If instead it is the opponent that is weaker locally, then there may be many value differences locally. There may be

A lives vs A is attacked vs ... vs A sacrificed vs A dies
A kills the neighbour opponent chain vs A leans on the opponent to build vs A has small centre potential

Then, the stronger side may need to balance many things. The largest term may be preventing the opponent's attack at all. That is the balance. This may mean you occupy that point even if it is close to your own group. This can be called removing aji, thick play, ready to attack any opponent group nearby. Then it may simply be endgame and territory in one sector may be miai with territory in another.

Evaluation of bonds (territory with group) (enemy groups)(friendly groups)

Here we think of binding board intersections to chains and moves. This is "territory", "development", "potential", "aji", etc. Binding to territory is stronger if the chains are more alive. It spreads further if the opponent finds it difficult to live locally. Moves are worth how much they change territory. They are worth most when affecting a large area and when at the boundary of life and death (perhaps for multiple fighting groups). The total binding strength can be thought of as the sum total of this control/influence. The marginal can be thought of as the value of the largest move there. i.e. what happens when a player makes a move that vacuums up value?

A weak group should normally expect a positive impact overall unless the negative impact dominates, namely losing the possibility of using that intersection as a liberty or eyespace. But merely from attack, a group should expect a positive impact compared to that group not being on the board, even if its value comes from being able to sacrifice it for some profit elsewhere. You can't get worse than being dead in Go, so if a group isn't worth saving, its owner simply won't play there and neither will the opponent. If an attacking fight occurs, the group must have positive enough value to be worth saving. However, the direction of impact may also be predictable. It will likely be forced to live in a relatively low value area where it isn't so hard to make eyespace, while the opponent gets influence around. We can say that a weak group may have a shell of positive influence (positive factor that is, so more akin to itself) on the most nearby intersections, but a negative influence on the places the opponent can secure by attacking it.

From this point of view, the analysis of A above can be though of as:
If nearby to A, the opponent is strong, this cancels out the negative influence of the weak group already. Each move may be fully positive, in proportion to how alive the group is rather necessarily about territory and potential.

If nearby to A, the opponent is weak, then A may be alive enough to get a large proportion of the positive influence locally with any move. The main additional value is preventing the opponent's sacrifice strategy and double threats by minimising their shell of negative influence.

When two friendly weak groups are nearby, the attacker may find value in cutting even if such a move is an overplay relative to each. We can say that the negative influence overlaps too much with the other group. In the most severe case, one weak group (the smaller) will have to be sacrificed for the other to live. That is a successful double attack. An overconcentration would be if the defender must play a slow move to connect the weak groups to prevent such a threat. If optimal this means there is no active way to defend. i.e. defending one's own weak point and claiming the extra potential by leaning on the attacker's weak points isn't worth the cost.

When attacking weak points where there is a cut, it may be that the defender already has reasonable eyespace on all sides. Then the attacker should consider balancing the direct cut with more control of the eyespace region. The cut probably should be the focus, as attacking can get the extra forcing moves to profit, but the largest threats arise from more immediate threats, so remaining within 1 or 2 moves of being able to cut is important.

I'm not sure if there is some kind of inverse square law of gravity/charge that says when friendly/enemy groups are attracted to the same territory/eyespace.

An ideal theory would be able to prove good enough approximations that in order to refute a move, all you would have to do is show the refuting move, count and compare. For example showing that 4-4 8-3 enclosure is too vulnerable to the 3-3 for this to be efficient.
In terms of automation, I think that at the least, each large weak group should have slots for the order the opponent is expected to attack in with a field of expected influence in that scenario. This guides maximin to which direction to play in globally to compensate local imbalances. If the group is so weak that a narrow 1d path to life needs to be found, then many slots may be needed. It may need miai at each stage between living with another eye or escaping towards a potential other eye.

I realise that this does end up fairly related to GnuGo, so perhaps I didn't give them enough credit. But this is also a guide for how humans should think when they play and evaluate in a review or use tewari.

What is a fight?
What are standard fighting possibilities? Why is there a windmill spiral wave?

A fight is probably one where all attacking possibilities are heavy. During a fight, the temperature may increase, with threats getting larger since they get closer to the boundary but also may concern the life and death of a larger area as the cutting stones involved become larger groups. However, in optimal play, the end result will be balanced overall (extra value for the side that spends more moves, that is close to the average temperature during the fight).
Evaluations and decisions may be as difficult as a semeai, which is basically a fight where there is no room for retreat and life and death is completely dependent on killing the opponent (even if full evaluation may allow sacrifice).

The choices are often about status and judgement. Which cut to take?

Just live vs Capture the weakest boundary stone vs Cut the weakest boundary stone first vs Cut into empty space with a stronger cutting stone.

Many modern 4-4 joseki have these choices repeatedly appearing. If the opponent is too strong nearby that you don't have much potential (i.e. little buffer room before you have to live small), you should try to live faster. If the opponent invades the 3-3, perhaps play the double hane which connects all your stones but takes gote and reinforces the opponent's side control on both sides. This can be slack, i.e. not attacking at the maximum and hence making the opponent lighter. However, your centre gets stronger and the opponent's isn't alive yet. It can be used to sacrifice an area where you are already weak.
Other 4-4, 3-3 invasion joseki flow into the outside, giving way in the corner. The most complicated ones refuse to let the corner out into the sides but leave two cuts on the outside (and two unsettled boundaries). The judgement is often that the attacker should choose the most valuable point and the defender should respond accordingly.
Some 4-4 joseki have the attacker getting eyespace on both sides by attacking the corner and sealing it. This is common to reduce the opponent's potential when there isn't a clear direction to build the 4-4's own (perhaps because building leads to balancing it against the corner's development which may be too large since the corner is alive). However, they become thin this way and the corner can live and cut the outside with local pressure on both sides from the centre which can be used to sacrificed on just live easily and eliminate the attacker's potential.

Other

In a cross fuseki, the value of a 3-3 point stands out as approach may be met with the small knight's move which costs you on the other side which was also your potential. However playing 3-3 directly loses you centre potential too, but you expect some return from being able to attack both sides.

Since eyespace has more influence than a weak group (which may have negative influence), then if when it comes to choosing which side to block for a 3-3 invasion to a 4-4, consider that the threat of a large territory also has influence but not as much as the factor a living group has on its potential. Normally don't block in a way that pushes the opponent into your territory because that may make it too small for too little gain. Be careful that the local situation may remain tense as weak points remain, especially if there is a cutting point. This can give one side the extra influence of several moves in a row locally.

The search for the "divine move" (in Hikaru no Go) perhaps asks for a fair result on the board. But biased in practice due to player's differing experiences.

What is "pressure" in Go? It is in proportion to strength perhaps. But increasing pressure locally is perhaps about asking opponent to fight more locally or else lose out locally. Like the 3-3 invasion, one space jump variations.

To summarise the key lesson of the lengthy post,

The opponent's influence near your group is related to how easy it is for each side to live there, and may be positive if you are weak. However their total influence should be less than if you didn't have a group there at all. This should inform my estimate of komi, though I don't know how.

I think it must be local balance equations but how exactly?

power

If power (Dahl 1957) is to do with A getting B to do something B would otherwise not do, perhaps weak points interact when one with high temperature (or at least larger follow ups, perhaps in a semeai) affects the local main line elsewhere (nearby, or perhaps a ladder breaker).

Note that the threat of follow ups is more powerful than the execution holds perhaps under this definition. The execution loses the aji of different threats.

tightness on weak points

by playing tighter on a weak point, you play closer but your own stones may be in the short term weaker. However, the opponent's weak point may remain weak if their whole group is weak unless they can sacrifice or cut through your weak point.

full tightness may capture their stones.

Slightly less than full tightness may be ok when capture doesn't work or isn't worth it, but often also helps the opponent reinforce their weak point. Consider the classic variation of the 3-3 invasion to the 4-4 with the hane but not double hane. The 3-3 lives in sente, but in return, the 4-4 still has some endgame on it and solidly poking into the side, which may be good if this damages stones their. However, if there is no centre/side support for the influence, this can lose control of shape and eyespace since the 4-4 still has a cutting point and the weak points of the 3-3 are much smaller (the corner can't be captured). Hence, the 3-3 invader may be able to take sente to reduce the side first and sometimes even force the 4-4 to live small.

This goes downwards towards just solid connected shape (as is more common today in 4-4 joseki) which asks the opponent to balance weak point defence with reducing the outside influence, so weak points will likely remain, but the influence of the outside may not be as extensive, but it will be stronger where it exists. And the 4-4 gets sente.

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When both have a weak point, consider the direction in which you can build a moyo. Try to make miai of a bigger moyo and playing closer to weak points.

If the opp has no weak point, avoid playing nearby, but defend if you must. If the opp has a weak point, try to defend actively by attacking to help connect to your weak groups or make moves smaller or first shoring up weak points if you expect to be able to make territory from attacking.

Don't overconcentrate by ignoring boundary weak points. If the opp is solid enough that you don't expect to be able to invade, then especially at the boundary of development, try to lean on them.

To say a move "doesn't work" normally refers to boundary plays that stretch too far, so that tactically the end result isn't good. It is normally a "fight" which increases the temperature, but the end result isn't positive for the starter, or perhaps only equivalent to a small gote.

how to play bad go.

ignore weak points. take the opponent's eyespace when they are safe in the centre and attacking your weak group. surround strong groups. play forcing moves that aren't actually forcing. only see that your moves are connected and your opponent's aren't locally, without appreciating double attacks.

heuristics for eyespace. How many stones are required to secure the centre?

IIRC Erdos's (not very good) heuristic for the maker-breaker game is that a set that requires n more elements selected to reach completion is worth 2^-n to the maker. The value of a move is accordingly the sum of various powers of 2.

https://en.wikipedia.org/wiki/Maker-Bre ... _for_Maker
In Go, d2=2, k=4, |F|~|X|, so this condition is impossible. Curiously when k=2 (i.e. in the corners), then this means Maker wins. In that case, in higher dimensions, perhaps Maker can't win, and hence my komi prediction was wildly inaccurate and should be much lower?

Another way to think about it is that if n moves are required to make an eye and each move threatens n eyes. Then initially the value of a point is 1/2^n. The first move has value n/2^n. If subsequent moves are of this sort, and 8n-4 moves are required in total, then?

NB in go, an advantage for the breaker is that eventually the maker may threaten eyes that the breaker may have already broken. An advantage for the maker is that to break an eye, the breaker also needs to be alive.

Hence it is much easier to make eyes that this suggests.

Perhaps sum over different possibilities? Must be less than 1/2 chance in total on a neutral board as an eyespace point needs to be empty and not dame (i.e. controlled by one side). Then we integrate over all possible. It might not be finite.

F(f(theta)) from 0 to 2pi. Where F is the integral of f(theta)dtheta. f is distance to eyespace point and should be continuous-like (i.e. step size at most one). Finally we want sum over 2^-(F) over all possibilities. Say f must be integer. And that f is constant for a range of pi/2f.

If a range of dtheta can be arbitrarily plugged into any integer value of f. Then contribution to F is accordingly 1,2,3,... . This can be factored out and the contribution is 1/2+1/4+1/8+ ... . This tells us to double the integral for every dtheta which will blow up. This is approximately continuous for dtheta small and f big already.

This all probably made no Go sense. The best heuristic so far is probably just 1/2^(-n) as originally or 1/16.

Flow of go

common fighting shapes in go are huxiangqianzhi. Mutual restriction. weak groups adjacent, fighting for boundary.

Go is about harmony, judgement. In empty space, the decision is about which weak point to take against valuable areas, and what to do when there is no big weak point.

If mutually weak, that "attracts moves by ?" unless repelled by difficulty of local life. every stone repels the opponent according to the cost of living, but its potential attracts them.

e.g. normally don't attach to strong wall as what are you trying to control? even if you aren't trying to control next to the wall, opp can counter easily and even if they can't take what you are trying to control, there will be a higher cost since your boundary stone is weak.

nb most valuable ladders arise from joseki that surround opp from centre but also try to block off a side.

myt: why is fight tense? every move is killing, needs to count liberties. playing a weak point loses at the next weak point if the opp must respond but may gain more beyond that with threats.

attraction of weak points

well value is linear so the value of a move is roughly sum of value on each 4 neighbours. We can say that moves are attracted to weak points. However, to what degree? How much does it change if the move is further away? Well, if it is about taking out the last eye or shorting a liberty, far away has zero effect. If it is about making partial surround, then perhaps less moves are required to kill and value is in proportion to that. If it is about supporting profit from attacking moves, then be careful not to overconcentrate, but the value will be in proportion to that area with little to no connection to the value of the weak group, beyond perhaps making moves forcing earlier (if they were forcing anyway, there might be no impact).

I doubt there is any particular inverse square law, though there might be if we consider this for all weak points around even summing the smallest no man's land intersections. It's just that weak points dominate the calculation, like black holes sucking up moves.

non-equilibrium thermodynamics. It is clear that this is deeply relevant to go, and probably any game, but I don't understand the theory well enough. Go is fairly general, so go concepts are likely to be relevant to other games (with some stretching, metaphors, etc.)

What is entropy in this case? The depth of the tree? my point about temperature being like extremes for W and B makes some sense. What are the appropriate extensive variables? The must be functions of stone positions but is just saying stone positions enough for a theory?

critical points correspond to my weak points notion, the most unstable areas where each move can affect a lot.
what is free energy then (if F+TS is conserved?). what are thermodynamic forces? probably linearish locally and strongest near weak points. If optimal play is considered the lowest energy state then, does the second law become minimising differences to this? probably free energy is like local temperature. and temperature in go doesn't quite correspond to thermodynamic temperature.

If a partition function can be found, much can be done.