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 Post subject: complexity of go positions
Post #1 Posted: Mon Jun 27, 2022 6:51 am 
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a few thoughts referencing https://www.lifein19x19.com/viewtopic.php?f=15&t=18601

"How should you measure the complexity of a go position?"

- Size of weak groups involved.
- Number of groups.
- Number of moves spent in the same local area. i.e. moves oscillating between attack and defence of multiple groups. For example this makes ko very complex, since to settle just a local area, perhaps more than say 6 moves are needed, since many are at the same point. Then reading is more difficult by which I think it means that in order to prevent losing a lot of points, a lot of reading is required.
- Often, there are simple variations without needing much reading that only lose a little, but in complex positions, there is no simple sacrifice or exchange or settling.
- The value of a small shape mistake. If the second best move is much worse than the best, we can say there are complications.

Note that the more weak groups there are, the more likely that sacrifice variations exist that only lose a little (like the averaging effect of miai groups). However, this can still very complex at a high level if neither side is willing to give up even a few points.

Overall, we have local life and death at a high level but which can still require detailed reading of weak points, and also reading of their knock on effects and influence on neighbouring groups. For example, attacking to get a sente seal is very valuable if there are neighbouring weak groups since they won't have valuable moves near your thickness, so you can settle your own weak groups by pushing towards your thickness. But how do you value such moves?

Taisha: A leaning move tries to get more forcing valuable forcing moves keeping the corner low on the 3rd line. Quickly there are 4 weak groups in a cross cut. With each group, there are several forcing moves and different defense options depending on what you want. It can get very complicated when the presser continues pressing from the centre, making the opponent heavier, which works well with support. It is then complicated because the corner side isn't sure which group to defend as moving into the presser's area with many follow ups is valuable, but the life and death isn't clear especially if you aren't clearly alive in the corner. Overall there are simply many variations (the legendary proverb said there were 1000 variations I think) and Japanese houses had their own secret research. Probably this is because there is a windmill of 4 groups, and developing any is very valuable, so tiny nuances can change the best variation, with sacrifice options. This is also when more local moves are likely to be valuable, since each group is close to able to live locally, but may be put under quite some pressure. However it isn't so complicated that it takes over an entire side of space, since the groups are on the 3rd line and such groups are either normally clearly dead or alive.
Avalanche: The outside tries to get the corner side to calmly defend the 3-3 as the corner side already surrounds the eyespace and territory, but the outside group has a hane at the head of 2 or 3 stones problem. The outside is only trying to force the corner low in sente with some forcing moves on the outside, and therefore needs to keep up the threat of playing 3-3. Cutting 3-3 from the inside gets many more liberties, and hence more forcing moves on the outside too, while the outside has enough space in the centre to be ok for now. Soon enough there are 6 weak groups.
Muramasa's sword: Very popular in the Nie Weiping era. AI has refuted many moves of this variation, and the corner group may be satisfied to just let the opponent cut off the centre. However, the outside has stones close to the corner's cutting point on the 4th line, so the cutting point can get more liberties but threatening to link up, as well as the cut in the corner. The corner is just trying to get sufficient space to push through a valuable area, making the outside's attack too close to strength. But the aji is bad. The corner must capture the cut or else they lose the corner and lots of territory, and capturing may get strength to counterattack both sides. Hence it is simply a case of reading if there are enough liberties and local space for the relevant groups. Exchange variations are possible since the cutting point needs to threaten to connect to the outside in order to live and the corner may have enough liberties to threaten seki or ko.
MYT flying dagger: AFAIK this was never seen pre-AI. One 3-3 corner group is surrounded by a group with 2 cutting points. Living directly is possible but submissive, so cutting tries to live more actively (either capturing without needing to play a small move), or damaging the outside. The 4-4 surrounder has simplifying variations, but can choose a very complicated line because the corner is weak enough on the boundary of life and death that the outside may be able to fight. There are many sacrifice variations. The cutter's forcing moves are in the centre, and is therefore unable to capture (though the aji is very bad if everything lives cleanly). The attacker's group can't easily live either but can get enough liberties in sente to make a seki in the corner as the outside gets solid enough from the corner trying to live that the corner runs out of forcing moves and must go back to the corner. Before or after a seki, the outside group will also need to live, but remains surrounded by virtue of having started off as a cutting stone.
There was a highly researched cousin variation pre-AI which cut the other of the outside's two cutting points. There were some very complicated variations, and I have a vague memory that some variations (perhaps not this joseki) would crawl on the first line across the entire side in order to make an eye in the other corner, after which this would win the capture race.

nb: if there is only one big key weak group, the owner of that group is unlikely to be able to cut the opponent, so everything will revolve around that group's life and death, which can be treated as a cutting group. This is the case for muramasa (though it is an optional cut) and MYT (the 3-3 corner), but for the avalanche, it is more about aji, and for the taisha it is more about development.

If there is a commonality to how these joseki start, it is probably that there is a valuable point (often 3-3) at the boundary of two weak groups, and yet it is inefficient to play there directly. Hence you try to seal the other group in hoping to leave 3-3 as a forcing move since it works with surrounding.

regarding the benefit of recognising good and bad shape, I think part is to do with separating the value of forcing moves in terms of gain in different directions. Leaving no forcing moves means the opponent is less interested in continuing locally, which is even better if they must anyway. Playing faster sometimes means the opponent is able to force you back to playing slow shape, but it is about consider the value difference of future moves.

20220822 based on my writing elsewhere. It seems that complexity can be considered a function: the time it takes for an algorithm that minimises error of evaluation to go below eps error. This can be arbitrarily high with large dragons in semeai.

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Last edited by dhu163 on Tue Aug 02, 2022 8:13 am, edited 4 times in total.
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 Post subject: Re: complexity of go positions
Post #2 Posted: Mon Jun 27, 2022 7:50 am 
Judan

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You have mentioned some practical aspects of complexity. From a mathematical POV, complexity of a position depends on, e.g., the minimum number of necessary decisions for followers, follow-up moves, the initial position and its moves to find and verify a) one correct sequence (intermediate solution) or b) all correct sequences (strong solution). (A weak solution is to say something about existence of solutions. If go is generalised to, e.g., arbitrary board sizes, one can try to determine complexity classes.)

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