It is currently Mon Feb 10, 2025 8:20 pm

All times are UTC - 8 hours [ DST ]




Post new topic Reply to topic  [ 8 posts ] 
Author Message
Offline
 Post subject: Go maths literature
Post #1 Posted: Fri Jan 24, 2025 9:05 am 
Lives in gote

Posts: 474
Liked others: 62
Was liked: 278
Rank: UK 2d Dec15
KGS: mathmo 4d
IGS: mathmo 4d
Hi people.

I've spent a few years thinking about applications of maths to Go. CGT provides THE framework for endgame, so I am more thinking about opening and middlegame, and extensions of endgame theory (this can work surprisingly well in the opening where there are a limited number of good variations locally which often settles quickly into territory, possibly with some weakness whose cost can be estimated, and the evaluation of the outside influence can be estimated by splitting into cases. Even middlegame fights have some possibility of being understood by endgame, but much more confusing, perhaps waves becomes more appropriate.)

(NB I think there are still feasible open questions in endgame though. For example the nature of the Go board leads to a certain distribution of endgames. Are all finite trees with integer score leafs possible? What is the distribution in a typical game? How does this change with a different graph?)

As far as I know there is very little written in Asia on the serious application of maths to go. I've only found some mysticism. There is western material on CGT, search algorithms, seki, ko and semeai. (Berlekamp and his students, Spight). However, I am thinking of more general things.

My ideas revolve around marginal value (e.g. height in wall), local minimax by expected temperature (how to modify strategy), fractals (energy, entropy), some endgame formulae that might be relevant (though I don't yet see how), criticality, topology.

Basically I want to see a more rigorous theory of Go optimal play, and I think AI makes it possible.

At the least I think that although the problem is difficult, fuzzy, lacking total orders, I think due to it being quite unexplored, there is much low hanging fruit, and sometimes the low hanging fruit is even the most useful in application.

My question is this. Is there any literature at all on these sorts of topics? I would be interested in reading what exists, and not reinventing the wheel. Does anyone know of anything on opening and middlegame in terms of maths or connections between Go and maths that I'm not aware of?

Top
 Profile  
 
Offline
 Post subject: Re: Go maths literature
Post #2 Posted: Fri Jan 24, 2025 11:46 am 
Judan

Posts: 6255
Liked others: 0
Was liked: 794
dhu163 wrote:
CGT provides THE framework for endgame


No. Rather, CGT provides ONE OF THE frameworks for endgame. Another framework for endgame and its application during opening and middle game is the non-CGT mathematics explored mostly by me and Spight, of which you find almost all in [22]

https://www.lifein19x19.com/forum/viewt ... 45#p143245

(I have also studied a bit of maths for non-endgame-like aspects of opening and middle game.)

Quote:
As far as I know there is very little written in Asia on the serious application of maths to go.


AFAIK, they also have a few CGT researchers but likely Western researchers have done the most.

Quote:
a more rigorous theory of Go optimal play, and I think AI makes it possible.


No. AI is not there yet. Humans can interpret AI samples to illustrate human theory but humans have to research the theory. AIs are not serious go theory researchers yet. And I think not for quite a few further years.

Strong AI play in itself has nothing to do with go theory research. (Of course, there was human research in informatics to create AI with strong play.)

Quote:
Is there any literature at all on these sorts of topics?


See [22] and much of the CGT literature.

Top
 Profile  
 
Offline
 Post subject: Re: Go maths literature
Post #3 Posted: Fri Jan 24, 2025 12:49 pm 
Lives in gote

Posts: 474
Liked others: 62
Was liked: 278
Rank: UK 2d Dec15
KGS: mathmo 4d
IGS: mathmo 4d
As far I understood, your framework is basically miai counting, which comes with some general bounds (orthodoxy, sentestrat etc which were part of CGT). But further analysis (creator, preventer, follower) is limited to relatively shallow trees with only one (or two) optimal local moves for each side. Correct me if I'm wrong.
A lot of the casework is working out what happens in certain ideal situations. To me that isn't an end in and of itself, but want to see if there are patterns that extend generally.

The miai counting system is powerful because it simplifies calculation (summing numbers rather than games) and typical Go endgame often isn't that deep. But not complete without infinite effort. However, even within this system, I think you (and I) missed ways to simplify. In particular I think the alternating sum can be just reduced to 1/2 the temperature difference between local moves (assume rich enough environment), and this is particularly useful for comparing variations in the opening. You can add a parameter for the temperature at which the opponent uses different aji, which changes depending on the environment (and hence is a variable affected by future strategy). I'm still not completely clear how it works though, especially as it is quite fuzzy.

Though I have a question. Do difference games count as part of CGT?

To me CGT has the right definitions to encapsulate the most general positions. It provides general operations with basic consequences including bounds. Anything else that works and is correct has to predict equivalent answers, seems to me within that framework. Maybe my philosophical thinking is flawed here though.
Or possibly I thought the basic parts of the theory were too "obvious" if you've read the sensei's library page on miai counting, and thought about difference games. Within the miai counting framework, I think I rediscovered the bounds of orthodoxy, before I knew it was a thing.
Though perhaps there are always more complexities to consider. Perhaps it isn't obvious what is necessary and sufficient. But it still isn't obvious to me what is necessary and sufficient.

AI not there yet: yes, agreed, I meant humans need to research, by which I mean, that is what I'm trying to do. Strong AI gives data to analyse.

I am aware of [22], though may have forgotten almost all of what I read.
The biggest non-trivial stuff I think is thermography. But even there it seems to me not easy to use directly, so I'm not sure it is more helpful than CGT.
There are almost certainly parts of CGT with applications that I am not aware of, but still it takes much work to consider consequences for opening, middlegame.

Regarding maths for opening and middle game, which is more what I'm getting at, could you hint at what sort of concepts you worked with.

--
still, at least you have something concrete, I am thus far unable to state my thoughts clearly enough in equations, and yet still feel as though equations are possible. I hope this doesn't go on too long.

Likely general equation impossible, but I think there are still regional equations and ways to detect edges of validity. Or maybe it turns out I am crazy, or that this is beyond me.

Top
 Profile  
 
Offline
 Post subject: Re: Go maths literature
Post #4 Posted: Fri Jan 24, 2025 9:12 pm 
Judan

Posts: 6255
Liked others: 0
Was liked: 794
The framework of my endgame theory you might call modern endgame theory, which uses miai counting, quite like that of CGT. However, from a view of mathematics, my framework is school mathematics often with linear algebra and mostly (except for ca. 7%) is not CGT with its infinitesemals. I often compare two or a few numbers (such as two move values) algebraically to see which is larger.

Much of my endgame theory studies the basic cases of simple local endgames or several such on the board. You refer to them as relatively shallow trees, i.e., no follow-ups, follow-ups of depth one or depth two. I often study a local endgame with simple follow-up(s) in an environment of simple gotes without follow-ups, such as an ideal environment. I have solved every such basic case. These are the cases that I could solve. Future research needs to expand to deeper follow-ups etc. Currently, we players can and should apply my theory for simple follow-ups also as a simplifying, approximating model for usually similarly behaving iterative follow-ups, more positions with several local endgames with follow-ups or more complicated environments with follow-ups, kos etc. Usually, realistic environments behave similarly to the solved ideal environments.

Yes, my theory is powerful because its study of the basic cases simplifies calculation and compares (not just sums up) numbers instead of the much more complicated CGT's combinatorial games with also their consideration of infinitesimals.

You must be right that we overlooked further ways of simplification! :) Your idea of approximating an alternating sum by 1/2 the difference of two move values (aka local temperatures) of local endgames sounds interesting and worth studying.

Difference [combinatorial] games (or positions denoted algebraically) belong to CGT, at least if done formally. If P and Q are positions, then P - Q is a difference game and needs its CGT analysis.

CGT is a fairly general framework but is not the only "most general framework" because, even with thermographic extensions, is not a generalisation of everything. Instead of CGT, school mathematics etc. can also be generalised. However, as you see in my maths, there are some mathematical relations between CGT and non-CGT. In particular, a few of my theorems needed proofs best done in CGT proof style.

Do not refer to Sensei's Library maths pages on miai counting as if they were correct! They are some introduction for beginners but contain a few mistakes. (Not to discuss them now, just pointing out.)

If you have forgotten most of my mathematical theory after reading once, study it several times! :) Such is just good practice. I also needed to study Berlekamp / Wolfe several times until some reasonable understanding.

Thermography is hard to apply. For every fairly simple local endgame, I need several hours to apply thermography. Thermography is a powerful mathematical theory but essentially inapplicable for go players during their games.

As to concepts of my endgame theory applicable also to opening and middle game, I will use another thread in the books subforum to comply by the L19 Terms of Service.

Top
 Profile  
 
Offline
 Post subject: Re: Go maths literature
Post #5 Posted: Fri Jan 24, 2025 9:22 pm 
Lives in gote

Posts: 474
Liked others: 62
Was liked: 278
Rank: UK 2d Dec15
KGS: mathmo 4d
IGS: mathmo 4d
I had a thought that maybe I should be more polite re 22 and just say I don't at the moment see why most of 22 is interesting to me.

I also had some defensive thoughts that maybe I want to see more rigour because I worry that any statement people make about go can be nonsense and it is hard to tell. Or at least hard to tell the region of applicability. e.g. proverbs.

Even if clarifying the logic might not strictly be mathematics, it is reasonable work for a mathematician. For example, I thought about what is the worst possible move on a go board? I think other than shorting libs, filling in eyes, worst moves are next to thickness of either side. So the proverb about play away from thickness is fairly general. This sort of context is a long way from a complete theory, but should perhaps a derivable from a good theory, so perhaps a sort of test.

edit: did you notice the flaw in my statement above? well, sometimes it is good to play next to your opponent's thickness, because it can be part of making your own territory that is thick enough despite being next to your opponent. Still, the value of the move is in a sense independent of any value from being next to thickness. In a sense thickness can be 1d, whereas the board is 2d. How easy to make plausible sounding statements, how hard to make exact.

re: maths for opening, middlegame.

I noticed you mentioned the idea of n-move territory. I dismissed this as uninteresting before. But now I think that it is a necessary part of the sort of theory that I am looking for, but it needs to combine efficiency, temperature and shape somehow.

Top
 Profile  
 
Offline
 Post subject: Re: Go maths literature
Post #6 Posted: Fri Jan 24, 2025 11:22 pm 
Judan

Posts: 6255
Liked others: 0
Was liked: 794
"sometimes it is good to play next to your opponent's thickness, because it can be part of making your own territory that is thick enough despite being next to your opponent"

I invented the related principle:

"Nearby reduction of influence and restriction of its use are possible if the reduction group spoils important development directions and is strong enough so that the opponent cannot profit well by attacking."

Top
 Profile  
 
Offline
 Post subject: Re: Go maths literature
Post #7 Posted: Fri Jan 24, 2025 11:40 pm 
Lives in gote

Posts: 474
Liked others: 62
Was liked: 278
Rank: UK 2d Dec15
KGS: mathmo 4d
IGS: mathmo 4d
Admittedly not what I was looking for, but there may still be gems in the follower details that hint how to generalise. I may ponder more at some point.

I suppose I should conclude that there is very limited public knowledge if any about applying maths to the opening and middlegame.

A note on a detail of endgame.
(Something I rediscovered when I was thinking about miai counting a few years. I also noticed the total order of simple gote, and I think there was something else with a total order, but I can't fully remember.)
I'm not sure if you are aware, it might be in your book(s). I know pros are, having heard it on a TV commentary. It came up recently, I explained this to a European pro.
Sente n is more often than not smaller than gote 2n. similarly komi is more often than not >T/2.

If you aren't, then treat it as a problem.

Re: rereading. Good point. I have noticed recently that rereading is more important than I realised. Probably because I read technical stuff too little, including in my degree. It feels like an accident not a plan that I tended towards mathematics in my life. Advanced maths always seemed magical and mysterious,
not for people like me. But at least at school maths was my strongest subject.

It takes time and detail to translate words into the correct mental picture. Though even then insight can be difficult. Rereading after several years still can make a difference.
For example the logical connection between Lagrangian and Newtonian dynamics still puzzles me. And it brings to mind probably not quite correct analogies to entropy and waves. I think I only just noticed how the terms relate in the derivation. But still interpreting it intuitively is difficult. I try to imagine the equations having moving parts but I'm still confused.
Perhaps analogously to this sort of thing, I think individuals collectively have a lot of knowledge of maths applications in Go, but converting it into communal language is awkward, especially when intuitions are different for different applications.

Top
 Profile  
 
Offline
 Post subject: Re: Go maths literature
Post #8 Posted: Sat Jan 25, 2025 2:07 am 
Judan

Posts: 6255
Liked others: 0
Was liked: 794
dhu163 wrote:
Sente n is more often than not smaller than gote 2n.


"smaller" is not the most suitable word here. You need more something like "to be played earlier". For the more accurate theory, see [22] etc.

Quote:
there is very limited public knowledge if any about applying maths to the opening and middlegame


I will discuss this in the other thread

https://www.lifein19x19.com/viewtopic.php?f=17&t=18385

Top
 Profile  
 
Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 8 posts ] 

All times are UTC - 8 hours [ DST ]


Who is online

Users browsing this forum: Google [Bot] and 1 guest


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Search for:
Jump to:  
Powered by phpBB © 2000, 2002, 2005, 2007 phpBB Group