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Classically, the sides (3rd line: the line of territory; 4th line: the line of “influence”), are valued highly. What I take from John's writings about the diagonals is that this is another line that should also be considered; maybe the “line of control”?
I have become increasingly aware that we may have become
trapped into a Japanese Weltanschauung of go. I have had this feeling for decades, partly because of go-specific comments made by Go Seigen, partly through looking at old Chinese games, and partly from reading about Chinese strategy and tactics outside of go. But the feeling has recently intensified. This was driven in the first instance by trying to look at go though AI eyes (hence, in a way, this whole thread), but mostly by looking at old Chinese go again.
First, I think Chinese go has been misrepresented as "all-out fighting" or "power go". Even the Chinese have said this in the past, but I think too they are now re-evaluating this. But for us especially, who have different associations for words like "fighting". When we hear "all-out fighting" I suspect that the sort of images that swim up in most of our minds is pub brawls, riots, or bench clearances in baseball. Very, very few of us have had actual experience of "proper" fighting (I too was a post-war baby) and there our images are formed (or misformed) by Hollywood.
But if you go back to the classical era, "proper" fighting was something that was always a strongly potential part of life, and even scholarly Chinese gentlemen-officials who tended to despise the military men made sure they read and even memorised books like Sun Zi's Art of War. The Go Classic in Thirteen Chapters is even based on it.
It has always been difficult to get to grips with Japanese go vocabulary because until the 20th century there were essentially no texts. Books of games and problems, yes. Books with words, no. But it was very different on old China. We have lots of old commentaries, often quite long, and by go masters not amateur amanuenses. They say eyes are the window of the soul. Well, so are words.
It is
very instructive to look at old Chinese go words. First of all they have been uncontaminated by Japanese vocabulary (unlike modern Chinese go terms). The palette is very, very different from Japan's. Not just in what is there but in what isn't there. No thickness, no moyos, no tewari, no miai - at least not expressed in the Japanese way. But some of the less obvious omissions (or near-omissions) may surprise you. "Attack" is actually quite a rare word. Instead they say an awful lot that a move is "severe". Is an AI bell tinkling already? Invasions, believe it or not, are rare. They like splitting moves - the whole development of Chinese fuseki, culminating in early/mid Qing times, can be seen as refinement of the 9-3 splitting move (in the area of a Go Seigen group incidentally

) - and they like encroachments. (Of course, invasions risk creating a separate group and nobody likes paying tax.)
But the one word that stands out is shi 势. Roughly translatable as power, it subsumes thickness and much else. So much else, that the US military has spent a fortune on trying to understand shi, which is still the basis of Chinese military strategy
and political strategy - and much else! Can you imagine the Pentagon spending money on studying thickness to understand Japanese politics. Old Chinese go is real hand talk - not a pale Japanese imitation.
And Sun Zi is always there in the background. The most eye-opening moment in go for me recently was discovering Huang Longshi's Five Grounds - based of course on Sun Zi's Nine Grounds.
Now, if you look at just these two elements - shi and grounds
through the prism of old Chinese go terms, you get a feel for the game that is very different from our traditional Japanese-based feel and one that, I believe, more accurately mirrors what is going on in AI go.
You need to read at least one big book to even get a tenuous feel for shi (Mott & Kim is the definitive one in English) but if I may be allowed to try to distil it in to one English word, it would be CONTROL. With some caveats - it is not just control in the sense of having your hands round the other guy's throat or other important anatomical parts but is also control in the sense of sheer overawing presence (and so, to go off at a slight tangent, this is why I keep saying that thickness with weaknesses is not thickness and that thickness is not thickness unless it functions as thickness - but function here means control rather than attacking).
Now consider this: one common name for the four starting stones of old Chinese go is "shi stones". Control stones if you will. And these stones are placed diagonally, so that each player starts with a LINE OF CONTROL, i.e. X marks the spot.
The other element of my attempt to model AI go was to map the board into overlapping areas. It seemed to work, although in a crude way. But I tried this before I discovered Huang's Five Grounds, and since then I have refined my thinking to reflect that. Think of grounds as a locale, an area where both players may be present and engaged in various ways (as defined by 'control' - battle areas, if you like). These grounds shift as stones are added, which makes them way, way beyond my ability to program, but one thing I noticed, by reading the old commentaries and analysing the words, is that time is an absolutely crucial element. Not in the ultra-crude way of sente and gote but in the wider (and subsuming) sense of initiative, in which you can play a gote move to keep the initiative (the Japanese gote no sente, of course, but bigger than that). The number of comments in old Chinese texts that describe moves as well timed or the in right order struck me as far higher than anything in Japanese texts. I think we tend to think of timing simply in terms of technique or tewari. In reality, as the old Chinese commentaries show, timing is about keeping CONTROL.
I have also been struck (as I was decades ago but didn't understand it then) by how often the old Chinese commentaries have recourse to terms such as 细,which gets translated in tone-deaf ways such as "meticulous", and 落得, also rendered in tonally deaf ways such as "is advantageous". I believe the go sense of 细 is better understood as "tactically precise", and once you get in that mindset you realise that this too is about keeping CONTROL (and, mutatis mutandis, defining the shifting grounds). 落得 seems to have made an impression too on the old Japanese players. They rendered it as 打得, both versions meaning literally "can play" but the reference is to the (early)
timing of forcing plays. Rather than the move being "advantageous" it is a no-loss move, which has very different connotations. And, in my new view of go, what that translates into is a way of defining GROUNDS. More bars tinkling on the AI Glockenspiel?
So, trying to put all that into a broken nutshell, I think a new theory of go in the AI era should look back at the old Chinese past (and/or Edo go, especially Dosaku, in Japan - but China mainly because of the huge volume of commentaries) and should emphasise primarily CONTROL and GROUNDS, with a strong secondary emphasis on severity, precision and timing as a way of keeping control and defining grounds.
And if you do revisit old Chinese games, think of them as battles and not fights. Even changing just one word changes the word associations and thus the perceptions. They do have highly developed strategy (as well as impeccable tactics!).