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Thank you, Robert, for adding a definition - that's something we need in many of our articles. Now, I have a question about it. It seems pretty wordy to me, as if we said "money transfers are payments of cents, payments of euros, receipts of cents, and receipts of euros." Is there a fundamental difference between the 4 options? If it's easy to make territory, doesn't that mean that one can also connect friendly stones, and that it's hard for the opponent to do the same?
For influence or thickness , each of its definition aspects 'connection', 'life' and 'territory' can vary in its degree. Some influence or thickness is strong at connection but weak(er) at life or territory, some other influence or thickness is strong at life or strong at territory or strong at both connection and life but less so at territory etc. Therefore a consideration of only one of the three basic aspects would be insufficient. E.g., if the opponent starts to reduce the player's moyo created especially by a high wall, then typically the reduction stone is already alive (due to its running potential towards the center) but is not already connected; its life aspect is better than its connection aspect. Likewise, degrees for the player can differ from degrees for the opponent.
Interesting discussion; I should try to better understand these other aspects of thickness, since I often only make good use of thickness when my opponent allows me to turn it into territory. It would be good to have an example; maybe I'll come back when I have one from my own games. (I currently don't play much online anymore, so I would have to search my archived games.)
One thing in your reply confuses me: Obviously, to be alive, a stone needs to be connected with other friendly stones. But you are using the terms in the opposite way; you say that a stone can be "alive" without being "connected". I think this is because "connection as used in go can have several distinct meaning", and so can "alive" (see Alive - Introductory). It seems what you mean is "potentially alive" versus "strategically connected with stones in another area of the board". If that is so, then the definition needs to be more specific as to which meaning of the term is used.
To live, a stone does not need to be connected with previously existing stones. Degrees of connection or life are checked in relation to previously existing stones. Therefore different degrees of connection and life are possible.
A stone can be alive but not connected and not connectable to previously existing stones; such a stone makes life together with newly played stones.
There are indeed different kinds of connection. In the theoretical models (which have the precision you are searching), I use a specific kind called n-connected, which has the feature to express the degree of connection by a positive, zero or negative natural number n. The defender playing second can pass his first n moves (zero or positive n) or playing first needs n plays before the opponent attacks (negative n) to establish a string connection (forms one string) or direct connection (cannot be cut).
In my book, you can see examples illustrating these terms.
n = -1 means, what is traditionally called, unsettled connection status
n = 0 means the opponent can threaten the connection and the player has to answer
n >= 0 means connected
n = -2 allows ordinary ko threats threatening to establish connection
n <= -2 means not connected
Usually consideration of values between -2 and 2 suffices.
In thickness groups early in the game, 0-connected is common for ordinary walls that are not just one string. 1-connected is a sign of pretty great thickness (WRT to connection).
The definition given on the parent page is not the theoretical model but the less formal impact model, where usage of n-connection is avoided. The precise theoretical model is useful in seemingly unclear situations or for meticulous comparison of qualities of different plays.
"n-connected" is a great idea; I wonder if this can be generalized to what we may call "n-ability", for lack of a better name. Some of what you're saying, such as use as ko threat, seems to be equally applicable to other things a player wants to achieve, such as getting life or snatching away a big endgame move. Maybe we can create an article about that general concept.
Which of your books are you referring to? Could you please also add the formal definition to the article?
I'm sorry, I realize I made a mistake in my last post. Connection and life are already combined in the first bullet of each pair in the definition (bullet #1 for friend, #3 for foe), so I shouldn't have contrasted connection with life, but connection and life with territory. Now, as for territory, you can't have that without connection or life, can you?
Several (or many?) concepts can be generalised in a similar manner. Start with n-atari: atari after n approach plays.
The book is JosekiVolume2Strategy.
I cannot copy the whole book here but maybe some days or weeks later I find time for adding more. Maybe somebody else is faster with the citations.
That connection and life are combined in the same bullet is only a linguistic convenience! You can make two separate bullets if that confuses you less. (Also for the opponent.)
Territory does not really require connection and the player's life (imagine a huge moyo) but requires the opponent's missing possibility for life. In practice there is often (or always?) territory based on connection and the player's life though. The crucial matter is, of course, again the degree of territory: Is it already territory (0-territory) or does the player need to make n more successive plays to turn it into territory (n-territory)? (In practice, usually 0- and 1-territory are all one needs to consider.)
Why not use surreal numbers/CGT values instead of integers? It fits in with their use in other places and if you get interactions between two different stones you want to connect to, you can find an infinitesimal to model that situation.
Models useful for players should first of all be applicable. Distinguishing the integers is good enough for practical purposes. It is very meaningful to assess how often a player can pass / play elsewhere or how often he needs to make successive plays for defense. Numbers of plays or passes are natural numbers; there is no need for fractions or even infinitesemals.
Influence can be different for each intersection. Determining the natural numbers for each can already be difficult enough and one determines them only where necessary for a better positional assessment. So asking for fractions or infinitesemals is asking for inapplicability. The power of my models lies in their applicability and in good meaning in case of approximative use such as "2-connected or better" or in case of partial use like when determining only an intersection's degree of connectivity but not the degrees of the life or territory.
The model answers questions like "Can a black stone placed there be connected?", "Can a white stone placed there be connected?", "Is connection of a black stone placed there better than connection of a white stone placed there?". E.g., if a black stone placed on a particular intersection would be 1-connected to some of his previous strings and a white stone placed there would be 0-connected, then we know, that with respect to connectivity the intersection is under greater black than white influence. If both degrees are 0-connected, then that intersection is neutral with respect to connectivity.
Now suppose that at the border line in between a black and a white moyo you have already determined the intersections of (almost) neutral connectivity. Your next question might be: Which black play provides the most additional territory? You test a play on each of those intersections and evaluate the territories. For your moyo, you determine the 0-territory (which counts 100%) and, if you like to be careful, also the 1-territory (which counts 50%) and do the same for the opposing moyo. You get an approximation of expected territories. Only an approximation but that is good enough for the opening or the middle game. Precise territory values like in the late endgame cannot be determined anyway because the opponent's plays elsewhere do have a small but relevant impact on local territory, too.
So although the model does not necessarily determine perfect play, it is (much) more precise than what I have seen before for opening and middle game. Previously we had only visual aids such as sector lines but now we can verify whether, e.g., a moyo border line is where the sector line suggests or closer to Black's or White's moyo. Also we do not need to guess the best move somewhere at the border line any longer but now we can associate each candidate with a territorial value. Such values are approximations, so if two candidates suggest almost the same territorial value, then further considerations might be needed for making a decision between them.
Similarly if we are searching for a good, stable reinforcement of a moyo of modest size, we would choose the 0-connected plays (or better connected, unless overconcentrated) and disregard the -1-connected plays. So we get candidates of 0-connected plays, then verify if they are also alive and choose, e.g., that 0-connected play providing the greatest territory increment to achieve also the greatest efficiency. (Or global relations let you choose slightly less territory but a better far distance development.)
For the sake of assessing a group's (e.g., wall's) aji, the natural numbers of connectivity in -1-connected versus 0-connected versus 1-connected are already good enough. -1-connected suggests a reinforcement unless the group is indirectly connected (the opponent can cut but currently cannot get an advantage from cutting). 0-connected means the group is connected but contains noteworty aji. If it is a wall, then very likely an extension is mandatory. 1-connected means that the group is as strong as having already been reinforced by a proper move. Fractions are not needed! If you really want more information, then determine also the degrees of m-alive and the intersections of 0-territory and (optionally) 1-territory in front of the group.
Having three natural number parameters per player to characterise thickness of a group or influence of an intersection is enough (and often more than enough) in practice. Fractions would only deceive us. Rather use (some of) the parameters to determine candidate plays, then apply also other (e.g., global) considerations for a) verifying whether the local region is the most urgent and b) which of the local candidates fits best in the global context.
We can start considering fractions when, centuries later, the endgame is understood so well that it will have been extended to the middle game:)
Superdave, are there any examples where surreal numbers/CGT values have been used successfully for move counts? I've only seen them used for points.
I suppose there must have been such things among researchers but I cannot point you to a place.