RobertJasiek wrote:
I expected gote with follow-up(s) to be easier than sente but, now that I study gote, I get the impression that we know much less and fall back to reading because we do not know better. Can't be! Bill, what do you or CGT theory know, what shape-independent, go-player-applicable theorems are known, how are they proven, and where to find them? Or is there really nothing of a kind we have touched for a sente in an environment? Do I have to start from scratch or which basic conceptual ideas should I rely on? Your conceptual ideas for sente have turned out to be very fruitful but so far I see nothing alike for gote with follow-up.
Thanks for asking.
Here is a little background.
CGT has two different approaches, both of which can be found in
Mathematical Go: Chilling gets the last play or in
Winning Ways. One is thermography, which matches traditional go evaluation and extends it with Berlekamp's concept of komaster and my treatment of multiple kos and superkos. The other is the theory of infinitesimals, which aims at getting the last play. Go players have long recognized the importance of getting the last play at various points in the game, but did not develop any theory, which is why some of Berlekamp's last play problems stumped 9 dans. I have covered both of these areas here and on Sensei's Library.
Neither of these approaches deals with gote with followers per se. There is no particular problem with evaluating them, and sometimes they act enough like infinitesimals that the theory of infinitesimals applies.
Now, my own approach, which I developed after learning traditional go evaluation, is to assume an environment of simple gote, and otherwise is straightforward comparison of results. It is easy to derive the results of thermography from it as an approximation. If you don't make approximations you can solve problems where getting the last play is important. However, I never developed anything like the theory of infinitesimals, and thermography is easier.
Let me give an illustration that is pertinent to your question.
The environment is a set of simple gote: {t0 | -t0}, {t1 | - t1}, {t2 | -t2}, . . , such that t0 >= t1 >= t2 >= . . . >= 0. (Note that this is not Berlekamp's universal environment, but is more general.)
How does Black play the sente, {2s | 0 || -r}, with r > 0, s >= t0 ? Obviously, Black can play it with sente. The result will be t0 - t1 + t2 - . . . . It is possible to show that that is the result with best play, so Black might as well play it now. If r = 0 then the play is an infinitesimal, which CGT says to play now. But, OC, in go we might want to save the play as a potential ko threat, and leaving it on the board might induce an error by the opponent by playing the reverse sente. So we might compare that result with that when Black takes t0 and White then takes the reverse sente. That comparison tells us to play the sente when r/2 > t1 - t2 + . . . . Or, approximately, when r > t1. (Note that that is close to the answer given by traditional go evaluation and thermography, but is not exactly the same.)
All of this holds true regardless of the relationship of r to s. If r < s, then the play is, as indicated, a sente, but if r > s it is gote. In either case the comparisons are the same. So, using my straightforward approach, it does not matter whether a play is a sente or a gote with a follower.
That said, in more complex comparisons it may be useful to make use of whether a play is sente or gote, to make things easier. But, strictly speaking, it is not necessary.