Sente, gote and endgame plays

[RobertJasiek]

We were talking about apples and oranges:) Now you clarify that you study a local sente with G as an alternative local option for Black and that so far you only study the case Big/2 > T, i.e., a special case of the follow-up move value aka Black's starting sente profit value being larger than T.

Thank you for your relevant research! I confirm the details of your message.

Your finding should be generalised a bit by not prescribing (all) the sizes of the leaves Ti, -Ti, Big, 0, maybe G and -R. Instead, use move values. Thereby you can describe a larger class of positions.

White to move, White's sente with White to move and White's sente with Black to move are "exercises for the reader";) I.e., if you do not provide that theory, I fear that I need to work it out some time.

My previous message is about my research in a different class of positions: a local sente WITHOUT local gote option at Black's start. (Same "exercises" ahead, OC.) No wonder that I got confused about some terms when they have not been applicable because you have studied something else:)

When I will have bitten myself through this, some other projects are a study of several local sentes of one player in an (ideal) environment and a study of several sentes of both players in an environment. Do you know how naive I were? I thought it was all easy but already the one sente studies are tougher than I imagined. Solving the early endgame (in parts) in general is not that easy after all. Uhm, I should have known:( I better do not mention simplifying assumptions, such as new influence having no value:) However, I do think that the classes of positions we are researching in are important ones for practical endgame application.

[RobertJasiek]

In a local sente with the initial position's count C and the sente follower's count S, we have C = S and accordingly the net profit 0 of the sente sequence. I have been wondering how this should be proven and have come up with a sketch of a proof using the assumption of a local sente being defined by its move value M being smaller than its follow-up move value F. However, then it occurred to me that C could be defined(!) as C := S. I guess that there could be more complicated proofs relying on min-max reasoning, temperature graphs and whatnot. If we forget about the combinatorial game theory origins and start introducing modern endgame theory afresh, which necessary definitions and which proofs are the best and why? I am not interested in subsets of local sentes, but want to discuss (simple) local sentes of arbitrary counts of its followers.

[Bill Spight]

In a local sente with the initial position's count C and the sente follower's count S, we have C = S and accordingly the net profit 0 of the sente sequence. I have been wondering how this should be proven and have come up with a sketch of a proof using the assumption of a local sente being defined by its move value M being smaller than its follow-up move value F. However, then it occurred to me that C could be defined(!) as C := S. I guess that there could be more complicated proofs relying on min-max reasoning, temperature graphs and whatnot. If we forget about the combinatorial game theory origins and start introducing modern endgame theory afresh, which necessary definitions and which proofs are the best and why? I am not interested in subsets of local sentes, but want to discuss (simple) local sentes of arbitrary counts of its followers.

Well, that equation antedates combinatorial game theory. It goes back in mathematics at least to the early 20th century and in go theory at least to the early 19th century. OC, it is possible to build theory entirely upon final scores, but counts are useful theoretical constructs. AFAIK, we do not have any records of the reasoning of go players of the 19th century or earlier about the equation, but it is not too difficult to show that evaluating the count of a sente position the same as the count of a gote position will lead you to play the reverse sente too early in many situations. For instance, consider this one point sente: {20 | 0 || -1}. Evaluating the count as 4.5 instead of 0 would suggest that White should play the reverse sente before this gote: {4 | -4}. While it is possible to construct a position where that is the case, it is almost always wrong.

it occurred to me that C could be defined(!) as C := S.

Well, that is basically what Conway did in his definition of thermography. But, harkening back to the early 20th century, the method of multiples shows that C -> S in the limit, as the number of multiples approaches infinity. And in my redefinition of thermography in terms of minimax play in Berlekamp's universal environment, you can show that C = S in such an environment, because it is wrong for White to play the reverse sente in that environment when the temperature of the environment is greater than 1, and when the temperature of the environment is 1, White is indifferent to playing the reverse sente or playing in the environment.

[RobertJasiek]

My sketch of a proof indeed also presumes some simplistic environment because you have trained us to think like that:) I have a problem with this approach though: it should be applicable not only in idealistic environments but always. Therefore, it seems that the definition approach is more according to my taste. A limes approach would be an overkill because I do not study infinitely many multiples. I mentioned a minmax approach because of your standard examples convincing(?) Bill Taylor of using sente counts instead of only gote counts but I am not sure whether that could be defined independently of environments, and how.

[Bill Spight]

My sketch of a proof indeed also presumes some simplistic environment because you have trained us to think like that:) I have a problem with this approach though: it should be applicable not only in idealistic environments but always.

The main value that I see for the non-terminal values for positions and plays is to provide a heuristic for selecting plays or candidate plays. The largest play is usually the best play. Now, in an ideal environment the largest play is always the best play. And the go board in a real game is so often close enough to the ideal environment that the largest play is the best play. Also, we know that drops in temperature may signal when the largest play is not the best play, because you want to get the last play before the temperature drops. Go players figured this out long before there was any such thing as combinatorial game theory. (OC, they did not talk about temperature, which was borrowed from CGT, but they had the general idea.) Go plays and positions are only partially ordered, so there is no reduction to any number that applies in all environments.

[RobertJasiek]

For which ideal environment is always the largest play the best play? This is not so if a finite ideal environment is defined as simple gotes without follow-ups, one such per positive move value, move values at constant drops incl. the last drop from the smallest move value to 0. Exercise: find a counter-example. Hint 1: the local endgame that is not part of the environment need not have move value and follow-up move value(s) that belong to the set of those in the environment! Hint 2: orders do not differ much from decreasing move values.

Do you have a proof for the go board in a real game being often close enough to the ideal environment that the largest play is the best play? I.e., how do we count frequencies of move value distributions for arbitrary positions? In practice, by experience, I agree, of course. Needless to say, rare worst cases can be really bad:)

What do you mean with "no reduction to any number that applies in all environments"?

[Bill Spight]

For which ideal environment is always the largest play the best play? This is not so if a finite ideal environment is defined as simple gotes without follow-ups, one such per positive move value, move values at constant drops incl. the last drop from the smallest move value to 0. Exercise: find a counter-example. Hint 1: the local endgame that is not part of the environment need not have move value and follow-up move value(s) that belong to the set of those in the environment! Hint 2: orders do not differ much from decreasing move values.

As you point out, if you start with a universal environment, it is possible to construct games for which, when added to that environment, the largest play is not always best. However, given a set of games (go positions in this case), it is possible to find a universal environment in which the largest play is always best. (BTW, in a universal environment there are multiple simple gote at every level, not just one.) That environment is ideal for those games. There exists an ideal environment for 19x19 go, we just do not know what it is.

Do you have a proof for the go board in a real game being often close enough to the ideal environment that the largest play is the best play? I.e., how do we count frequencies of move value distributions for arbitrary positions? In practice, by experience, I agree, of course. Needless to say, rare worst cases can be really bad:)

That is an empirical question, so the proof is in the pudding. Go players have found the size of plays to be a useful heuristic, and have identified three points in the game where it often is not, where it may be important to get the last play before a temperature drop. One is at the transition from the opening to the middle game, another is at the transition from the middle game to the endgame, and the third is at the end of play. (Berlekamp, et al., showed that temperature 1 is the key level at the end of play.) Also, go players know that both local fights and ko fights can raise the temperature temporarily and lead to significant temperature drops. It can also be advantageous to delay winning a ko in order to take advantage of a temperature drop. You and I have both studied such situations.

What do you mean with "no reduction to any number that applies in all environments"?

Here is an example.

$$W White to play. Outer stones alive.
$$ ---------------------------------------
$$ | O O O O . O O a O . . X b X X X X X X |
$$ | X X X X X X X X O . . X O O O O O O O |
$$ | . . . . . . . . . . . . . . . . . . . |

Click Here To Show Diagram Code

[go]$$W White to play. Outer stones alive.
$$ ---------------------------------------
$$ | O O O O . O O a O . . X b X X X X X X |
$$ | X X X X X X X X O . . X O O O O O O O |
$$ | . . . . . . . . . . . . . . . . . . . |[/go]

"a" gains 4.5 points on average, while "b" gains 6 points on average. However, there are environments in which "a" is best and environments in which "b" is best. The two plays are not strictly ordered, so no matter what numbers you assign to their values (as long as they are different ), the smaller valued play will be best in some environments.

[RobertJasiek]

Thank you for the clarifications!

Before the game end's last move issue, last move maxims are too simplistic. What appears to be a 'last move before temperature drop' issue might in fact depend on other aspects, such as parity of a specific number of available moves.

[Bill Spight]

Thank you for the clarifications!

Before the game end's last move issue, last move maxims are too simplistic. What appears to be a 'last move before temperature drop' issue might in fact depend on other aspects, such as parity of a specific number of available moves.

Are you talking about miai? Berlekamp and Wolfe address that issue, as does the traditional go literature, although less clearly.

[RobertJasiek]

Equal options are just a side issue.

I mean real problems. E.g., consider one local sente in an environment during the late endgame. Study T > F, where T is the temperature and F is the follow-up move value. You will rediscover some parity issues and other surprises:) E.g., T ignored, the parity of the number of the environment's moves larger than F determines the player to take the follow-up with profit value F. (You need not waste time on duplicate effort and proofs if you can wait a few months.)

[Bill Spight]

Equal options are just a side issue.

I mean real problems. E.g., consider one local sente in an environment during the late endgame. Study T > F, where T is the temperature and F is the follow-up move value. You will rediscover some parity issues and other surprises:) E.g., T ignored, the parity of the number of the environment's moves larger than F determines the player to take the follow-up with profit value F. (You need not waste time on duplicate effort and proofs if you can wait a few months.)

What you are talking about sounds familiar. You forget that I have been at this for a long time. Since the 1970s.

[RobertJasiek]

Of course:) But I know too little about what your knowledge is in detail. It might well be that I am the one reinventing:) We will see... But more likely, each of us has discovered something new because the topic is deep even for the basic types of positions.

[RobertJasiek]

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.

[Bill Spight]

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.

[RobertJasiek]

Thank you for the motivation! I need to think how far just the same approach brings me for gote.