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.

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.



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.

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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.

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.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

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"?

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.



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.



Here is an example.

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.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

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.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

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.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

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.

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.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

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

Bonne chance!

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

1) Some proofs for a local sente presume F > M (larger follow-up move value) so do not work for a local gote with its F < M. Either new proofs are required or local gotes have different behaviour for certain value classes.

2) The starting follow-up player's choice between starting in the environment or locally cannot rely on a sente sequence argument, or profit values taken F - F for a move (action) and its immediate reply.

3) The conceptual approach of comparing sequences cannot be applied without modifications if the local gote has follow-ups after each player's starting move.

Correction for (2): A local gote can be a global sente, in which case F - F is possible.

What is the definition of Berlekamp's universal environment? Have you used this term with a different definition?

You emphasise last move so much that I wonder why. I understand that infinitesemals asssess the last move of the endgame and that last move is an issue for the dame parity endgames or pass-fights. However, for earlier aspects of the endgame, I have yet to be convinced about any importance of last move before a significant value drop. So far, for every such case, I could construct example positions in which correct play is to give tedomari to the opponent. Hence, where are the conditions for the early / intermediate endgame under which choosing to get tedomari oneself is necessarily correct?

You mention that {2s | 0 || 0} was an infinitesemal. Can you please shortly explain the relevance?

Having checked my general proofs for local sente, luckily they are independent of F > M so can be generalised to also cover local gote with one follow-up. Only some more specialised proofs for ideal environments do presume the local sente condition. I still need to check whether methods (related to comparing results) for sente also apply to gote.

Might be straightforward. Local gote with follow-ups for both players' srarting moves will require greater effort.

EDITs

I believe that you are familiar with the stack of coupons environment, from which on his turn a player may take a coupon instead of making a board play, in which the difference between consecutive coupon levels is a constant. A universal environment is like that, but with many coupons (an odd number of them) at each level. I used the concept to redefine thermography in terms of play. Have I used the term differently? Maybe, casually.

The term is of some theoretical interest, because ideal environments are typically universal environments. A universal environment with a difference between levels of 0.01 point, 101 coupons per level, and a top level (temperature) of 20 points, is almost certainly nearly ideal for 19x19 go, starting from an empty board. (But who would want to play with such an environment? )



As a practical matter, the main time that the hottest play is not best and humans can detect that fact is when there is a temperature drop, and you want to get the last play before the drop.



Sure. There was a time when I used to compose problems where best play depended on features of the environment that were hidden from view deep in the game tree. Well, that might make for clever problems, but it doesn't make for useful ones. Getting the last play before a temperature drop is a useful idea.



Even though it is an infinitesimal, in a non-ko situation it is correct to play it when it is global sente. (In a ko situation doing so gives up a ko threat, OC. ) It is simply a special case where r = 0. Knowing how to play infinitesimals tells you something about how to play {2s | 0 || -r} when r > 0.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Nope. It still works.



Yup. Have fun!

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Universal environment: I suppose the constant drop also applies from the smallest positive value to 0. Even numbers of same values in the environment would add nothing to non-ko situations because of being equal options. When you specify odd numbers of same values in the environment, I understand this to mean that there is the difficult case of even numbers of same values, of which an odd number are in the environment and an odd number are on the board. However, why does this not cover well the case of an even number of same values on the board? Surely, I must be overlooking some intention here.

Ideal environment: For my applications, I have defined this to be on the board, simple gotes without follow-ups (need not be (Tx|-Tx) but can be (L|R) = Tx), finite, each value exactly once, constant drop D, smallest value being D. You seem to use the phrase informally for environments of kinds including universal environment, my ideal environment and other environments being "ideal" in some suitable, good sense. My definition has the advantage of being relevant for practical positions and the disadvantage that it does not cover the also relevant special environments with constant drops and some multiple values (possibly different numbers of multiples for different values). Extensions to that enrichment are possible but require further research.

Getting tedomari: Sure, it is a useful idea. But - the question remains which advice to give. If one gives advice of standard play being to get the last value before a large drop, one must also study and explain conditions for or frequencies of this being correct.

Without modifications applied to follow-ups for both: Uhm, you say it still works. Without modifications. Huh! Astonishing. The problem is: I do not believe it yet. Need to study... Seriously, SOME modifications are needed, and if only to distinguish the second follow-up move value from the first. Will I have fun? Yes, and much research time spent:)