Forum for Miai counting
Relation between Area and Territory Miai Values and Counts [#792]
The existence of a relation between area and territory miai values and counts is not new. However, having now studied a few particular classes of positions, I wonder just how general the relations are. Here is what I suspect:
Definitions
P := a local part of the position, independent from the rest of the board
B := number of black stones in P
W := number of white stones in P
Ma(P) := area miai value in P
Mt(P) := territory miai value in P
Ca(P) := area count of P
Ct(P) := territory count of P
Equations
Ma(P) = Mt(P) = 0 | if not playing in P is best under either scoring
Ma(P) = Mt(P) + 1 | if playing in P has some value
Ca(P) = Ct(P) + B - W
Questions
- How general are these equations?
- What are formal, general proofs for an arbitrary P?
One problem I see is that the territory score isn't determined sufficiently by P alone. I am not sure whether or not your definition of Ct(P) is meant to take account of prisoners.
For truely independent P (where not even ko threats are shared with the outside, so (hyper-)activity can be resolved within P), area scoring assigns unique values to both the score and move sizes.
As far as I can see, territory scoring generally does this for move sizes only, unless
- the "local prison" (containing all prisoners that stem from P) is considered part of P, or
- the rules require pass stones and the local move count is even.
(By the way, you might be interested in jiàn hé values (see Rubilia/NormalValues).)
I do not speak of the score. I speak of the count. See the parent page and other pages explaining miai counting. The count includes the prisoners. If prisoners are immaterial for P, then including prisoners is equal to adding 0.
Yes, I do mean truely independent P.
What do you mean by "area scoring assigns unique values to both the score and move sizes"? I hope that you do not confuse scoring with count here. Maybe you want to say: "area counting assigns unique values to both the count and move sizes". Inhowfar unique? Why not also territory miai counting? Territory miai counting also considers both a count and a miai value (aka move size). So why do you say that "territory scoring (to be: counting) generally does this for move sizes only"? Of course, "unless 1. the "local prison" (containing all prisoners that stem from P) is considered part of P" is fulfilled.
Currently I do not consider rules with pass stones (nor chilling).
I will read rubilia later.
I do not speak of the score. I speak of the count.
Oh indeed, I was wondering if you really meant score, and erroneously assumed you did. Sorry.
I am not used to "counts", nor to the very narrow meaning of "score" provided at that page, but rather understand "score" as a generalization of "final score" to not fully played out positions. Thereby, under area scoring, there is no need to think about "counts", since they equal the scores of the concerned P's.
The count includes the prisoners.
Doesn't the territory count include future prisoners only?
Consider this 3x1 position A (0 komi, no pass stones):
![[Diagram]](diagrams/24/f57aa6860bfa407779b2cfbaefd4e000.png){kind=small}
If I am not completely mistaken, the territory count is 2 moku in favor of White.
The territory score, however, depends on the path that led to here:
![[Diagram]](diagrams/26/6f4b369b661cc5ba14bc86c23d385819.png){kind=small}
![[Diagram]](diagrams/27/d89bdbf3910a702256bf2c734d362d1f.png){kind=small}
: pass
What do you mean by "area scoring assigns unique values to both the score and move sizes"? I hope that you do not confuse scoring with count here.
See above. Where someone speaks of "counting", I think of an (quite arbitrary) procedure to determine the score.
Inhowfar unique?
When there are no ko threats in the environment that could affect P, the area score (or count, if you like) of hyperactive subpositions of P in principle can be exactly determined. Likewise, the move size (or miai value) according to area scoring of active subpositions of P then can be exactly determined. (Of course, neither is currently practical for the vast majority of positions where P is e. g. 9x9 or larger.)
Why not also territory miai counting? (...) Of course, "unless 1. the "local prison" (containing all prisoners that stem from P) is considered part of P" is fulfilled.
This is not clear to me. What happens to already captured prisoners?
It is useful to speak of counts instead of scores for many typical stable followers because a) there might still be later endgame about the dame, b) actual removals might still have to take place b1) during (a), b2) during an agreement phase after removals, or b3) during hypothetical analysis after the later end of the alternation.
The territory count includes future prisoners, but depending on definition one might also include the current prisoners of the starting position. For the sake of simplicity, so far I consider only positions where the latter number is 0.
I have no idea why you let scores (!) depend on game history. Anyway, I do not speak of scores, but of counts.
Counting: No, please do not confuse "miai counting" with "counting mechanics for the score at the game game". They have nothing to do with each other, except that unfortunately both use the same word "counting". Here, I speak only about "miai counting".
Currently, what I do is determining exact counts of subpositions in general (i.e. for arbitrary numbers of involved stones and liberties). So far I have not reached hyperactive positions, but for them I can at least determine a formula for every assumed class of ko threat environments. So when you say "in principle can be exactly determined", this is what I am doing in general for the types of semeais that I am studying.
For I study only semeais, it is practical to do so general because semeais have the great advantage that the internal game tree is comparatively straightforward. It is much easier to analyse than in many endgame shapes, where there can be many, rather arbitrarily looking follow-up plays. In semeais, there is not that much followup because, once a semeai's major life and death is settled, the only things that remain are dame or basic endgame kos. Size of the semeai is immaterial; greater size merely increases the numbers of involved stones or liberties but does not make the semeai more complicated in its structure, simply speaking.
RobertJasiek wrote:
It is useful to speak of counts instead of scores for many typical stable followers because a) there might still be later endgame about the dame, b) actual removals might still have to take place b1) during (a), b2) during an agreement phase after removals, or b3) during hypothetical analysis after the later end of the alternation.
This sounds like, by "scores", you are referring to final scores. I agree that such, in general, cannot be sensibly assigned to not fully played out positions. It seems to me that "count" means something very similar, if not the same, as the mean score with the main lines' final scores weighed according to
w = color_sign/(move_tally(T_leaf) - move_tally(current_node)) ,
so it might be in fact just a naming issue. ("T_leaf" denotes what would be a leaf in the T_tree. The T_tree consists of all nodes that can be reached from the current node by sequences of plays bigger than the current temperature T.)
I begin to realize though, that using the term "score" for the result of literally scoring the current position "as is" (zero in variant 2 at NormalValuesPresentationWays), rather than to the mean score, may make sense.
I have no idea why you let scores (!) depend on game history. Anyway, I do not speak of scores, but of counts.
Huh, as long as we ignore already captured prisoners, the territory score (even in the narrow meaning) does depend on the game history, don't you think so? Consider position A the end of the agreement phase, where both sides agree that the single white stone is alive. How should we determine the final score (and then, the result) when neither the game history nor the amount of previously captured prisoners was known?
The score of the final position does not really depend on the game history any longer because it can be analysed statically without having to worry about ko restrictions. The only remainder is the previous number of prisoners.
How to score the final position? Apply the scoring definition to it!
The only remainder is the previous number of prisoners.
I agree that the numbers (correct: tally) of prisoners is the only aspect of the game history that still matters at the end of agreement phase, when the final territory score is to be determined.
However, where that remainder is unknown because we "ignore already captured prisoners", it's not easy to apply a scoring definition that is based on, amongst others, the number of prisoners.
Cher Robert,
There are indeed positions where pass is uniquely best under both types of scoring. For instance, any legal board play may put one's own living stones in atari, permitting them to be killed. Here it would make sense to define the area miai value as -1. However, I do not believe that anything would be gained from doing so.
My current interest is partitioning the board, determining the count of each part, and - if the alternation may end except for an agreement phase - forming the sum of the counts over all parts to get the score. For this purpose, defining an area count as 0 is very practical.
Can you help me about the other two equations? Are there any (related) proofs by Conway, Berlekamp, or you? Until I see some, I need to prove the same result for each type of semeai I am studying afresh. This takes quite some possibly unnecessary extra time. (My intention is to get a formula for each area count, territory count, area miai value, and territory miai value for each type and subtype of semeais. As you might guess, there are more subtypes than we would wish. Hunter has scratched only the surface.)
Cher Robert,
In general, the relationship is easy to show, because the miai value simply tells how much a play gains. By area scoring you gain a point for the stone played which you do not gain by territory scoring.
However, as you know, there may be other differences, depending on seki. For instance, a "button" position has a miai value by area scoring of 1/2. However, a territory miai value of -1/2 implies a fractional territory score (on the board). No known form of territory scoring has fractional scores. Also, consider this well-known position.
![[Diagram]](diagrams/39/b175b2ad9bf58a20f917a81e787e7628.png)
Let's assume that neither player can afford to fight the ko. Aside from the question of whether to count a or not, how much is 
worth?
By area scoring it gains nothing, because under those circumstances the point belongs to Black anyway. So under territory scoring its miai value should be -1. Under Lasker-Maas or Spight rules that is so, but under Japanese rules it is not.
For me, the relationship is not easy to show in general, even though by area scoring you gain a point for the stone played which you do not gain by territory scoring. My difficulty is that I do not know what exactly the plain, precise, formal definitions are of "area miai value" and "territory miai value". I might create my own ad hoc definitions, but then I am not sure if we speak of the same. So what are the generally used precise and formal definitions of these two terms? How then is it easy to prove?
Of course, I am aware of some special scoring differences between area scoring and territory scoring in special positions. I am glad to ignore those so far. However, since they exist, my equations do not always hold. This is my question: Can we define characteristics of a (an as complete as possible) class of "normal" (local) positions where my equations hold?
Why, in a button position, does a territory miai value of -1/2 imply a fractional territory score (count) on the board?
In the seki or ko position, why do you say that under territory scoring its miai value should be -1? Of course, that is elegant in a sense, but currently I am interested not in elegant rulesets but in explaining things for real world scoring rulesets, i.e. in particular Traditional Territory Scoring. I want to enable players to determine biggest moves and scores in practical games, not to provide the most elegant theory for theoretically the most elegant (here: territory) rules possible.
Cher Robert,
For me, the relationship is not easy to show in general, even though by area scoring you gain a point for the stone played which you do not gain by territory scoring. My difficulty is that I do not know what exactly the plain, precise, formal definitions are of "area miai value" and "territory miai value".
Formally, the miai value corresponds to the temperature of a position, in CGT terms. It is determined by the thermograph of the position. Each thermograph has a vertical mast at its top, which indicates the count of the position. The temperature of the position is the temperature at the base of that mast. (For more details, see On Numbers and Games, Winning Ways, or my paper on Extended Thermography.)
I might create my own ad hoc definitions, but then I am not sure if we speak of the same. So what are the generally used precise and formal definitions of these two terms? How then is it easy to prove?
Well, it's not easy to prove, because it is not true. There are exceptions involving seki. (That is true even for my territory rules, because they do not have fractional scores.) It is easy to show in general, because miai value represents how much a play gains, for gote or reverse sente, or how much the reverse sente would gain, for sente. Counting the played stone gains one more point by area scoring.
Of course, I am aware of some special scoring differences between area scoring and territory scoring in special positions. I am glad to ignore those so far. However, since they exist, my equations do not always hold. This is my question: Can we define characteristics of a (an as complete as possible) class of "normal" (local) positions where my equations hold?
Why, in a button position, does a territory miai value of -1/2 imply a fractional territory score (count) on the board?
At temperature -1, all we have left are integer scores. If a play to one of those score loses 1/2 point, then the position it was played in has a count of an integer +/- 1/2 at higher temperatures. Temperature 0 is the temperature of territory scores, and that position has the same fractional count at that temperature.
In the seki or ko position, why do you say that under territory scoring its miai value should be -1?
All I had in mind was, "Given an area miai value of 0, the territory miai value should be -1." Under my territory rules that is so. White, if forced to make a local play or to give up a pass stone ("gaining" -1), will prefer to give up the stone.
Of course, that is elegant in a sense, but currently I am interested not in elegant rulesets but in explaining things for real world scoring rulesets, i.e. in particular Traditional Territory Scoring. I want to enable players to determine biggest moves and scores in practical games, not to provide the most elegant theory for theoretically the most elegant (here: territory) rules possible.
For non-kos, that is easy. As I said, the miai value is how much a gote or reverse sente gains. To find that out, you have to figure the values (counts or scores) of the positions involved. For gote plays, the value of the original position is the average of the values of the position resulting after Black plays first and the one after White plays first. For sente plays, the result after the sente has the same value as the original position.
Kos are trickier, and I do not think are easy to understand fully without thermography. However, for regular kos you can find the miai values by dividing the difference in values between resulting positions by the number of net plays between them. E. g., if Black can win a ko in one play, but it takes two plays for White to take and win it, the miai value is the difference in results divided by three.
Thermography and formal CGT are overkills for the ordinary go players' practical determination of values in his games and positions. Therefore I think that miai value and count should be redefined generally in a much more practical manner along the iterative lines of stable followers and numbers of net plays.
Exceptions for sekis: Ok, we include the differences between area / territory for eye intersections or one-sided intersections. Is there more to consider?
Not true in general: right. But this simply means that we should define a (or better: the most general) class of positions for that it is true.
So I understand that no one has defined that class of positions yet?
If it is "easy to show for gote, reverse sente, and - as you seem to say - sente", how do you show it?
Fractional values: Ok, they occur for counts, of course. I thought you had been saying they would also occur for some scores of final positions. Or do they occur since you consider a sum of a final position and a fractional button?
You say: "For sente plays, the result after the sente has the same value as the original position." Earlier you explained why sente is (can be) ambiguous. For which unambiguous kinds of sente does your statement hold? How do we determine that something is sente of such a kind?
Kos studied without thermography: I am not convinced that thermography is necessary. After all, it is a graphical representation of values. So studying only values should be sufficient. Since I have not studied values of arcane kos yet, I might not appreciate (yet) why thermography could be convenient (?) there. As long as only ordinary n-kos, n-flower-kos, and similar semeias with several basic ko shapes are studied, I hope that thermography can be avoided completely.
Robert: Thermography and formal CGT are overkills for the ordinary go players' practical determination of values in his games and positions. Therefore I think that miai value and count should be redefined generally in a much more practical manner along the iterative lines of stable followers and numbers of net plays.
Bill: I agree. I like O Meien's emphasis on how much a play gains. That's how people want to think about the value of a play, anyway.
Robert: Exceptions for sekis: Ok, we include the differences between area / territory for eye intersections or one-sided intersections. Is there more to consider?
Bill: Gramlich's button included a ko threat that was unremovable at territory temperature 0 because it would cost a point locally to remove it.
Robert: Not true in general: right. But this simply means that we should define a (or better: the most general) class of positions for that it is true.
Bill: For teaching purposes, I think that restricting the class to positions without seki or ko now or in the followers covers the main ideas. Refinements can come later.
Robert: So I understand that no one has defined that class of positions yet?
Bill: The most general class? Not to my knowledge. Aside from fractional scores, I think that we can say that sekis are included in the class unless there is unequal territory or a one-sided dame in the seki. For fractional scores we can say that the exceptions require unremovable ko threats at temperature 0.
Robert: If it is "easy to show for gote, reverse sente, and - as you seem to say - sente", how do you show it?
Bill: If a player makes a play from position A to position B, the area difference between the positions (in the normal case) is equal to the territory difference plus one point for the stone played. From that fact it follows that the amount gained by a gote or reverse sente under area scoring is the amount gained under territory scoring plus one point.
Robert: Fractional values: Ok, they occur for counts, of course. I thought you had been saying they would also occur for some scores of final positions. Or do they occur since you consider a sum of a final position and a fractional button?
Bill: A territorial button might be worth 1/2, which means that if White plays first in gote the result is worth 1 point for Black, while if Black play first the result is 0. Neither player will wish to play, which is why we can score the position. Under Japanese rules such positions are typically regarded as worth 0. However, under area scoring they have a miai value of 1/2 point, since if White plays first the position is worth 0, while if Black plays first it is worth 1. The only way we can assign it a territorial miai value of -1/2 is if it is worth 1/2 at territory temperature 0, which means that that is its territorial score.
Robert: You say: "For sente plays, the result after the sente has the same value as the original position." Earlier you explained why sente is (can be) ambiguous. For which unambiguous kinds of sente does your statement hold? How do we determine that something is sente of such a kind?
Bill:
![[Diagram]](diagrams/33/3cc5dae677b27ae1599093c923395052.png)
This is the sense of sente I meant. With correct play Black will not always reply locally to 
. However, the local situation is such that 
is the normal and very likely reply.
The game tree looks like this.
A
/ \
5 \
B
/ \
4 0
( / represents a play by Black; \ represents a play by White.)
A play at B is obviously gote. The count at B is 2, and each play gains 2 points.
If White's play at A were gote, the count at A would be 3.5, and each play would gain 1.75 points. Normally, White's play would be made when the biggest play on the board would gain that amount. But then Black's reply would be the biggest play, gaining 2 points, so White's play would be sente. Starting with the assumption that White's play is gote, we conclude that it is actually sente. All we have to postulate is that the players should make the largest play. That is normally the case.
(For reference, that person's name is Gramlich, not with "ae".)
I cannot restrict the class by exchluding seki or ko because I write about semeais, where both occur frequently. However, so far I exclude all abnormal ko types and restrict myself to x-kos.
Your sketch of a proof is canonical, but I am worried about how to actually prove in general "If a player makes a play from position A to position B, the area difference between the positions (in the normal case) is equal to the territory difference plus one point for the stone played.". It is tempting to assume that to be correct, but I do not trust sketches of supposedly correct proofs, I am afraid.
(BTW, hopefully my recent email to you has not been lost in some filter?)
Cher Robert,
RobertJasiek wrote:
(For reference, that person's name is Gramlich, not with "ae".)
Thanks. :-) Corrected.
Your sketch of a proof is canonical, but I am worried about how to actually prove in general "If a player makes a play from position A to position B, the area difference between the positions (in the normal case) is equal to the territory difference plus one point for the stone played.". It is tempting to assume that to be correct, but I do not trust sketches of supposedly correct proofs, I am afraid.
What is the definition of area? Territory plus living stones.
(BTW, hopefully my recent email to you has not been lost in some filter?)
No. Thanks for the email. I just haven't gotten around to replying yet.
The problem is not the definition of area but the definition of the most general class of positions (or a good approximation) so that the proof holds, i.e. to prove that the propositions do apply to each position in the class.
Bill wrote:
Here it would make sense to define the area miai value as -1. However, I do not believe that anything would be gained from doing so.
Whilst I am focussing on getting the definitions straight (see above), I also try to follow this subthread. I can't conceive of a position where the area miai value (based on temperature, not excitation) might be considered -1 moku. Could you provide an example for clarification, please?
Hi, blubb!
![[Diagram]](diagrams/51/8cdda3d080cfcc31a7d5ca9fc2a101da.png)
Oh well, so we are getting back to the obscure floor thingy. :)
Let me think about it a bit more.
(If anything, I'd understand a floor of -1 moku for territory miai values, but not for area.)
Well, yes. It's better, I think, to have a floor of 0 for area scoring. But a floor of -1 is conceivable.
Then again, isn't a floor of -17 moku conceivable, too?
It depends upon what you want to model. With a floor of -17 this game, { 0 | }, is worth 17 points, whereas in CGT it is worth 1 point. (It is worth 1 point in go, too, under Lasker-Maas or Spight territory rules, where it represents a seki where Black has a one-sided dame.)
I do see that, and why, a lower bound of -1 makes sense for CGT temperatures. Neither area nor territory scoring really comply with genuine CGT scores (i. e. Conway integers) though, don't they? For area scoring, any floor other than either at 0 moku or at -boardsize seems arbitrary to me. At Environmental No Pass Go, I've tried to give a more detailled reasoning.
CGT scores can be fractions, not just integers. CGT scoring is closest to territory scoring with group tax, which seems to be the oldest known form of go scoring. Curious. ;-)
Thanks for your patience, Bill.
CGT scores can be fractions, not just integers.
Ah right, I forgot about open positions like {0|1}. Anyway, aren't scoring rules normally applied to integer positions (modulo infenitesimals)? I am neglecting former ING dame sharing here. :-)
CGT scoring is closest to territory scoring with group tax, which seems to be the oldest known form of go scoring. Curious. ;-)
Hmm, let me try to put up an example. Starting with 8 moves each so far and no prisoners, all scores given in moku, treating W as positive (left):
![[Diagram]](diagrams/44/6038ea6d9336d175ebe8ffcf07cc53fa.png)
Actual scores (moku) according to M= ( 8 - 8) = 0 move tally S= ( 8 - 8) = 0 Stone scoring A= (15 -10) = 5 Area scoring T= ( 7 - 2) = 5 Territory scoring P= ( 0 - 0) = 0 Prisoner scoring (as in Zaru Go)
(I have chosen a position where a group tax would cancel out.)
![[Diagram]](diagrams/18/34709a5332bec09f8d56d68f0855b99b.png)
![[Diagram]](diagrams/32/a375a53b0f5af0096cf1cf5a88aff5d2.png)
![[Diagram]](diagrams/50/3aa74f1a6779a9bc0f291ed31e0addf2.png)
Without pass stones With pass stones M= (13 -11) = 2 M= (13 -11) = 2 S= (13 - 8) = 5 S= (13 - 8) = 5 A= (15 -10) = 5 A= (15 -10) = 5 T= ( 5 - 2) = 3 T= ( 7 - 2) = 5 P= ( 3 - 0) = 3 P= ( 5 - 0) = 5
Apparently, B can limit W's gain to two surplus moves. W cannot force B to pass more than twice before W also runs out of good moves. Hence, with CGT-like "move scoring", the initial position simplifies to {{0|}|} = 2.
Territory scoring (regardless of a group tax) gives a score 3 or 5 moku though, depending on pass stones. I do not see how W would substantiate such a margin in terms of Conway integers, unless B passed earlier than necessary.
As far as I can see, territory scores with group tax to some extent "behave like" Conway numbers, but as indicated by the ending position above, {|} in general does not correspond with a territory score of zero.
CGT games are no pass. The form of no pass go that is close to territory scoring with group tax is no pass go with prisoner return. Instead of making a board play a player may return a prisoner to the opponent. Without prisoner return, no pass go is a different animal. ;-)
![[Diagram]](diagrams/25/83d0d4b8d106e5cf65e3cdc9a0bfad5f.png)
In no pass go with prisoner return, this position is worth -5. To indicate that, let's play it out at temperature -1.
![[Diagram]](diagrams/42/75f55cdd28c13b6e7ce57d3f6bd813b2.png)
resigns. But the position after 
should still be worth -5. That is so, since it is worth { | -4}. To indicate that, let's play it out with White playing first.
![[Diagram]](diagrams/4/514374f8d25fbfdcba6cc9cf062db815.png)
@ 
. 
captures. 
resigns.
The position after should be worth -4. That is so, since White has 4 moves: 1 on the board plus 3 prisoner returns.
![[Diagram]](diagrams/21/879f3bf80e07e9f1c85c1d721952ace9.png)
Now let White play first in the original position. resigns. The position after is worth -4, as the reader may verify. Since White has lost 1 point in play, the original position is worth -5.
![[Diagram]](diagrams/26/ad6c2f9a2df69e394ff0053c66b5e9fa.png)
I think that the original position is worth -2 1/2 | -3v (-3 plus down) in straight no pass go. The miai value of a play is 1/4 and the count is - 2 3/4. After we do have a scorable position, and White wins by 2 1/2. (As the reader may verify. <hehe>)
CGT games are no pass.
CGT, at least, is clear about that. There are so many possible Go scoring rules, each of which asking for its own (more or less satisfying) CGT mapping ...
I think that the original position is worth -2 1/2 | -3v (-3 plus down) in straight no pass go. (...) As the reader may verify.
I am a reader of your post, and I get {3^|5/2} (YinYang color assignment), so I guess I can verify your result. :-)
Doing this calculation, I was surprised that in pure no-pass Go, even the score of
![[Diagram]](diagrams/28/0a188e4c768b130eb8ba9526ad6e5055.png)
![[Welcome to Sensei's Library!]](images/stone-hello.png "Sensei's Library")