This thread is for our study of microendgame and infinitesimals, as introduced in Mathematical Go Endgames, other texts or sources. I have made several attempts to understand infinitesimals and think others share the frustration. Rather soon one meets a wall when something essential remains unclear and deeper learning is blocked. I start with my current difficulties of understanding, will add more later and invite you to do alike or clarify.
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QUESTION 1
Why is the chilled count of the following position 2UP * ?
This has the chilled count 0.
This game is a chilled {0|*}, which is defined to be UP.
From the followers, it follows that the initial position has the chilled count {0 | {0|*}} = {0|UP}.
However, I do not understand how we get 2UP *.
***
QUESTION 2
Mathematical Go Endgames distinguishes incentive and temperature. I have not really tried to understand the mathematical definitions but wonder what is the practical difference between the two terms as used in the book?
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QUESTION 3
Now I study table E11, example 3.
The book specifies the chilled count as 0, adds one black chilling mark in the diagram and mentions the incentive -1. I do not care about the incentive yet but first try to understand: Why is the chilled count 0?
In unchilled go, the count is B = 0.5 + 0.75 = 1.25 and the move value is 0.75.
In unchilled go, the count is W = 0.5.
From the counts of the followers, we get the initial position's tentative gote move value (B - W) / 2 = (1.25 - 0.5) / 2 = 0.75 / 2 = 0.375. The move values from the initial position to the black follower increase so the initial position is a local sente. This means its move value is the sente move value but we first need to consider the sente follower:
The unchilled count of the sente follower is S = 0.5 + 0.5 = 1.
The white follower's count is the reverse sente follower's count R = W = 0.5.
The initial position's sente move value is S - R = 1 - 0.5 = 0.5.
However, I am more interested in the initial position's unchilled count C, which is the inherited count of the sente follower C = S = 1.
Now, I recall that the book added a tax mark in the diagram, so the chilled count C1 (chilled by 1) of the initial position is C1 = 0.
Ok, if I did it right, I seem to have answered my own question but - did I do everything right? What bugs me even more is infinitesimals: why do infinitesimals not occur in the chilled count of the initial position? Do they occur in the unchilled count of the initial position?
Studying Microendgame and Infinitesimals
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Re: Studying Microendgame and Infinitesimals
3.RobertJasiek wrote:This thread is for our study of microendgame and infinitesimals, as introduced in Mathematical Go Endgames, other texts or sources. I have made several attempts to understand infinitesimals and think others share the frustration. Rather soon one meets a wall when something essential remains unclear and deeper learning is blocked. I start with my current difficulties of understanding, will add more later and invite you to do alike or clarify.
***
QUESTION 1
Why is the chilled count of the following position 2UP * ?
3. In chilled go, the Black play,
, costs one point. So the local count is 4 - 1 = 3.In Mathematical Go Berlekamp and Wolfe are interested in infinitesimals and fractions, so they mostly ignore integers. It is not that a position is worth 3 + ^^*, it is that the infinitesimal is ^^*, so that's what they call it. They don't really mean that the count is 0. They just don't care which integer it is, for the purposes of the book.This has the chilled count 0.
The marked White play costs White one point, so the chilled count is 2 + 1 = 3. The value of the position is then 3 + ^. And the chilled value of the initial position is 3 + ^^*.This game is a chilled {0|*}, which is defined to be UP.
From the followers, it follows that the initial position has the chilled count {0 | {0|*}} = {0|UP}.
However, I do not understand how we get 2UP *.
Each move before the end in such a corridor gains v*. But that still does not show how we get ^^*.
What we have here is ^^* + v + v + * = 0. I have even made it so that the actual count is 0. No matter who plays first in these corridors, the result with correct play is 0.
QED.***
An incentive is the difference between a position and one of its followers. Temperature is equivalent to miai value. Each tells us something about the value of a play. An incentive gives more information than the temperature, however.QUESTION 2
Mathematical Go Endgames distinguishes incentive and temperature. I have not really tried to understand the mathematical definitions but wonder what is the practical difference between the two terms as used in the book?
It's not, really. The black mark indicates that the count is actually 1.QUESTION 3
Now I study table E11, example 3.
The book specifies the chilled count as 0, adds one black chilling mark in the diagram and mentions the incentive -1. I do not care about the incentive yet but first try to understand: Why is the chilled count 0?
The temperature is -1. Which means that, if we wished, we could leave this position as is at the end of the game and count it as one point of territory for Black. But modern rules do not allow that, so we play it out.
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins
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Everything with love. Stay safe.
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Re: Studying Microendgame and Infinitesimals
Bill, I find the book's approach to ignore integers and only consider fractions and infinitesimals useful. We can use common methods to calculate integers and fractions. Ignoring the integers eases study of infinitesimals, IMO.
Why can we not say that a position is worth 3 + ^^*? Is this different to saying that it is worth 3 and has the infinitesimals ^^*? Uhm, do I get this right: the value of a position is 3 but the CHILLED value of that position is 3 + ^^*? Can we not just abbreviate this by saying "the position is worth 3^^*"?
As soon as we know the chilled counts, we can derive the infinitesimal gains of moves and say "each move before the end in such a corridor gains v*. So am I right that we cannot do vice versa?
Your proof is convincing, thanks. However, I would really prefer to derive ^^* for only the initial position itself from its followers. Is this impossible? Having to prove via an imagined position with a contrieved environment is not so convincing for the typical go player.
IIUYC, incentive and temperature can differ for infinitesimals because Black and White can have different incentives, such as v for Black or ^* for White (see Figure 2.8 in Mathematical Go Endgames). So incentives do not necessarily describes mean values, as from temperatures.
EdLee, I hope you know how to read a combinatorial game in {L|R} annotation and are aware of miai counting (per move value counting). The game star is * := {0|0} (like a territory scoring dame or a chilled 1 point simple gote, which is capturing or connecting one stone). The game UP is ^ := {0|*} (example see earlier messages, the symbol is an arrow upwards). The game DOWN is v := {*|0} (example see earlier messages, the symbol is an arrow downwards).
A TINY is written with a thick font + . It occurs when White might connect more than a stone at the end of an empty corridor. The points connected minus 2 (to ignore the leading stone itself) is the value. E.g., connecting p points at the end of a corridor is +p-2, where p-2 is written as a index of the + symbol. So let me write TINYp-2. E.g., if p = 3, we write TINY1.
If White may make successive local plays and finally connect the stone(s) at the end of the corridor...
...he connects 3 points, of which we ignore 2. So the index of the TINY is 1 and we write TINY1 or +1 (where the 1 is an index).
MINY is for the colour-inverse shapes of those generating a TINY. So Black attacks the corridor and we write a thick font - with the appropriate index.
Now, obviously, the empty heads of corridors have different lengths. The length L is the number of the corridor's empty head's intersections minus 2 and is annotated as 0^L (this time, it is annotated like a power and, if the text formatting allows it, without the power sign ^ especially because we might confuse it with an UP; this is not an UP here). The following captions express TINYs and MINYs as + or - (imagine thick font).
So far the definitions. Next, you need to learn arithmetics with infinitesimals.
EDIT
Why can we not say that a position is worth 3 + ^^*? Is this different to saying that it is worth 3 and has the infinitesimals ^^*? Uhm, do I get this right: the value of a position is 3 but the CHILLED value of that position is 3 + ^^*? Can we not just abbreviate this by saying "the position is worth 3^^*"?
As soon as we know the chilled counts, we can derive the infinitesimal gains of moves and say "each move before the end in such a corridor gains v*. So am I right that we cannot do vice versa?
Your proof is convincing, thanks. However, I would really prefer to derive ^^* for only the initial position itself from its followers. Is this impossible? Having to prove via an imagined position with a contrieved environment is not so convincing for the typical go player.
IIUYC, incentive and temperature can differ for infinitesimals because Black and White can have different incentives, such as v for Black or ^* for White (see Figure 2.8 in Mathematical Go Endgames). So incentives do not necessarily describes mean values, as from temperatures.
EdLee, I hope you know how to read a combinatorial game in {L|R} annotation and are aware of miai counting (per move value counting). The game star is * := {0|0} (like a territory scoring dame or a chilled 1 point simple gote, which is capturing or connecting one stone). The game UP is ^ := {0|*} (example see earlier messages, the symbol is an arrow upwards). The game DOWN is v := {*|0} (example see earlier messages, the symbol is an arrow downwards).
A TINY is written with a thick font + . It occurs when White might connect more than a stone at the end of an empty corridor. The points connected minus 2 (to ignore the leading stone itself) is the value. E.g., connecting p points at the end of a corridor is +p-2, where p-2 is written as a index of the + symbol. So let me write TINYp-2. E.g., if p = 3, we write TINY1.
If White may make successive local plays and finally connect the stone(s) at the end of the corridor...
...he connects 3 points, of which we ignore 2. So the index of the TINY is 1 and we write TINY1 or +1 (where the 1 is an index).
MINY is for the colour-inverse shapes of those generating a TINY. So Black attacks the corridor and we write a thick font - with the appropriate index.
Now, obviously, the empty heads of corridors have different lengths. The length L is the number of the corridor's empty head's intersections minus 2 and is annotated as 0^L (this time, it is annotated like a power and, if the text formatting allows it, without the power sign ^ especially because we might confuse it with an UP; this is not an UP here). The following captions express TINYs and MINYs as + or - (imagine thick font).
So far the definitions. Next, you need to learn arithmetics with infinitesimals.
EDIT
Last edited by RobertJasiek on Tue Sep 26, 2017 12:15 am, edited 1 time in total.
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Bill Spight
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Re:
See https://senseis.xmp.net/?ChilledGo on SL.EdLee wrote:Hi Robert, Bill ( or anyone with access to the relevant source materials):
Is it possible to have a very quick cheat-sheet style summary of the most basic terms ? ( UP, chilled, etc. ) Thanks.
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Re: Studying Microendgame and Infinitesimals
QUESTION 4:
Chilling loses all *s, says Mathematical Go Endgames. Why? What does this mean in practice?
Chilling loses all *s, says Mathematical Go Endgames. Why? What does this mean in practice?
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Re: Studying Microendgame and Infinitesimals
There it is using * to represent a dame in territory scoring. In practice, it means that we can ignore the dame.RobertJasiek wrote:QUESTION 4:
Chilling loses all *s, says Mathematical Go Endgames. Why? What does this mean in practice?
OC, in chilled go * represents the game, {1 | -1} + X, where X is a number, in territory scoring.
Similarly, a dame in territory scoring is {1 | -1} in area scoring.
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At some point, doesn't thinking have to go on?
— Winona Adkins
Visualize whirled peas.
Everything with love. Stay safe.
At some point, doesn't thinking have to go on?
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Re: Studying Microendgame and Infinitesimals
Above is a prototypical UP (written ^), below is a prototypical DOWN (written v). We can ignore the integer values.
In territory scoring this UP is written {3 || 2 | 0}, or 2 + {1 || 0 | -2}. In chilled go it is written 2 + {0 || 0 | 0}, or 2 + ^.
There are other forms. For instance,
UP and DOWN are significant because in a fight to get the last play they are analogous to outside liberties in a semeai.
In territory scoring this UP is written {3 || 2 | 0}, or 2 + {1 || 0 | -2}. In chilled go it is written 2 + {0 || 0 | 0}, or 2 + ^.
There are other forms. For instance,
UP and DOWN are significant because in a fight to get the last play they are analogous to outside liberties in a semeai.
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins
Visualize whirled peas.
Everything with love. Stay safe.
At some point, doesn't thinking have to go on?
— Winona Adkins
Visualize whirled peas.
Everything with love. Stay safe.
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Re: Studying Microendgame and Infinitesimals
My experience with Conway's ONAG is that all is exceptionally clear.RobertJasiek wrote:This thread is for our study of microendgame and infinitesimals, as introduced in Mathematical Go Endgames, other texts or sources. I have made several attempts to understand infinitesimals and think others share the frustration. Rather soon one meets a wall when something essential remains unclear and deeper learning is blocked.
No frustration at all.
(I am a mathematician, if you are not, you may have a different opinion.)
Reading Berlekamp-Wolfe on the other hand is not pleasant.
Lots of fuzzy talk without precise definitions.
Only later in the book the precise definitions are given,
but one cannot start at the point where the text gets more precise
because for some concepts it refers back to the fuzzier part.
I do not know any books on the application of combinatorial game theory to Go
that are precise, and pleasant reading. Bill Spight should write one.
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Re: Studying Microendgame and Infinitesimals
Endgame values are a very rich topic already without entering the microendgame. Writing about it precise and pleasant for reading can require spending a whole book, or two, on only the microendgame. I share your desire to have such books but they are not the first priority. After all, they are only about the last point, so to say. Getting the larger endgames explained is more important. Nevertheless, I do want to get that last point, ugh:)
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Re: Studying Microendgame and Infinitesimals
Thank you for the vote of confidence.vier wrote:I do not know any books on the application of combinatorial game theory to Go that are precise, and pleasant reading. Bill Spight should write one.
I know some people who find my writing unpeasant. 
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins
Visualize whirled peas.
Everything with love. Stay safe.
At some point, doesn't thinking have to go on?
— Winona Adkins
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Everything with love. Stay safe.
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RobertJasiek
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Re: Studying Microendgame and Infinitesimals
QUESTION 5:
Suppose the example {100||50|4} and the task of determining any infinitesimals. Can we simply chill to {0||0|0} and identify UP? Or is this not UP but chilled to {99||50|6} and is without infinitesimals?
Suppose the example {100||50|4} and the task of determining any infinitesimals. Can we simply chill to {0||0|0} and identify UP? Or is this not UP but chilled to {99||50|6} and is without infinitesimals?
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Re: Studying Microendgame and Infinitesimals
Yes, {100||50|4} chills to {99||50|6}.RobertJasiek wrote:QUESTION 5:
Suppose the example {100||50|4} and the task of determining any infinitesimals. Can we simply chill to {0||0|0} and identify UP? Or is this not UP but chilled to {99||50|6} and is without infinitesimals?
You can generalize infinitesimals, however. For instance, around temperature 36.5 {100||50|4} can behave like 63.5 + *. See https://senseis.xmp.net/?EndgameProblem24 for a low temperature example of non-infinitesimals acting like infinitesimals.
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins
Visualize whirled peas.
Everything with love. Stay safe.
At some point, doesn't thinking have to go on?
— Winona Adkins
Visualize whirled peas.
Everything with love. Stay safe.
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Re: Studying Microendgame and Infinitesimals
QUESTION 6:
I am trying to understand Mathematical Go Endgames, E.12 example 4.
The book states the chilled count 1/2v and shows one black dot so the unmarked chilled count is 1 1/2v. How to verify this? Let's see how far I get:
We know e = 1/2 and f = 2^ so the black follower's count is B = 1/2 + 2^ = 2 1/2^.
The white follower's count is W = 1/2.
Therefore, the initial local endgame is {2 1/2^|1/2} = 1 1/2 + {1^|-1}.
To get the chilled count, do we have to chill {1^|-1}?
How to chill {1^|-1}?
Suppose chilling gives {^|0}. Can this game be simplified and how?
Similarly for the colour-inverse case: can {0|v} be simplified and how?
I am trying to understand Mathematical Go Endgames, E.12 example 4.
The book states the chilled count 1/2v and shows one black dot so the unmarked chilled count is 1 1/2v. How to verify this? Let's see how far I get:
We know e = 1/2 and f = 2^ so the black follower's count is B = 1/2 + 2^ = 2 1/2^.
The white follower's count is W = 1/2.
Therefore, the initial local endgame is {2 1/2^|1/2} = 1 1/2 + {1^|-1}.
To get the chilled count, do we have to chill {1^|-1}?
How to chill {1^|-1}?
Suppose chilling gives {^|0}. Can this game be simplified and how?
Similarly for the colour-inverse case: can {0|v} be simplified and how?