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 Post subject: Re: Sente, gote and endgame plays
Post #141 Posted: Fri Nov 03, 2017 1:03 am 
Tengen

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RobertJasiek wrote:
Code:
     B(-4.5)
    / \
D(1)   E(-10)
      / \
F(-4)   G(-13)
    / \
H(2)   I(-10)


B has the tentative gote move value 5.5 and the tentative sente move value D(1) - F(-4) = 5. The tentative gote move value 5.5 is larger than the tentative sente move value 5. This condition identifies a local sente. So B is White's local sente.


This is still wrong, is it?

If indeed B is White's local sente, then B has the tentative sente count -4.

Code:
     B(-4)
    / \
D(1)   E(-10)
      / \
F(-4)   G(-13)
    / \
H(2)   I(-10)


Saying that B is White's local sente with the tentative sente move value 5 overlooks that, after the sente sequence from B to F, the follow-up gote move value at F is 6. The move value increases, that is, the tentative sente move value of B is smaller than the follow-up gote move value at F, that is, 5 < 6. Therefore, if the sente sequence from B to F is played, play does not stop but proceeds in alternation to I.

Since B is not a local sente but a traversal (that is, besides the types local gote, local sente, ambiguous and maybe local double sente, there is another type of local endgames: traversal), we discard the the tentative sente move value of B.

Code:
     B
    / \
D(1)   E(-10)
      / \
F(-4)   G(-13)
    / \
H(2)   I(-10)


B, being a traversal, has as its count the average of the counts of its followers D and I, that is, (1 + (-10)) / 2 = -4.5.

Code:
     B(-4.5)
    / \
D(1)   E(-10)
      / \
F(-4)   G(-13)
    / \
H(2)   I(-10)


We had that number for the count earlier but it was wrong nevertheless because it was associated with the wrong meaning of tentatively being a local gote. Now the count -4.5 of B is correct because it is associated with the correct meaning of being a traversal. (We must not take as a count what deceiptively looks right because of accidentally having the same number.)

***

Since B is a traversal, we may simplify the game by pruning the traversed nodes to get B':

Code:
     B'
    / \
D(1)   I(-10)


Note that B' does not have assigned its count yet. We determine it only now. B' is a simple gote so has the gote count -4.5.

Code:
     B'(-4.5)
    / \
D(1)   I(-10)


CGT tells us that B = B' from a value perspective if there are / will be no kos. Therefore, if first we determine the gote count -4.5 of B', we may then also use this value as the count of B. However, in B, this same count has a different meaning! Whilst in B' the count -4.5 is a gote count, in B the count -4.5 is a traversal count!

To recollect, suppose we are here when analysing B:

Code:
     B
    / \
D(1)   E(-10)
      / \
F(-4)   G(-13)
    / \
H(2)   I(-10)


From the count -4.5 of B', we know that this is also the count of B:

Code:
     B(-4.5)
    / \
D(1)   E(-10)
      / \
F(-4)   G(-13)
    / \
H(2)   I(-10)


From the earlier analysis of the game B (yes, B, not B'), we also know that B is a traversal so the assigned count of B is a traversal count derived from the counts of D and I.

Despite numerical equality, it would be wrong to say that the count of B could simply be calculated as the average of the counts of D and E. This would miss the meaning of the count, which is NOT a gote count but is a traversal count.

Did I say traversal was difficult? It is! :)

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 Post subject: Re: Sente, gote and endgame plays
Post #142 Posted: Fri Nov 03, 2017 1:23 am 
Judan

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RobertJasiek wrote:
From the earlier analysis of the game B (yes, B, not B'), we also know that B is a traversal so the assigned count of B is a traversal count derived from the counts of D and I.

Despite numerical equality, it would be wrong to say that the count of B could simply be calculated as the average of the counts of D and E. This would miss the meaning of the count, which is NOT a gote count but is a traversal count.


Right you are. :D

Quote:
Did I say traversal was difficult? It is! :)


Right again. :)

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 Post subject: Re: Sente, gote and endgame plays
Post #143 Posted: Sun Nov 12, 2017 8:48 pm 
Judan

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RobertJasiek wrote:
Reading https://senseis.xmp.net/?Count I thought I would have understood traversal (what confusingly that page and CGT calls reversal). However, when trying to apply it to the following examples, I notice that I have understood nothing.


Code:
     B
    / \
D(1)   E
      / \
     F   G(-13)
    / \
H(2)   I(-10)



Code:
          A
         / \
        B   C(-13)
       / \
   D(1)   E
         / \
        F   G(-13)
       / \
   H(2)   I(-10)



The second example is one move earlier than the first example. The second example can be represented as what looks like a hane-and-connect "sente" sequence.

How to distinguish and identify local gote, simple local sente and traversal from each other? What is the exact general procedure? Which tentative or final - gote or sente - counts and move values to calculate for which nodes? How and procedurally when? Which conditions determine the initial positions' types? What distinguishes a long sente sequence (more than 2 moves) from a traversal sequence?

I understand the conditions for simple local gote and simple local sente. However, when I try to apply them to the examples, I am confused.


Earlier, I said that you should always check for reverses. IMX with calculating positions and plays from the scores, reverses can lead to errors. However, that is not so with thermography, where everything comes out in the wash, at least for non-ko positions.

If we think in terms of thermography, we may be able to reduce our efforts. :)

Let's look at the first example.

Code:
     B
    / \
D(1)   E
      / \
     F   G(-13)
    / \
H(2)   I(-10)


For non ko positions the maximum lines of the thermograph are either of the form v = x - t or v = x, and similarly, the minimum lines are of the form v = x + t or v = x. That means that we can tell the maximum and minimum possibilities for the count from following alternating lines of play to the end. For B the maximum value is 1 and the minimum value is -10.

For the left wall of the thermograph of B, the line is v = 1 - t, up to the local temperature, above which it is v = b, for some value of b. For the right wall there are also three lines which may be relevant, v = -10 + t, v = 2 - t, and v = -13 + 2t. The coefficient of t for the line for G is 2 because it takes two plays by White to get to a score of -13. The line for G will not form part of the thermograph for B, but it may determine a relevant line where it intersects with the line for H or I. The line for H will not form part of the thermograph for B, because it is to the left of the line for D, which indicates the maximum possible values.

The intersection of v = 1 - t and v = -10 + t occurs at v = -4.5 and t = 5.5. That gives us possible values for the count and miai value of B. The intersection of v = 2 - t and v = -10 + t, which gives us the count and miai value of F, will occur with greater values for v and t and is thus irrelevant.

The intersection of v = -10 + t and v = -13 + 2t occurs at v = -10 and t = 3. We know that the miai value of F is greater than 5.5, which is greater than 3, so E is a Black sente with a count of -10.

All of this confirms that the count of B is -4.5 and its miai value is 5.5.

Note that we can, as mentioned above, ignore the H score because it is greater than the D score, which means that we can treat E as sente, even if it is not.

Now let's take a look at A.

Code:
          A
         / \
        B   C(-13)
       / \
   D(1)   E
         / \
        F   G(-13)
       / \
   H(2)   I(-10)


The count of A lies between -13 and -10. The basic thermographic lines are v = -13 + t, v = -10, v = 1 - 2t, and v = 2 - 2t. Note that G and C generate the same thermographic line. Note also that the maximum count of both G and C is -10. Since the intersection of v = 1 - 2t and v = -10 occurs at t = 5.5, which is greater than 3, which is the temperature at which v = -10 and v = -13 + t intersect, we conclude that C is a Black sente. And since H(2) is greater than D(1), E is also a Black sente with a count of -10, which is then the count of A. :)

Edit: Note that if either D or H had a value of -8 while the other stayed the same, then the count of A would be -11.

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 Post subject: Re: Sente, gote and endgame plays
Post #144 Posted: Mon Nov 13, 2017 11:13 pm 
Tengen

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Before entering study of linear algebra called thermography with 1000 questions, I have more fundamental questions relevant for validity of count / miai value calculations both without or with linear algebra.

We suspect non-existence of local double sente but what is a proof for this? Maybe it is straightforward, but I have not tried it yet. I have, however, recognised a necessity of clarification. If there is ambiguity between local gote and local double sente (as there is for certain shapes, such as mutual sacrifice of 3 stones and recapture), we need not speak of an optional strategy of local double sente because we can still treat it as local gote. In general, when distinguishing local gote, ambiguous, local sente and maybe local double sente, we use the comparison Mgote ? Msente for these tentative move values. More specifically, we must be considering a particular player's Msente. If it is Black's tentative local sente, we have the comparison Mgote ? M_B_sente. If it is White's tentative local sente, we have the comparison Mgote ? M_W_sente. A non-ambiguous local sente has Mgote > Msente. Therefore, Black's tentative local sente has Mgote > M_B_sente and White's tentative local sente has Mgote > M_W_sente. To possibly prove the non-existence of local double sente, we have to show that Mgote > M_B_sente AND Mgote > M_W_sente <=> (B - W) / 2 > Sb - W AND (B - W) / 2 > B - Sw is FALSE for all values. Do we have B >= Sb >= Sw >= W to help us proving?

My second problem is from which counts along "long" alternating sequences started by Black or White to calulate the count of the initial position. Sometimes, for either particular alternating sequence, we need the count of the direct follower. Sometimes we need the count of the "last" interesting follower. When to use which and why? To emphasise again, this choice occurs for both alternating sequences.

My third problem: Earlier I was naive to assume that we could distinguish local gote and sente by one condition Mgote ? Msente. Now, I think that (unless for the trivial cases of short alternating sequences before encores), we always need to check the two conditions Mgote ? M_B_sente and Mgote ? M_W_sente. Am I right?

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 Post subject: Re: Sente, gote and endgame plays
Post #145 Posted: Tue Nov 14, 2017 1:53 am 
Judan

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RobertJasiek wrote:
Before entering study of linear algebra called thermography with 1000 questions, I have more fundamental questions relevant for validity of count / miai value calculations both without or with linear algebra.

We suspect non-existence of local double sente but what is a proof for this?


Every finite combinatorial game has a mean value; if the game is not a number, there is a non-negative temperature at which the result of play will be equal to the mean value, no matter who plays first. For these statements let me refer you to On Numbers and Games.

Every sente sequence produces a vertical thermographic line, v = x, where x is the numerical result of the sequence. Suppose that there is a local double sente, such that when Black plays first the result is b and when White plays first the result is w, and b > w. In that case there is no non-negative temperature at which the results of play will be equal to the mean value, no matter who plays first. (You will have two vertical thermographic lines which do not intersect.) Therefore there can be no such double sente. There may be local double sente where b <= w. And I do not think that anyone has proven that there may not be local double sente with kos or superkos. In 1998 I showed how to find the thermographs of multiple kos and superkos, but I assumed that they had a mast value. Kos and superkos are basically non-finite games for which the rules may or may not produce a value. If the rules do not do so, then they cannot be scored, and they could be local double sente. In the original draft of my 1998 paper I pointed out that my method would not work for molasses ko unless the rules gave it a value. AGA rules do, other rules may not do so.

Quote:
My second problem is from which counts along "long" alternating sequences started by Black or White to calulate the count of the initial position.


You stop when you reach a number.

Quote:
My third problem: Earlier I was naive to assume that we could distinguish local gote and sente by one condition Mgote ? Msente. Now, I think that (unless for the trivial cases of short alternating sequences before encores), we always need to check the two conditions Mgote ? M_B_sente and Mgote ? M_W_sente. Am I right?


Since there is no local double sente that gains any points, you can start out by assuming that the position is gote and then possibly disprove that. :)

Edit: Or you can use thermography. You may start out with two vertical lines for best play, but if the position is not a number, at some temperature another line will intersect one of those lines and you then draw a gote mast line for higher temperatures.

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 Post subject: Re: Sente, gote and endgame plays
Post #146 Posted: Tue Nov 14, 2017 5:18 am 
Tengen

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In Black's tentative local double sente where b <= w, Black prefers to a) pass instead of playing his "sente sequence" or b) play his longer gote sequence to achieve some b' >= w so in practice, there is no local double sente.

I need think through your other explanations.

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 Post subject: Re: Sente, gote and endgame plays
Post #147 Posted: Tue Nov 14, 2017 9:02 am 
Judan

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RobertJasiek wrote:
In Black's tentative local double sente where b <= w, Black prefers to a) pass instead of playing his "sente sequence" or b) play his longer gote sequence to achieve some b' >= w so in practice, there is no local double sente.

I need think through your other explanations.


Code:
            A
           / \
          /   \
         B     C
        / \   / \
      13   5 7  -3


            D
           / \
          /   \
         E     F
        / \   / \
       7   3 3  -5


            G
           / \
          /   \
         B     C
        / \   / \
       5  -3 2  -8


A, D, and G are all numbers. Numbers of type D occur frequently in go. Numbers of type G occur occasionally. Number of type A are rare.

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 Post subject: Re: Sente, gote and endgame plays
Post #148 Posted: Tue Nov 14, 2017 10:29 am 
Tengen

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Bill Spight wrote:
Code:
            A
           / \
          /   \
         B     C
        / \   / \
      13   5 7  -3


            D
           / \
          /   \
         E     F
        / \   / \
       7   3 3  -5


            G
           / \
          /   \
         H     I
        / \   / \
       5  -3 2  -8


A, D, and G are all numbers. Numbers of type D occur frequently in go. Numbers of type G occur occasionally. Number of type A are rare.


[Edited your letters in the citation.]

Let me see:

Code:
            A
           / \
          /   \
         B     C
        / \   / \
      13   5 7  -3


Code:
            A
           / \
          /   \
         B(9)  C(2)
        / \   / \
      13   5 7  -3


Mgote = 5.5. M_B_sente = 3. M_W_sente = 2.

We have Mgote > M_B_sente, M_W_sente.

Therefore, from either player's sente perspective, it is his local sente. The starting Black achieves 5. The starting White achieves 7. White's start is more favourable for Black than Black's start therefore Black wants to pass and let White start. White's start results in a positive count favouring Black and White prefers to pass, let Black start and let Black achieve the smaller positive count. So White does not help Black. If White starts, he passes and lets Black start his sente sequence to 5. Therefore, this is not a local double sente but is Black's local sente.

Since A = {5|7} and not 5 >= 7, A is a number. But what number? I forgot.

Code:
            D
           / \
          /   \
         E     F
        / \   / \
       7   3 3  -5


Code:
            D
           / \
          /   \
         E(5)  F(-1)
        / \   / \
       7   3 3  -5


Mgote = 3. M_B_sente = 4. M_W_sente = 1.

We have Mgote < M_B_sente so it is not Black's local sente. We have Mgote > M_W_sente so it is White's local sente. The sente count 3 of D is inherited from the leaf 3 of F.

Since D = 3, D is a number.

Code:
            G
           / \
          /   \
         H     I
        / \   / \
       5  -3 2  -8


Code:
            G
           / \
          /   \
         H(1)  I(-3)
        / \   / \
       5  -3 2  -8


Mgote = 2. M_B_sente = 0. M_W_sente = 1 - 2 = -1 is correctly calculated as Black's minus White's value and is negative so White's start is his loss.

We have Mgote > M_B_sente, M_W_sente.

Therefore, from either player's sente perspective, it is his local sente.

However, since White's sente move value M_W_sente = -1 is negative, he prefers to pass. He does not use his loss-making sente sequence. Therefore, we do not have a local double sente.

Black may, but need not, start his sente sequence because the move value is 0. G is Black's local sente so the sente count of G is -3. Black's sente sequence transforms the count -3 of G into the count -3 of the black sente follower.

What are sample go positions for these games?

How many, and what, mistakes have I made?

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 Post subject: Re: Sente, gote and endgame plays
Post #149 Posted: Tue Nov 14, 2017 1:38 pm 
Judan

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Code:
            A
           / \
          /   \
         B     C
        / \   / \
      13   5 7  -3


            D
           / \
          /   \
         E     F
        / \   / \
       7   3 3  -5


            G
           / \
          /   \
         B     C
        / \   / \
       5  -3 2  -8


A, D, and G are all numbers. Numbers of type D occur frequently in go. Numbers of type G occur occasionally. Number of type A are rare.

A = 5

D = 3

G = 0

Numbers of type D are frequent. They are miai, and may be double ko threats.

G is a seki.

Here is an example of type A.



When Black plays first, we may assume the best result for Black, i.e., that White cannot win the potential ko. It is still not as good for Black as when White plays first.

Since this is a number, we could in theory just score it as 3 pts. for White, but the threat to score it as 0 forces White to play first, except perhaps in extraordinary circumstances.

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 Post subject: Re: Sente, gote and endgame plays
Post #150 Posted: Tue Nov 14, 2017 10:30 pm 
Tengen

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In your three-points-without-capturing shape, scoring rules can define whatever score to the initial position... So if the scoring rules are CGT territory scoring defined for terminal positions, the score and therefore counts of the black and white terminal followers are -5 and -3, respectively. So in CGT the initial position's count is -3.

You suggest that other scoring rules could score the initial position as 0, but why 0? Why not, e.g., 1 or -1 or -4? How do you derive the 0? By rules-style scoring definitions of the kind "territory only if surrounded by one player's independently alive stones and scoring dead stones only as prisoners or in territory"? This would alter tree analysis. We would not any longer derive counts of non-terminal nodes from counts of terminal nodes but all nodes would get their counts as the scores of the current nodes' positions!

E.g., with area scoring, the initial local score is 4. If Black starts, the terminal score is -1; this is 5 worse for Black. If White starts, the terminal score is 1; this is 3 better for White or 3 worse for Black. For Black, the optimal strategy is to let White achieve "3 worse for Black".

***

I am still trying to understand whether a local double sente with b <= w exists. Your example trees are not local double sentes. For simple tentative local double sentes with simple gote follow-ups, your example trees represent all cases 1) b, w > 0 (or by symmetry b, w < 0), 2) b = w, 3) b < 0 and w > 0 (or by symmetry b > 0 and w < 0). My reasoning when analysing your example trees applies regardless of the values (provided b <= w and the black-black follower being >= b and the white-white follower being <= w). Therefore, I think that also for b <= w there is no local double sente.

However, you write "There may be local double sente where b <= w.". Why? Which? With which example position? Why is my reasoning (kos excluded) insufficient for excluding the possibility of local double sente in which either starting player gains? Or have you just meant to say that, for b <= w, there may be local double sentes in which at least one player does not gain (but loses or keeps constant) from starting?

Ah, then you also write: "Since there is no local double sente that gains any points". So has my reasoning been a) right and b) what you have wanted to imply?

Even so, how about long-sequence local double sentes? Might they exist even with simple local double sentes not existing?

***

"Every finite combinatorial game has a mean value;"

Ok.

"if the game is not a number, there is a non-negative temperature at which the result of play will be equal to the mean value, no matter who plays first."

This description itself is easy to understand, thank you for the translation! I am still struggling though with relating the theorems in ONAG to your translation because I am not used to understanding the ONAG annotation. For now, I need to believe it.

"Every sente sequence produces a vertical thermographic line, v = x, where x is the numerical result of the sequence. Suppose that there is a local double sente, such that when Black plays first the result is b and when White plays first the result is w, and b > w. In that case there is no non-negative temperature at which the results of play will be equal to the mean value, no matter who plays first. (You will have two vertical thermographic lines which do not intersect.) Therefore there can be no such double sente."

With "sente sequence", I think you do not just mean an alternating 2-play sequence but additionally presume a requirement of it belonging to a simple sente.

I do not exactly understand the annotation v = x yet. Is this an abbreviation for v(t) = x, with t >= 0 being the temperatures? I.e., v is a mapping (abbildung)?

Your proof is by contradiction, ok.

"In that case there is no non-negative temperature at which the results of play will be equal to the mean value, no matter who plays first."

This summarises two cases: Black plays first; White plays first. For each case, we have to prove "there is no non-negative temperature at which the results of play will be equal to the mean value".

"In that case there is no non-negative temperature at which the results of play will be equal to the mean value"

So this is just a sketch of what must be worked out as a proof in detail.

Part I: We have to calculate the mean value. Since it is defined for some multiple of the local endgame, we have to study multiples of it until we find a suitable number of multiples of it to get the mean value.

Part II: Suppose a temperature t >= 0. We have the ensemble of the local game tree and an environment with the temperature t, that is, playing first in the environment gains t/2 for the player playing first in it. So we have to study the local lines of play with the addition of accounting the gain of the first play in the environment. For each such sequence of "global" play, we have to calculate the result and find that it is unequal to the mean value.

Ugh. Much work to be done to transform the sketch of a proof into a proof!

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 Post subject: Re: Sente, gote and endgame plays
Post #151 Posted: Wed Nov 15, 2017 8:56 am 
Judan

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RobertJasiek wrote:
In your three-points-without-capturing shape, scoring rules can define whatever score to the initial position... So if the scoring rules are CGT territory scoring defined for terminal positions, the score and therefore counts of the black and white terminal followers are -5 and -3, respectively. So in CGT the initial position's count is -3.

You suggest that other scoring rules could score the initial position as 0, but why 0?


That's what current Japanese rules do. OC, White can reopen play. Black will have the move, but will pass, and then White will play and get 3 pts. :)

Quote:
I am still trying to understand whether a local double sente with b <= w exists.


Plainly they do when b = w = m, where m is the mean value. We just do not call them double sente. ;) We do not call them double sente when b < w, since at least one player will not play. But we recognize other losing sente, so why not call these double sente? ;)

Quote:
With "sente sequence", I think you do not just mean an alternating 2-play sequence but additionally presume a requirement of it belonging to a simple sente.


By "sente sequence" I just mean a sequence that is played with sente. The original position may be a gote.

Quote:
I do not exactly understand the annotation v = x yet. Is this an abbreviation for v(t) = x, with t >= 0 being the temperatures? I.e., v is a mapping (abbildung)?


Every line has an equation. The axes are v (for value) and t (for temperature), if you will. For the line, v = x, x is the intercept on the v axis.


Quote:
So this is just a sketch of what must be worked out as a proof in detail.

Part I: We have to calculate the mean value. Since it is defined for some multiple of the local endgame, we have to study multiples of it until we find a suitable number of multiples of it to get the mean value.


We do not have to calculate a mean value, m. We only have to know that one exists, and that it is calculable. (See ONAG.)

In the case of an apparent double sente, we also have to know that b >= m >= w. If it is really a double sente, then m is incalculable.

Edit: Here is a true double sente where b > w.

Code:

            D
           / \
          /   \
         E     F
        / \   / \
       ∞   b w  -∞

:mrgreen:

OC, D does not have a mean value.

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Last edited by Bill Spight on Wed Nov 15, 2017 10:06 am, edited 2 times in total.
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 Post subject: Re: Sente, gote and endgame plays
Post #152 Posted: Wed Nov 15, 2017 8:59 am 
Judan

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Here is a "one pt. double sente". What is it, really? :)

Click Here To Show Diagram Code
[go]$$B Double sente?
$$ ----------------------
$$ | . O . O . . . O X . .
$$ | . . . X O . . O X . .
$$ | X X . X O . O O X . .
$$ | . X . X X O . O X , .
$$ | X X X X O O O O X . .
$$ | O O O O X X X X X . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .[/go]


The outer stones are alive, OC. :)

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 Post subject: Re: Sente, gote and endgame plays
Post #153 Posted: Sat Nov 18, 2017 11:46 pm 
Tengen

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Remarks: I have proven the following proposition and corollary, which we use to interpret the following theorem, which works without referring to the mean value and theorems in On Numbers of Games. This avoids a gap in your sketch of a proof of using, but not proving, "every sente sequence produces a vertical thermographic line, v = x".

Proposition: A local endgame is Black's local sente iff Csente < Cgote.

Corollary: A local endgame is White's local sente iff -Csente < -Cgote.

Remarks: -Csente and -Cgote are white-count values. -Csente < -Cgote <=> Csente > Cgote.


Theorem: A local endgame does not exist without kos (other than basic endgame kos before encores), with the black sente follower's count b, the white sente follower's white-count -w, b > w, b < Cgote and -w < -Cgote.

Remarks: By the proposition, b < Cgote identifies Black's local sente and -w < -Cgote identifies White's local sente so both conditions together identify a local double sente. The elegance of the theorem lies in the use of Cgote without spelling it out in detail in the form (L + R) / 2. This simplifies the proof. The theorem does not make requirements for the lengths of the sente sequences to the sente followers so it also expresses non-existence of long double sentes.

Proof by contradiction: Assume such a local endgame exists.

-w < -Cgote <=> w > Cgote. Together with the presupposition b < Cgote, this implies b < Cgote < w. This contradicts the presupposition b > w and therefore such a local endgame does not exist.

Remarks: Csente' is not Csente but occurs during a second application of the proposition in its variant as the corollary. By the definition of sente count, b is Black's sente count (Csente = b) and -w is White's sente white-count (-Csente' = -w). The theorem and its proof do not use sente counts, the proposition and corollary explicitly but the theorem interpreted in their context expresses non-existence of a local double sente with b > w.

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