Sente, gote and endgame plays

Bill Spight · Post by **Bill Spight** » Wed Nov 15, 2017 8:56 am

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.

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?

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.

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.

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: Select all


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

OC, D does not have a mean value.

Bill Spight · Post by **Bill Spight** » Wed Nov 15, 2017 8:59 am

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.

RobertJasiek · Post by **RobertJasiek** » Sat Nov 18, 2017 11:46 pm

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.

RobertJasiek · Post by **RobertJasiek** » Sat Dec 09, 2017 10:57 am

Studying ambiguous local endgames, I have come up with the following queries for examples or proof of their non-existence. Being busy with other things, I have not looked much for such examples and do not know if finding them is easy or hard. Anyway, this I search:

1) Ambiguous for both players: the gote move value, Black's sente sequence's sente move value and White's sente sequence's sente move value are all equal and larger than 0.

2) Sente for Black, ambiguous for White: the gote move value a) is larger than Black's sente sequence's sente move value and b) equals White's sente sequence's sente move value.

3) Like (2) but colour-inverse.

EDIT: I look for such local endgames whose short sequences are 1 or 2 moves long.

Bill Spight · Post by **Bill Spight** » Sun Dec 10, 2017 9:50 pm

Here is a doubly ambiguous position.

Click Here To Show Diagram Code: [go]$$ $$ ------------------- $$ . X O O . . X X O . $$ . X X X X O O O O . $$ . . . . . . . . . .[/go]

RobertJasiek · Post by **RobertJasiek** » Sat Feb 17, 2018 12:03 pm

Decreasing move values?

In a local endgame without ko threat play, let
BSG = Black's short gote sequence of 1 move
WSG = White's short gote sequence of 1 move
BSS = Black's short sente / ambiguous sequence of 2 moves
WSS = White's short sente / ambiguous sequence of 2 moves
BLG = Black's long gote sequence of 3+ moves reversing to BSG
WLG = White's long gote sequence of 3+ moves reversing to WSG
BLS = Black's long sente sequence of 4+ moves reversing to BSS
WLS = White's long sente sequence of 4+ moves reversing to WSS

Consider these non-trivial cases of concatenated sequences (of which some do not alternate):
BSG - BSG
BSG - BSS
BSG - WSS
BSG - BLG
BSG - WLG
BSG - BLS
BSG - WLS
WSG - WSG
WSG - BSS
WSG - WSS
WSG - BLG
WSG - WLG
WSG - BLS
WSG - WLS
BSS - BSG
BSS - WSG
BSS - BSS
BSS - WSS
BSS - BLG
BSS - WLG
BSS - BLS
BSS - WLS
WSS - BSG
WSS - WSG
WSS - BSS
WSS - WSS
WSS - BLG
WSS - WLG
WSS - BLS
WSS - WLS
BLG - BSG
BLG - WSG
BLG - BSS
BLG - WSS
BLG - BLG
BLG - WLG
BLG - BLS
BLG - WLS
WLG - BSG
WLG - WSG
WLG - BSS
WLG - WSS
WLG - BLG
WLG - WLG
WLG - BLS
WLG - WLS
BLS - BSG
BLS - WSG
BLS - BSS
BLS - WSS
BLS - BLG
BLS - WLG
BLS - BLS
BLS - WLS
WLS - BSG
WLS - WSG
WLS - BSS
WLS - WSS
WLS - BLG
WLS - WLG
WLS - BLS
WLS - WLS

For each case I ask:
- Do move values at the start of the first and at the start of the second concatenated parts of a sequence decrease or are constant?
- What are proofs for this?
- Which counter-examples exist?

RobertJasiek · Post by **RobertJasiek** » Mon Feb 19, 2018 1:49 am

The citations applicable to short (finite, cycleless) games are from Combinatorial Game Theory by Aaron N. Siegel, 2013:

"Theorem 1.20. G >= H if and only if no G_R <= H and G <= no H_L." (p. 58)

"Theorem 1.30. Every G_L <| G and every G_R |> G." (p. 62)

"Proposition 3.18. Let G be a short game.
(a) L(G) >= R(G).
(b) R(G_L) <= L(G) for every G_L and L(G_R) >= R(G) for every G_R, even if G is equal to a number. [...]" (p. 76)

I suspect that these theorems imply decreasing-or-constant move values from one part to the next concatenated part of a move sequence. Do they? How and why?

Bill Spight · Post by **Bill Spight** » Mon Feb 19, 2018 3:31 am

RobertJasiek wrote:The citations applicable to short (finite, cycleless) games are from Combinatorial Game Theory by Aaron N. Siegel, 2013:

"Theorem 1.20. G >= H if and only if no G_R <= H and G <= no H_L." (p. 58)

"Theorem 1.30. Every G_L <| G and every G_R |> G." (p. 62)

BTW, theorem 1.30 follows from theorem 1.20, since G = G.

"Proposition 3.18. Let G be a short game.
(a) L(G) >= R(G).
(b) R(G_L) <= L(G) for every G_L and L(G_R) >= R(G) for every G_R, even if G is equal to a number. [...]" (p. 76)

I suspect that these theorems imply decreasing-or-constant move values from one part to the next concatenated part of a move sequence. Do they? How and why?

They do not. Consider the game, G = J + K, where J ={3|-3} and K = {2|-2}. Black can move first in K to J + 2, and then White can move in J to -1. White's move value is greater than Black's.

RobertJasiek · Post by **RobertJasiek** » Mon Feb 19, 2018 5:05 am

I forgot that a short game can be a sum of several local endgames. What I had in mind was one local endgame consisting of initially connected intersections. Can we at least say that within one local endgame move values decrease or are constant from part to part, or what is a counter-example (without ko threats)? I imagine early parts partially settling the local endgame and cannot find a counter-example.

There has been much talk about dropping move values but has it just deceived us?

I have been shocked to find that 3-move gote traversal parts can have increasing move values within them.

Bill Spight · Post by **Bill Spight** » Mon Feb 19, 2018 9:36 am

Click Here To Show Diagram Code: [go]$$ Distant sente $$ ----------------- $$ . X . . . X X O . $$ . X O O O O O O . $$ . X . . . . . . . $$ . . . . . . . . .[/go]

RobertJasiek · Post by **RobertJasiek** » Mon Feb 19, 2018 11:40 pm

Bill Spight wrote:
$$ Distant sente
$$ -----------------
$$ . X . . . X X O .
$$ . X O O O O O O .
$$ . X . . . . . . .
$$ . . . . . . . . .
Click Here To Show Diagram Code
[go]$$ Distant sente $$ ----------------- $$ . X . . . X X O . $$ . X O O O O O O . $$ . X . . . . . . . $$ . . . . . . . . .[/go]

Move values M of the initial position, M1 after Black 1, M2 after Black 1 - Black 2.

Mgote = (-4 - (-6)) / 2 = 1 = - 5 - (-6) = Msente (ambiguous)
M1sente = 1
M2gote = 2

If we perceive Black's ambiguous short sequence as a 1-move gote sequence, we have M = 1 of part I of the sequence Black 1 - Black 2 - White 3 and M1sente = 1 of part II (Black 2 - White 3) so the move values are constant.

If we perceive Black's ambiguous short sequence as a 2-move sente sequence, we have M = 1 of part I (Black 1 - White 2).

With neither perception, the move values increase from part I to part II. So your example is not a counter-example. Oh, wait. Did I allow constant for increasing-or-constant? It is, of course, enough hint to reveal the following counter-example:

Click Here To Show Diagram Code: [go]$$ Increasing move values $$ ----------------------- $$ . X . . X . X X X . O . $$ . X O O O O O O O O O . $$ . X . . . . . . . . . . $$ . . . . . . . . . . . .[/go]

M2gote = 3.5.

M1sente = 3 (Black's simple sente, serves also as follow-up move value for the initial position).

Tentatively, Mgote = (-7 - (-11)) / 2 = 2 and Msente = -10 - (-11) = 1 so
M1sente > Mgote > Msente <=> 3 > 2 > 1 so
M = Msente = 1 (Black's simple sente).

We have M < M1sente <=> 1 < 3. The increasing move values let this be a counter-example.

Note that we do not have traversal / reversion because there is no alternating 3-move sequence of moves with move value larger than 0.

Bill Spight · Post by **Bill Spight** » Tue Feb 20, 2018 12:35 am

Click Here To Show Diagram Code: [go]$$ Distant sente $$ ----------------- $$ . X 1 3 4 X X O . $$ . X O O O O O O . $$ . X . . . . . . . $$ . . . . . . . . .[/go]

elsewhere

gains 1 pt.

gains 2 pts.

Bill Spight · Post by **Bill Spight** » Tue Feb 20, 2018 12:41 am

Click Here To Show Diagram Code: [go]$$ Increasing move values $$ ----------------------- $$ . X . . X . X X X . O . $$ . X O O O O O O O O O . $$ . X . . . . . . . . . . $$ . . . . . . . . . . . .[/go]

This is a 1 pt. sente. The threat is a 3 pt. sente.

RobertJasiek · Post by **RobertJasiek** » Tue Feb 20, 2018 12:41 am

The question was not whether (in your example) White 4 gains 4 points but whether Black 3 - White 4 as a unit (sente sequence) gains more than Black 1.

Bill Spight · Post by **Bill Spight** » Tue Feb 20, 2018 1:37 am

RobertJasiek wrote:The question was not whether (in your example) White 4 gains 4 points but whether Black 3 - White 4 as a unit (sente sequence) gains more than Black 1.

That was far from clear. How do you decide how to divide up a sequence of play into units?

gains 1 pt.,

and

each gain 2 pts. We could have had a sequence, which could occur with correct play in a real game,

,

where

gains 1 pt. while

and

each gain 2 pts.

Edit:

In your example we could have a sequence,

,

where

gains 3 pts. while

and

each gain 3.5 pts.

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Sente, gote and endgame plays

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Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays

Re: Sente, gote and endgame plays