Life In 19x19

RobertJasiek wrote:The context of each proof is the proposition and its presuppositions, possibly together with applied earlier axioms, definitions and propositions. The context is the task of assigning the correct count, move value, gains and type to a local endgame (of a certain class). Now that you have possibly understood it, what do you think of my proof? Is it - for its declared scope - correct? Can you appreciate the elegant constructions of proofs by contradiction? The case "b = w" is the most difficult and beautiful part. Bill, did you want me to rediscover its beauty or have you just overlooked it?

Oops I only understand that you take into account trees like

......A......
...../.\.....
..../...\....
...B.....C...
../.\.../.\..
.l...b.w...r.

I can try to undestand your proof but you have to help me because I have first to understand your notation.

Let's take the beginning:

Remarks
CGOTE is a gote count. CSENTE is a sente count. C is a white-count.
MGOTE is a gote move value. MB,SENTE is Black's sente move value.
MW,SENTE is White's sente move value. M is a tentative move value.
FB is Black's follow-up move value. FW is White's follow-up move
value.
PA is the position of the following game tree's node A etc.
[Proposition 11 and related propositions say that Black's local sente
is characterised by these equivalent, alternative value conditions:
CSENTE < CGOTE <=>
MGOTE < FB <=> MB,SENTE < FB <=> MB,SENTE < MGOTE.

What does mean CGOTE, CSENTE, MGOTE, MSENTE? Maybe:
CGOTE = (l+b)/2 ?
MGOTE = (l-b)/2 ?
CSENTE = b ?
MSENTE ?
As soon as I understand the notation I am sure I will be able to progress quickly through your proof.

Gerard, yes, of course, as usual. Where Csente has the context Black's sente count. And

Mb,sente = count of Black's sente follower minus count of White's reverse sente follower,

Mw,sente = count of Black's reverse sente follower minus count of White's sente follower.

White-count = count from White's value perspective = negation of the count [from Black's value perspective].

EDIT: name.

"managed to show how a double sente (with our definition) move may appear when temperature drops?"

What, please explain how it is local!

RobertJasiek wrote:The case "b = w" is the most difficult and beautiful part. Bill, did you want me to rediscover its beauty or have you just overlooked it?

Well, it seems to me that the common understanding of double sente assumes that b > w. Cases where b = w are usually called miai, or possibly a double ko threat. Everything I said is a theorem in CGT, proven long before I learned about it, except for the statement about double sente, which is not part of CGT , because CGT has no concept of double sente.

It goes too far to call it common understanding that double sente assumed b > w, b = w was usually called miai, or possibly a double ko threat. There is no need to restrict b ? w. In your preliminary case examples study of double sente, you studied b>w and b<w.

Bill, what CGT theorem are you referring to?

RobertJasiek wrote:It goes too far to call it common understanding that double sente assumed b > w, b = w was usually called miai, or possibly a double ko threat. There is no need to restrict b ? w. In your preliminary case examples study of double sente, you studied b>w and b<w.

I don't remember what I studied before submitting my double sente article to The Go World many years ago. Nowadays I just rely upon CGT. When b < w the original game is a number, which I am sure I did not consciously study way back when.

Please explain why the following game is a number for b<w:

RobertJasiek wrote:Bill, what CGT theorem are you referring to?

When w > 0 > b then neither player will play in the original game and it equals 0. Otherwise you derive the simplest number as explained in On Numbers and Games. E.g., {0|1} = ½. IIUC, b and w are each the result of an even number of plays, i.e. a sente sequence. In that case the number is 0 or the closest one to 0.

RobertJasiek wrote:"managed to show how a double sente (with our definition) move may appear when temperature drops?"

What, please explain how it is local!

look at Bill post : viewtopic.php?p=260826#p260826

After a white play in the game {21 ||| 18 | 4 || 0 | -14} it remains {18|4||0|-14} and the thermograph drawed by Bill shows clearly it is a double sente according to our current defintion.

So, at least for some value cases, there are several approaches to prove non-existence of local double sente:)

Gérard TAILLE wrote: look at Bill post : viewtopic.php?p=260826#p260826

After a white play in the game {21 ||| 18 | 4 || 0 | -14} it remains {18|4||0|-14} and the thermograph drawed by Bill shows clearly it is a double sente according to our current defintion.

I wrote:

{18|4||0|-14}.

Let me study the latter summand.

Fb = (18-4)/2 = 7.
Fw = (0-(-14))/2 = 7.

We have these intermediate counts:

Cb = (18 + 4) / 2 = 11.
Cw = (0 + (-14)) / 2 = -7.

Tentative Mgote = (Cb - Cw)/2 = (11 - (-7)) / 2 = 18/2 = 9.

Let us verify this:

Mgote > Fb, Fw <=> 9 > 7, 7 is fulfilled. Therefore, the initial local endgame is a local gote with
M := Mgote = 9.


Local gote - not local double sente!

Thermography studies more in terms of environmental temperature. Such a characterisation of double sente by parallel walls in the thermograph is GLOBAL because temperature is an abstraction for available other plays elsewhere on the board. Such does not define local double sente.

Bill Spight wrote:

Gérard TAILLE wrote:

Bill Spight wrote:
OK. We can write G this way, using more slashes.

G = {21 ||| 18 | 4 || 0 | -14}

I'll draw the thermograph. Be back soon.

I'm back. Here is the thermograph.

Surely you know where I encountered difficulties Bill

Starting from G = {21 ||| 18 | 4 || 0 | -14}
if use tax = 7 then the game becomes
G = {14 ||| 11 | 11 || 7 | 7}

Now what will happen with tax = 8?
I see that the games {11 | 11} and {7 | 7} are blocked and I do not know how to go to tax = 8.
Can you help me Bill?

Sure. Replace {11|11} and {7|7} with their mean (mast) values. That corresponds with using the mast instead of the lower walls. That yields.

G = {14 || 11 | 7}

When t = 8 we have

G = {13 || 11 | 9}

When t = 9 we have

G = {12 || 11 | 11} -> {12 | 11}

When t = 9½ we have

G = {11½ | 11½}

The tax method works quite well Bill.

Now what about the ideal environment method? How to proceed to analyse temperature t = 10?
When I try a white move at t=10 the move lead now to the game
{18 | 4 || 0 | -14} and with temperature t=10 I have to stop here.
How now calculate the minimax value when white plays first at temperature 10.
The point is the following : I know that when temperature will decrease this local game {18 | 4 || 0 | -14} will become double sente but I do not know which side will take this double sente.
Certainly I will replace the game {18 | 4 || 0 | -14} by the game {11|-7} but I have the feeling that I failed to take into account the real behaviour of the game {18 | 4 || 0 | -14}.
Curiously tax method seems not to have the same problem.
What is your feeling Bill.

RobertJasiek wrote:Thermography studies more in terms of environmental temperature. Such a characterisation of double sente by parallel walls in the thermograph is GLOBAL because temperature is an abstraction for available other plays elsewhere on the board. Such does not define local double sente.

I was not refering to the vertical lines but only to the fact that the conditions: M < Fb, Fw are true for the game {18 | 4 || 0 | -14}.