Life In 19x19

Bill Spight wrote: I expect that this is obvious, but what are the minimax results when Black plays first and when White plays first?

{{+22|-6},-3|-7} tax = 0
black plays first => minimax = -3
white plays first => minimax = -7

{{+20|-6},-4|-6} tax = 1
black plays first => minimax = -4
white plays first => minimax = -6

{{+18|-6},-5|-5} tax = 2
black plays first => minimax = -5
white plays first => minimax = -5

Gérard TAILLE wrote:

Bill Spight wrote:

Gérard TAILLE wrote:My feeling : a "double sente" exists as soon as you accept to define it!

How about this?

A double sente is a combinatorial game such that both sides of its thermograph are vertical.

The definition should be as near as possible as the common understanding of go players (which is not really defined in real life is it?).

We can also take this other suggestion:

double sente.png

Other possible definition : by definition we have a double sente area if x,y >= n

Surely, with your fine knowledge of the problem, you will find more easily than me, what definition will fit the unclear but common understanding of double sente move in real life of go players.

Oh, double sente move is easy. I have already defined that.

I am quite convinced that, by just adding such defintion, a lot of new players will adhere to the theory, especially if you find some interesting behaviour of such double sente move. Because adding such pure defintion cannot harm why not trying our best?

No harm done? What do you think of the two double sente examples from Kano, 9 dan's, Yose Dictionary? Do you think they are good examples upon which to understand double sente?

In thermography a necessary condition for sente is a vertical wall, because that's is what indicates that the second player made a local reply to the first playe's move. A double sente therefore requires two vertical walls, one for each player. True, most go players are not familiar with thermography, but explain why walls are vertical or inclined, and they will get it. So what is the problem with requiring two vertical walls for double sente? Sine qua non, n'est-ce pas?

Gérard TAILLE wrote:

Bill Spight wrote: I expect that this is obvious, but what are the minimax results when Black plays first and when White plays first?

{{+22|-6},-3|-7} tax = 0
black plays first => minimax = -3
white plays first => minimax = -7

{{+20|-6},-4|-6} tax = 1
black plays first => minimax = -4
white plays first => minimax = -6

{{+18|-6},-5|-5} tax = 2
black plays first => minimax = -5
white plays first => minimax = -5

Bueno.

What about this combination?

{+22|-6},-3|-7} + {10|0}

Bill Spight wrote:

Gérard TAILLE wrote:

Bill Spight wrote: I expect that this is obvious, but what are the minimax results when Black plays first and when White plays first?

{{+22|-6},-3|-7} tax = 0
black plays first => minimax = -3
white plays first => minimax = -7

{{+20|-6},-4|-6} tax = 1
black plays first => minimax = -4
white plays first => minimax = -6

{{+18|-6},-5|-5} tax = 2
black plays first => minimax = -5
white plays first => minimax = -5

Bueno.

What about this combination?

{+22|-6},-3|-7} + {10|0}

well I would say:
black plays first => minimax = +3 = 10 + (-7)
white plays first => minimax = -3 = 0 + (-3)

Bill Spight wrote:Bueno.

What about this combination?

{+22|-6},-3|-7} + {10|0}

Gérard TAILLE wrote:well I would say:
black plays first => minimax = +3 = 10 + (-7)
white plays first => minimax = -3 = 0 + (-3)

Black can do better at temperature 0.

Bill Spight wrote:

Bill Spight wrote:Bueno.

What about this combination?

{+22|-6},-3|-7} + {10|0}

Gérard TAILLE wrote:well I would say:
black plays first => minimax = +3 = 10 + (-7)
white plays first => minimax = -3 = 0 + (-3)

Black can do better at temperature 0.

Oops yes Bill, I am a stupid boy!
(-6)in sente and then (+10) => +4

Gérard TAILLE wrote:

Bill Spight wrote:

Bill Spight wrote:Bueno.

What about this combination?

{+22|-6},-3|-7} + {10|0}

Gérard TAILLE wrote:well I would say:
black plays first => minimax = +3 = 10 + (-7)
white plays first => minimax = -3 = 0 + (-3)

Black can do better at temperature 0.

Oops yes Bill, I am a stupid boy!
(-6)in sente and then (+10) => +4

Now how about integer temperatures up to 5?

Bill Spight wrote:
What about this combination?

{+22|-6},-3|-7} + {10|0}

Now how about integer temperatures up to 5?

I can see that the game {+22|-6},-3|-7} becomes {+20|-6},-5|-5} at temperature 2 and the minimax -5 cannot change above that temperature.
For the game {10|0} it becomes {+5|+5} at temperature 5.
As a conclusion the minimax at temperature 5 is 0 whatever player is playing first and the minimax cannot change above temperature 5.

OK. Thermographs do two things. The thermograph of a game finds the mean value of the game, and it finds the result of minimax play in the game, starting with each player, at each temperature.

What else does your skeptic desire?

Gérard TAILLE wrote:The definition [of a double sente] should be as near as possible as the common understanding of go players

Since not all players use values but players with a weak understanding of endgame only use an informal understanding, the common go players' understanding of double sente would be informal. However, some players have not reflected yet that local versus global considerations of double sente differ. Therefore, the common go players' understanding of double sente does not exist. Concerning global considerations, some players are aware that one should not always play a double sente immediately because it might be relatively small while other players (with a weak understanding of endgame) are not aware of that and instead believe overly simplistic traditional advice to play in double sente as early as possible. Only for local considerations, we can identify some common go players' understanding of double sente: that either player's local play is sente meaning an immediate reply by the opponent. In only informal terms, we cannot better characterise why an immediate local reply should be necessary.

In terms of values, we can characterise why an immediate local reply should be necessary: after either player's local play, the reply is more valuable. That is, the move value in the initial local endgame position is smaller than both replies' follow-up move values. Let us use these variables:

M := the move value in the initial local endgame position.
Fb := the move value in the follow-up position created after Black's start.
Fw := the move value in the follow-up position created after White's start.

Now, we can characterise a local double sente endgame be these value conditions:

M < Fb, Fw.

(This annotation abbreviates "M < Fb and M < Fw".)

However, simply speaking, the mathematically proven theorem says:

A local double sente endgame with M < Fb, Fw does not exist.

The common go players' understanding did not know this yet:)

RobertJasiek wrote:

Gérard TAILLE wrote:The definition [of a double sente] should be as near as possible as the common understanding of go players

Since not all players use values but players with a weak understanding of endgame only use an informal understanding, the common go players' understanding of double sente would be informal. However, some players have not reflected yet that local versus global considerations of double sente differ. Therefore, the common go players' understanding of double sente does not exist. Concerning global considerations, some players are aware that one should not always play a double sente immediately because it might be relatively small while other players (with a weak understanding of endgame) are not aware of that and instead believe overly simplistic traditional advice to play in double sente as early as possible. Only for local considerations, we can identify some common go players' understanding of double sente: that either player's local play is sente meaning an immediate reply by the opponent. In only informal terms, we cannot better characterise why an immediate local reply should be necessary.

In terms of values, we can characterise why an immediate local reply should be necessary: after either player's local play, the reply is more valuable. That is, the move value in the initial local endgame position is smaller than both replies' follow-up move values. Let us use these variables:

M := the move value in the initial local endgame position.
Fb := the move value in the follow-up position created after Black's start.
Fw := the move value in the follow-up position created after White's start.

Now, we can characterise a local double sente endgame be these value conditions:

M < Fb, Fw.

(This annotation abbreviates "M < Fb and M < Fw".)

However, simply speaking, the mathematically proven theorem says:

A local double sente endgame with M < Fb, Fw does not exist.

The common go players' understanding did not know this yet:)

Sorry Robert to be here a player with a weak understanding of endgame but let me try to treat myself.
Well, let's take the definition M < Fb, Fw.
As soon as you decide to define what is a double sente it exists doesn'it?

Click Here To Show Diagram Code

[go]$$W
$$ -----------------
$$ | . . . . . . . . |
$$ | . . . X O . . . |
$$ | . . . X O X . . |
$$ | X X X X 1 X . . |
$$ | . . . . O O O O |
$$ | . . . . . . . . |
$$ | . . . . . . . . |
$$ | . . . . . . . . |
$$ -------------------[/go]

In the diagram above you can see that the move is a quite big gote move and an answer by a black hane may be not big enough if it exists some other big gote moves on the board.
That means that creates a double sente area (OC in the sense of the definition you proposed above).
Can you explain with this exemple how mathematics can prove theorem: A local double sente endgame with M < Fb, Fw does not exist.

Bill Spight wrote:OK. Thermographs do two things. The thermograph of a game finds the mean value of the game, and it finds the result of minimax play in the game, starting with each player, at each temperature.

What else does your skeptic desire?

Click Here To Show Diagram Code

[go]$$B
$$ -----------------
$$ | . . b a X . . X |
$$ | X X X O O . O O |
$$ | . . X X O . O X |
$$ | . . . X X O O X |
$$ | . . . . X X X X |
$$ | . . . . . . . . |
$$ | . . . . . . . . |
$$ | . . . . . . . . |
$$ -----------------[/go]

The game above looks like G1 = {{+22|-6},-3|-7}
And my goal is to compare this game G1 with the game G2 = {-3|-7}

In order to do this comparaison I decide to build various environments E_i made of simple gote areas.
Formely you can write E_i = {g_i,1|-g_i,1} + {g_i,2|-g_i,2} + ...
In addition because I do not need high temperatures I may assume g_i,j <= 5

Now to compare G1 and G2 I decide to compare the score of the games
{G1 + E_i} and {G2 + E_i}
Finaly, in order to get a good result I take a very large number of E_i, one million if you want but a finite number to be able to calculate a mean value.

The point is the following: unless I am wrong you have
∀i, score(G1 + E_i) >= score(G2 + E_i)
and ∃i, score(G1 + E_i) > score(G2 + E_i)
and this implies that
mean(score(G1 + E_i)) > mean(score(G2 + E_i))
and as a consequence I expected to see meanValue(G1) > meanValue(G2)
but we have meanValue(G1) = meanValue(G2) = 2

I know that with the ideal (monster?) environment E_ideal we have
score(G1 + E_ideal) = score(G2 + E_ideal)
but when I take the average on a very large number of real environments G1 looks stricly better than G2.

I do not feel it is skepticism Bill. Maybe it is only that I expected more from the theory ?

Unlike proofs by counter-example, the theorem cannot be proven by example because it applies to all such examples. Instead, the proof is by abstract verification. More later.

Can you please simplify your example so that there are not many follow-up moves and safely alive surrounding strings?

Gérard TAILLE wrote:As soon as you decide to define what is a double sente it exists doesn'it?

A double sente by definition is a finite combinatorial game whose thermograph has a left wall, L = u, and a Right wall, R = v, such that u > v.

Does such a double sente exist?

Gérard TAILLE wrote:

Bill Spight wrote:OK. Thermographs do two things. The thermograph of a game finds the mean value of the game, and it finds the result of minimax play in the game, starting with each player, at each temperature.

What else does your skeptic desire?

$$B
$$ -----------------
$$ | . . b a X . . X |
$$ | X X X O O . O O |
$$ | . . X X O . O X |
$$ | . . . X X O O X |
$$ | . . . . X X X X |
$$ | . . . . . . . . |
$$ | . . . . . . . . |
$$ | . . . . . . . . |
$$ -----------------

Click Here To Show Diagram Code

[go]$$B
$$ -----------------
$$ | . . b a X . . X |
$$ | X X X O O . O O |
$$ | . . X X O . O X |
$$ | . . . X X O O X |
$$ | . . . . X X X X |
$$ | . . . . . . . . |
$$ | . . . . . . . . |
$$ | . . . . . . . . |
$$ -----------------[/go]

The game above looks like G1 = {{+22|-6},-3|-7}
And my goal is to compare this game G1 with the game G2 = {-3|-7}

To do so, you subtract G2 from G1. I. e., you consider the sum,

H = {{22|-6},-3|-7} + {7|3}

If White (Right) plays first, White cannot win. That is, if White plays to -7 in the game on the left, Black (Left) replies to 7 in the other game, for jigo (0). If White plays to 3 in the game on the right, Black replies to -3 in the other game, also for jigo.

If Black plays first she wins. She plays to {22|-6} + {7|3}. White's best play is to -6 on the left, after that Black plays to 7 in the other game, for a score of 7 - 6 = 1.

Therefore G1 > G2.

This question is answered with a difference game, not with thermography.

I do not feel it is skepticism Bill. Maybe it is only that I expected more from the theory ?

As my shop teacher used to tell us, use the right tool for the job.