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 Post subject: Re: This 'n' that #941 Posted: Wed Jun 16, 2021 4:17 pm
 Honinbo

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Gérard TAILLE wrote:
`[go]\$\$B\$\$ --------------\$\$ | X . X O X . .\$\$ | . X O O X . .\$\$ | X . O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

Obviously we are now in line for the analyse and the thermograph of the above position. That's fine because it was not that obvious.

Now you propose the following position:

Bill Spight wrote:

`[go]\$\$B\$\$ --------------\$\$ | X O . O X . .\$\$ | . X O O X . .\$\$ | X . O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

The thermograph for this position, after White has captured one Black stone has the inclined mast when 6½ ≥ t ≥ 1, along the line, s = -5 + t. At t = 4, s = -1, which is 4 points better for White than s = +3. That is why capturing the stone gains, on average, t.

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

This white move (takes ko) looks strange and I hardly see a gain with this move.
The point is the following : instead of the white can afford to play in the environment and he will be still able to defend the corner.

`[go]\$\$W\$\$ --------------\$\$ | X 1 X O X . .\$\$ | . X O O X . .\$\$ | X . O X X . .\$\$ | . O O X . . .\$\$ | 2 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

After the exchange the result is quite different. If now white plays in the environment black can kill the corner. I conclude reverses and white must continue in the corner then
`[go]\$\$W\$\$ --------------\$\$ | X 1 X O X . .\$\$ | 3 X O O X . .\$\$ | X . O X X . .\$\$ | . O O X . . .\$\$ | 2 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`
tenuki
After in the corner and tenuki white still cannot play in the environment which is a quite bad news for white.
The all squence is then
`[go]\$\$W\$\$ --------------\$\$ | X 1 X O X . .\$\$ | 3 X O O X . .\$\$ | X 5 O X X . .\$\$ | . O O X . . .\$\$ | 2 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`
tenuki
tenuki

Let's us compare with the sequence after in the environment

`[go]\$\$W\$\$ --------------\$\$ | X 3 X O X . .\$\$ | 5 X O O X . .\$\$ | X 7 O X X . .\$\$ | . O O X . . .\$\$ | 2 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`
tenuki
tenuki
tenuki
tenuki

When comparing the two sequences we can see that the corner is the same but the moves in the environment are different:
ScoreWhiteTakesKo : g1 + g2 - g3 + g4 - g5 + g6 ....
ScoreWhiteTenuki : -g1 + g2 + g3 + g4 - g5 + g6 ....
White should prefer taking the ko if:
ScoreWhiteTakesKo < ScoreWhiteTenuki <=> 2(g1 - g3) < 0 which is never true.
Taking the ko in the corner is not a good idea isn'it? OC in a real game with potential black ko threats then you change all the assumptions.

Where is the gain you founnd by taking the ko?

If you want to find the gain from taking the ko, start by taking the ko.

If you want to find the gain from playing in a non-thermographic environment, make that play part of the game. That is, part of the ko ensemble.

For instance, if you add the simple gote, {u|-u}, 6½ ≥ t > 1, to the corner position. we can try to find the thermograph of that ko ensemble. Let's find the walls when 6½ ≥ u ≥ t.

`[go]\$\$B Black takes u first\$\$ --------------\$\$ | X 2 X O X . .\$\$ | 4 X O O X . .\$\$ | X 6 O X X . .\$\$ | . O O X . . .\$\$ | 5 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

= u, = t

s = -5 + u + t

Now let's try playing the hane first.

`[go]\$\$B Black plays the hane first\$\$ --------------\$\$ | X 2 X O X . .\$\$ | 4 X O O X . .\$\$ | X 6 O X X . .\$\$ | . O O X . . .\$\$ | 1 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

= u, = t

s = -5 + u + t

All same same.

Now let's try White first.

`[go]\$\$W White takes -u first\$\$ --------------\$\$ | X 3 X O X . .\$\$ | 5 X O O X . .\$\$ | X 7 O X X . .\$\$ | . O O X . . .\$\$ | 2 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

= -u, , , = t

s = -5 - u + 3t

`[go]\$\$W White takes ko first\$\$ --------------\$\$ | X 1 X O X . .\$\$ | 3 X O O X . .\$\$ | X 5 O X X . .\$\$ | . O O X . . .\$\$ | 2 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

= u, = t

s = -5 + u + t

Taking -u first is better. Big duh.

The walls meet when

s = -5 - u + 3t = -5 + u + t , that is, when

t = u

When u > t > 1 , then the White wall follows the line, s = -5 - u + 3t and the Black wall follows the line, s = -5 + u + t.

To get this far. we assumed that u > t. But when 6½ ≥ t ≥ u the thermograph is the same as above, when u was out of the picture. The thermograph has an inclined mast following the line s = -5 + 2t starting at (s,t) = (-5 + 2u, u) and a vertical mast at rising from (s,t) = (8,6½).

----

When u > t, then White does better to take -u. But when t ≥ u, White does at least as well to take the ko.

_________________
At some point, doesn't thinking have to go on?

Visualize whirled peas.

Everything with love. Stay safe.

Last edited by Bill Spight on Wed Jun 16, 2021 11:27 pm, edited 2 times in total.
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 Post subject: Re: This 'n' that #942 Posted: Wed Jun 16, 2021 11:19 pm
 Honinbo

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OK. We know that Black first can not do better than to play the hane, with some possible but now unknown exceptions. What if White plays first and takes the ko?

Let us add this game to the corner to get the ko ensemble.

{u||||2u|0||-u|||-2u}, where 6½ > u > t > 1

Variation 1.

White first takes ko. (Written territorial values below are averages.)

`[go]\$\$W White first takes ko\$\$ --------------\$\$ | X 1 X O X . .\$\$ | 3 X O O X . .\$\$ | X 5 O X X . .\$\$ | . O O X . . .\$\$ | 2 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

= u = t

s = -5 + u + t

Now let take -u. There are two variations.

Variation 2.

`[go]\$\$W White takes -u\$\$ --------------\$\$ | X 3 X O X . .\$\$ | 5 X O O X . .\$\$ | X 7 O X X . .\$\$ | . O O X . . .\$\$ | 2 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

, , = u

s = - 5 + 2u

Taking the ko is better for White than variation 2 by u - t. How does that happen?

plays to {2u|0||-u|||-2u}

After plays the hane, cannot, on average, afford to play to -2u, because will kill the corner for

s = 21 - 2u > 8.

After takes the ko, plays to {2u|0||-u}. must still take the ko.

After takes the ko, plays to {2u|0}. must win the ko.

Now plays to 2u, for s = -5 + 2u.

Variation 3.

What if does not play the hane now but follows Berlekamp's Sentestrat strategy by playing to {2u|0||-u}, instead? Now let play to -u.

`[go]\$\$W White takes -u\$\$ --------------\$\$ | X 5 X O X . .\$\$ | 7 X O O X . .\$\$ | X 9 O X X . .\$\$ | . O O X . . .\$\$ | 4 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

= -u, = u, = -u, , . = t

s = -5 - u + 3t

This result is worse for White than taking the ko because

s = -5 - u + 3t < -5 + u + t , that is, when

t < u , which is presumed.

But that simply means that should not play Sentestrat.

Edit: I think that Berlekamp would object to my calling in this case an example of Sentestrat, because does not actually raise the local temperature, it keeps it the same. But it is like Sentestrat in that it replies to your opponent's play instead of making your own play.

_________________
At some point, doesn't thinking have to go on?

Visualize whirled peas.

Everything with love. Stay safe.

Last edited by Bill Spight on Thu Jun 17, 2021 8:21 am, edited 1 time in total.
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 Post subject: Re: This 'n' that #943 Posted: Thu Jun 17, 2021 3:51 am
 Lives in gote

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Bill Spight wrote:
OK. We know that Black first can not do better than to play the hane, with some possible but now unknown exceptions. What if White plays first and takes the ko?

Let us add this game to the corner to get the ko ensemble.

{u||||2u|0||-u|||-2u}, where 6½ > u > t > 1

Variation 1.

White first takes ko. (Written territorial values below are averages.)

`[go]\$\$W White first takes ko\$\$ --------------\$\$ | X 1 X O X . .\$\$ | 3 X O O X . .\$\$ | X 5 O X X . .\$\$ | . O O X . . .\$\$ | 2 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

= u = t

s = -5 + u + t

Now let take -u. There are two variations.

Variation 2.

`[go]\$\$W White takes -u\$\$ --------------\$\$ | X 3 X O X . .\$\$ | 5 X O O X . .\$\$ | X 7 O X X . .\$\$ | . O O X . . .\$\$ | 2 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

, , = u

s = - 5 + 2u

Taking the ko is better for White than variation 2 by u - t. How does that happen?

plays to {2u|0||-u|||-2u}

After plays the hane, cannot, on average, afford to play to -2u, because will kill the corner for

s = 21 - 2u > 8.

After takes the ko, plays to {2u|0||-u}. must still take the ko.

After takes the ko, plays to {2u|0}. must win the ko.

Now plays to 2u, for s = -5 + 2u.

Variation 3.

What if does not play the hane now but follows Berlekamp's Sentestrat strategy by playing to {2u|0||-u}, instead? Now let play to -u.

`[go]\$\$W White takes -u\$\$ --------------\$\$ | X 5 X O X . .\$\$ | 7 X O O X . .\$\$ | X 9 O X X . .\$\$ | . O O X . . .\$\$ | 4 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

= -u, = u, = -u, , . = t

s = -5 - u + 3t

This result is worse for White than taking the ko because

s = -5 - u + 3t < -5 + u + t , that is, when

t < u , which is presumed.

But that simply means that should not play Sentestrat.

Oops, surely you are teasing me Bill.

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

I showed the white move is bad in a pure gote environment {g1|-g1}, {g2|-g2}, {g3|-g3} ....
and now you find an environment {u||||2u|0||-u|||-2u}, where 6½ > u > t > 1 and where is better than playing -u. Yes OC Bill it is possible to find a special environment (even without ko!) in which could be better.

I understood from your post https://lifein19x19.com/viewtopic.php?p=265525#p265525 (where you used the acronym "KISS") you prefer to keep things simple by avoiding adding even simple gote. Yes your are teasing me Bill don't you?

Though taking the ko by in the diagram above might be good in certain circumtances (maybe also if black has ko threats available but it is the another context), I consider that this move is in general a bad move.
Did you find an environment with only simple gote {u|-u} in which is better than playiing in the environment?

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 Post subject: Re: This 'n' that #944 Posted: Thu Jun 17, 2021 7:35 am
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Gérard TAILLE wrote:
Bill Spight wrote:
I know I did some work on this position some years ago, because I commented on it on SL, at https://senseis.xmp.net/?L2GroupWithDescent#toc4 . But it doesn't ring a bell. There are a number of kos in the game tree, but I am not well, and I am not interested in doing a full analysis, at least not now.

`[go]\$\$ Corner ko\$\$ --------------\$\$ | . . . . X . .\$\$ | . . O O X . .\$\$ | . . O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

However, a few days ago I found some interesting things with the following first 4 plays.

`[go]\$\$ Corner ko\$\$ --------------\$\$ | . 2 . . X . .\$\$ | 4 1 O O X . .\$\$ | 3 . O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

At this point White threatens to win the ko in 1 move. We assume that t > 1.

`[go]\$\$ Corner ko\$\$ --------------\$\$ | . 2 . . X . .\$\$ | 4 1 O O X . .\$\$ | 3 6 O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

= t

Since Black played first we can leave it at that.

Result: -5½ + t

Or Black can kill the corner in 3 net moves.

`[go]\$\$ Corner ko\$\$ --------------\$\$ | 5 2 7 9 X . .\$\$ | 4 1 O O X . .\$\$ | 3 . O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

, , = t

Result: 21 - 3t

The temperature of indifference occurs when

-5½ + t = 21 -3t, or when

t = 6⅝

`[go]\$\$ Corner ko\$\$ --------------\$\$ | . 2 . . X . .\$\$ | 4 1 O O X . .\$\$ | 3 . O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

After this sequence you studied tenuki in the environment
I am wondering if this could really by a good sequence (I mean in a simple environment, either ideal or made of only simple {u|-u} gote points.
My feelig is that both and reverses. If it is true either black has to continue in the corner or black should prefer tenuki directly instead of
What is your feeling?

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 Post subject: Re: This 'n' that #945 Posted: Thu Jun 17, 2021 8:14 am
 Honinbo

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Gérard TAILLE wrote:
Bill Spight wrote:
OK. We know that Black first can not do better than to play the hane, with some possible but now unknown exceptions. What if White plays first and takes the ko?

Let us add this game to the corner to get the ko ensemble.

{u||||2u|0||-u|||-2u}, where 6½ > u > t > 1

Variation 1.

White first takes ko. (Written territorial values below are averages.)

`[go]\$\$W White first takes ko\$\$ --------------\$\$ | X 1 X O X . .\$\$ | 3 X O O X . .\$\$ | X 5 O X X . .\$\$ | . O O X . . .\$\$ | 2 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

= u = t

s = -5 + u + t

Now let take -u. There are two variations.

Variation 2.

`[go]\$\$W White takes -u\$\$ --------------\$\$ | X 3 X O X . .\$\$ | 5 X O O X . .\$\$ | X 7 O X X . .\$\$ | . O O X . . .\$\$ | 2 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

, , = u

s = - 5 + 2u

Taking the ko is better for White than variation 2 by u - t. How does that happen?

plays to {2u|0||-u|||-2u}

After plays the hane, cannot, on average, afford to play to -2u, because will kill the corner for

s = 21 - 2u > 8.

After takes the ko, plays to {2u|0||-u}. must still take the ko.

After takes the ko, plays to {2u|0}. must win the ko.

Now plays to 2u, for s = -5 + 2u.

Variation 3.

What if does not play the hane now but follows Berlekamp's Sentestrat strategy by playing to {2u|0||-u}, instead? Now let play to -u.

`[go]\$\$W White takes -u\$\$ --------------\$\$ | X 5 X O X . .\$\$ | 7 X O O X . .\$\$ | X 9 O X X . .\$\$ | . O O X . . .\$\$ | 4 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

= -u, = u, = -u, , . = t

s = -5 - u + 3t

This result is worse for White than taking the ko because

s = -5 - u + 3t < -5 + u + t , that is, when

t < u , which is presumed.

But that simply means that should not play Sentestrat.

Oops, surely you are teasing me Bill.

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

I showed the white move is bad in a pure gote environment {g1|-g1}, {g2|-g2}, {g3|-g3} ....
and now you find an environment {u||||2u|0||-u|||-2u}, where 6½ > u > t > 1 and where is better than playing -u. Yes OC Bill it is possible to find a special environment (even without ko!) in which could be better.

It's not just better, it's thermographically better. t remains the same while the ko ensemble is played out.

I say that because you seem to be resisting the idea of including moves in the environment in the ko ensemble. I have sympathy for your approach, because it is one that I took for some 25 years.

Gérard TAILLE wrote:
Though taking the ko by in the diagram above might be good in certain circumtances (maybe also if black has ko threats available but it is the another context), I consider that this move is in general a bad move.

Bad move is a strong judgement, which, IMHO, is unwarranted. You haven't even shown that there is a loss at all from playing in the ko. (Not that it is a good idea. It is obvious in general that White should wait for Black to play the hane.)

Gérard TAILLE wrote:
Did you find an environment with only simple gote {u|-u} in which is better than playiing in the environment?

In the thermographic environment, which is composed of simple gote, the gain from White taking the ko is equal to the gain from playing in the environment. That is unusual, since we are at a lower temperature than that of the mast value, above which neither player is likely to play locally. It is a result of the inclined mast.

So we have two moves that are thermographically equal in value. Since thermography plays the averages, that does not mean that are exactly equal. We do expect that normally any difference between them will be slight, so that even if one is wrong, it may not make any difference in the final score. That is not the stuff of bad moves. It also means that we cannot say, a priori, which move is better. Now we know, as a practical matter, that it is in general better for White not to take the ko, but thermography tells us that there are potentially cases where taking the ko is better, and that often it does not matter which play is made.

_________________
At some point, doesn't thinking have to go on?

Visualize whirled peas.

Everything with love. Stay safe.

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 Post subject: Re: This 'n' that #946 Posted: Thu Jun 17, 2021 9:50 am
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Bill Spight wrote:
In the thermographic environment, which is composed of simple gote, the gain from White taking the ko is equal to the gain from playing in the environment. That is unusual, since we are at a lower temperature than that of the mast value, above which neither player is likely to play locally. It is a result of the inclined mast.

So we have two moves that are thermographically equal in value. Since thermography plays the averages, that does not mean that are exactly equal. We do expect that normally any difference between them will be slight, so that even if one is wrong, it may not make any difference in the final score. That is not the stuff of bad moves. It also means that we cannot say, a priori, which move is better. Now we know, as a practical matter, that it is in general better for White not to take the ko, but thermography tells us that there are potentially cases where taking the ko is better, and that often it does not matter which play is made.

No doubt that by adding the area {u||||2u|0||-u|||-2u}, where 6½ > u > t > 1, to the ko ensemble taking the ko gains u -t.
But the area {u||||2u|0||-u|||-2u} is in fact very subtil in order to reach this goal.
If instead you take only the simple gote {u|-u} always with 6½ > u > t > 1, then taking the ko will lose 2(u - t) and here also playing in the environment for white is not just better it's thermographically better!

That's my point : unless you add to the ko ensemble a very subtle area, playing in the environement for white is thermographically better than taking the ko.

Look at my previous calculation with an environment g1 ≥ g2 ≥ g3 ≥ g4 ...
ScoreWhiteTakesKo : g1 + g2 - g3 + g4 - g5 + g6 ....
ScoreWhiteTenuki : -g1 + g2 + g3 + g4 - g5 + g6 ....
White should prefer taking the ko if:
ScoreWhiteTakesKo < ScoreWhiteTenuki <=> 2(g1 - g3) < 0

Now I like to use all the power of thermography tool. My understanding is that an ideal environment is in fact a strange mathematical object with proporties that looks contradictory.
In one hand you can consider g1 = g2 = g3 = g4 ... but at the same time you can also wait for a drop of the temperature!
IOW you may consider 2(g1 - g3) = 0 and at the same time you can assume 2(g1 - g3) > 0 but very small.
That the reason why I claim that playing in the environment is in general better than taking the ko.
BTW by taking g1 = u and g2 = t then you have 2(g1 - g3) = 2(u - t).

It is the same thing in the following position

`[go]\$\$B\$\$ --------------\$\$ | X . X O X . .\$\$ | . X O O X . .\$\$ | X . O X X . .\$\$ | . O O X . . .\$\$ | 1 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

In one hand you can delay without harm the black hane but in the other hand you must play hane immediatly.

It just a matter of using this famous ideal environment taking into account all its power.

As far as I am concern I like to say in an ideal environment we have g1 = g2 + ε with ε equal to infinity small but ε > 0

OC, with specific environment anything may happen.

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 Post subject: Re: This 'n' that #947 Posted: Thu Jun 17, 2021 10:36 am
 Honinbo

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Gérard TAILLE wrote:
Bill Spight wrote:
I know I did some work on this position some years ago, because I commented on it on SL, at https://senseis.xmp.net/?L2GroupWithDescent#toc4 . But it doesn't ring a bell. There are a number of kos in the game tree, but I am not well, and I am not interested in doing a full analysis, at least not now.

`[go]\$\$ Corner ko\$\$ --------------\$\$ | . . . . X . .\$\$ | . . O O X . .\$\$ | . . O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

However, a few days ago I found some interesting things with the following first 4 plays.

`[go]\$\$ Corner ko\$\$ --------------\$\$ | . 2 . . X . .\$\$ | 4 1 O O X . .\$\$ | 3 . O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

At this point White threatens to win the ko in 1 move. We assume that t > 1.

`[go]\$\$ Corner ko\$\$ --------------\$\$ | . 2 . . X . .\$\$ | 4 1 O O X . .\$\$ | 3 6 O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

= t

Since Black played first we can leave it at that.

Result: -5½ + t

Or Black can kill the corner in 3 net moves.

`[go]\$\$ Corner ko\$\$ --------------\$\$ | 5 2 7 9 X . .\$\$ | 4 1 O O X . .\$\$ | 3 . O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

, , = t

Result: 21 - 3t

The temperature of indifference occurs when

-5½ + t = 21 -3t, or when

t = 6⅝

Gérard TAILLE wrote:
`[go]\$\$ Corner ko\$\$ --------------\$\$ | . 2 . a X . .\$\$ | 4 1 O O X . .\$\$ | 3 . O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

After this sequence you studied tenuki in the environment

Actually, no. I studied the position after without regard to whose turn it is. OC, it arises after those 4 plays, but to analyze the position before is a big job.

Gérard TAILLE wrote:
I am wondering if this could really by a good sequence (I mean in a simple environment, either ideal or made of only simple {u|-u} gote points.
My feelig is that both and reverses. If it is true either black has to continue in the corner or black should prefer tenuki directly instead of
What is your feeling?

Well, threatens to kill, so it is sente. And it seems like returns to the mast value of the original position. That is not a reversal, strictly speaking, because 1) sente are not, ipso facto, reversals; and 2) ko positions do not fit CGT, of which reversals are a part. Some replies are so bad that it can make sense to speak of reversals in ko positions, but that is really only informal speech, and is not the case here.

Earlier you had found that is good in no ko threat positions at low temperatures, contrary to common wisdom that the descent to a is correct. Bravo! For one thing, I was interested in whether it was also good at the temperature of the mast value.

I can't say with absolute certainty, but it appears that the throw-in is White's best play in that case, as well. That is because it yields a mast value of 1⅛, while the descent yields a mast value of 1½, which is ⅜ point worse for White. It does so at temperature 6⅝, where it appears to be neither sente nor a reversal, but gote.

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 Post subject: Re: This 'n' that #948 Posted: Thu Jun 17, 2021 11:32 am
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Gérard TAILLE wrote:
Bill Spight wrote:
In the thermographic environment, which is composed of simple gote, the gain from White taking the ko is equal to the gain from playing in the environment. That is unusual, since we are at a lower temperature than that of the mast value, above which neither player is likely to play locally. It is a result of the inclined mast.

So we have two moves that are thermographically equal in value. Since thermography plays the averages, that does not mean that are exactly equal. We do expect that normally any difference between them will be slight, so that even if one is wrong, it may not make any difference in the final score. That is not the stuff of bad moves. It also means that we cannot say, a priori, which move is better. Now we know, as a practical matter, that it is in general better for White not to take the ko, but thermography tells us that there are potentially cases where taking the ko is better, and that often it does not matter which play is made.

No doubt that by adding the area {u||||2u|0||-u|||-2u}, where 6½ > u > t > 1, to the ko ensemble taking the ko gains u -t.
But the area {u||||2u|0||-u|||-2u} is in fact very subtil in order to reach this goal.
If instead you take only the simple gote {u|-u} always with 6½ > u > t > 1, then taking the ko will lose 2(u - t) and here also playing in the environment for white is not just better it's thermographically better!

That is true if {u|-u} with u > t is in the ko ensemble, not the environment.

Gérard TAILLE wrote:
That's my point : unless you add to the ko ensemble a very subtle area, playing in the environement for white is thermographically better than taking the ko.

Probably so.

Gérard TAILLE wrote:
Look at my previous calculation with an environment g1 ≥ g2 ≥ g3 ≥ g4 ...
ScoreWhiteTakesKo : g1 + g2 - g3 + g4 - g5 + g6 ....
ScoreWhiteTenuki : -g1 + g2 + g3 + g4 - g5 + g6 ....
White should prefer taking the ko if:
ScoreWhiteTakesKo < ScoreWhiteTenuki <=> 2(g1 - g3) < 0

Are g1, et al., in the ko ensemble, or in the environment? It matters when you draw the thermograph.

Gérard TAILLE wrote:
Now I like to use all the power of thermography tool. My understanding is that an ideal environment is in fact a strange mathematical object with proporties that looks contradictory.
In one hand you can consider g1 = g2 = g3 = g4 ... but at the same time you can also wait for a drop of the temperature!

This is a problem of temporal logic. Logic has trouble dealing with change. This is hardly a new problem, OC. Zeno based his paradoxes upon that fact. Thermographs represent the territorial values of the mast and walls of the thermograph at all values of the temperature, t. They do not represent changes in t. But they do represent the different values at t0, before the change in temperature, and at t1, after the change.

Gérard TAILLE wrote:
IOW you may consider 2(g1 - g3) = 0 and at the same time you can assume 2(g1 - g3) > 0 but very small.
That the reason why I claim that playing in the environment is in general better than taking the ko.

On that we agree. Where we have disagreed is on whether taking the ko is bad.

Gérard TAILLE wrote:
It is the same thing in the following position

`[go]\$\$B\$\$ --------------\$\$ | X . X O X . .\$\$ | . X O O X . .\$\$ | X . O X X . .\$\$ | . O O X . . .\$\$ | 1 O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

In one hand you can delay without harm the black hane but in the other hand you must play hane immediatly.

In real life the contrast is not usually so stark. Even if you make the wrong play, if it is within ½ point of the right play the difference seldom affects the result of the game. You can see this with proessional play. If a popular joseki move is suboptimal by ½ point or so, it may remain joseki, but pros usually start avoiding it with a decade or so.

Gérard TAILLE wrote:
It just a matter of using this famous ideal environment taking into account all its power.

Well, yes. If you embrace thermography.

Gérard TAILLE wrote:
As far as I am concern I like to say in an ideal environment we have g1 = g2 + ε with ε equal to infinity small but ε > 0

Then Berlekamp's environment based upon ε = 0.01 should appeal. If the hundredths place is even, then it is ideal. And if not, 1.99 - 1.98 + ... = 1.00, for instance, with an error of only 0.005.

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 Post subject: Re: This 'n' that #949 Posted: Thu Jun 17, 2021 12:53 pm
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Here is how I handled things 50 years ago, with an environment of simple gote, {g0|-g0}, {g1|-g1}, ..., such that g0 ≥ g1 ≥ ... ≥ 0. After all that was left was the environment, at gi, I simply estimated gi - gi+i + ... as gi/2. In effect, all of the previous gs were in the game, not the environment. I didn't play the averages with them.

That's pretty much how thermography works. Thermography plays the averages.

Edit: I do play the averages with all of the gs, in the sense that they stand in for what else is on the board. They do not represent it accurately, only approximately. But it is only with the later gs that my method played the averages in the thermographic manner.

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 Post subject: Re: This 'n' that #950 Posted: Fri Jun 18, 2021 10:38 am
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Here is how I handled global sente with my old method.

Black's global sente: {2b|0||-a}, a > 0, b > g0

1) Black plays to {2b|0}, White replies to 0, then Black plays to g0 - g1 + ...

s = g0 - g1 + ...

2) Black plays to g0, White plays reverse sente to -a, then Black plays to g1 - g2 + ...

s = g0 - a + g1 - ...

Since g1 is the residual g, we estimate g1 - g2 + ... as g1/2. That makes the estimated results for the different options as follows.

1) s = g0 - g1/2

2) s = g0 - a + g1/2

So when do we estimate we should play the global sente? When

g0 - g1/2 > g0 - a + g1/2 , that is , when

a > g1

That's interesting for two reasons. First, it justifies the traditional evaluation of the value of the sente play as a. Second, we do not compare a with the largest play in the environment, but with the second largest play. That is different from what the textbooks suggested. But it is consistent with pro play. I had observed that when to play kikashi was a matter of style. This was apparent in the games between Sakata, who played kikashi early, and Takagawa, who played kikashi late. Still, I thought, and still think, that the textbooks erred by not comparing the sente with the second largest play to determine when to play it.

----

Global reverse sente: {a||0|-2b}, a > 0, b > g0

1) Black plays to a, White replies to -g0 + ...

s = a - g0 + ...

2) Black plays to g0, White plays to {0|-2b}, Black replies to 0, then Black plays to g1 - ...

s = g0 + 0 - g1 + ... = g0 - g1 + ...

So Black should play the reverse sente when

a - g0/2 > g0/2 , that is, when

a > g0

This was consistent with what the textbooks said.

Taking these two results together, we see that for a particular global sente, it is more urgent to play the sente than the reverse sente. Something that the textbooks did not say, but that pros understood.

More later.

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 Post subject: Re: This 'n' that #951 Posted: Fri Jun 18, 2021 10:43 am
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`[go]\$\$W\$\$ --------------\$\$ | X 1 X O X . .\$\$ | . X O O X . .\$\$ | X . O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

Let's try to understand the remaining misunderstanding : is white move a bad (or whatever you want) move?

Strictly speaking the answer is quite simple : we cannot say white move is a bad move because a "bad" move has simply never been defined!

Can we go further and try to define what kind of move can be said as "bad"?

Considering the position P above, we can probably try the approach used by CGT by using various environments. Lets imagine a very large ensemble of environment E = {E1, E2, E3 ...}.

a strong definition of a bad move may be the following:
If it does not exist an environment in which is one of the best move, then is a bad move for this set of environment E.
Unless E is restricted to a small set of environment (for example you may decide to put in E only the ideal environments for every t values) this defintion seems too strong:

Let's for example take as position P the following position
`[go]\$\$W\$\$ --------------\$\$ | X a X O O . .\$\$ | X O O O O . .\$\$ | X b X X O . .\$\$ | X O O O O . .\$\$ | X X . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

We certainly would like to say that a black move at "a" is bad (seeing the possibility to play at "b") but if E contains all environments made of pure {u|-u} gote points then, by taking the environment {2|-2} a black move at "a" becomes correct! With such defintion it becomes difficult to have a bad move and for me such defintion becomes completly unuseful.

Bill, do you have any idea for defining an interesting "bad" move notion or do we have to give up and forget this notion?

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 Post subject: Re: This 'n' that #952 Posted: Fri Jun 18, 2021 12:18 pm
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Gérard TAILLE wrote:
Bill, do you have any idea for defining an interesting "bad" move notion or do we have to give up and forget this notion?

Well, to borrow from the chess literature, there are a range of errors from inaccuracies to blunders. I am not exactly sure what an inaccuracy is, but I expect that blunders are very bad and inaccuracies are not bad. Where the threshold to bad is, I don't know. There are tactical, strategical, and psychological aspects to the question.

For instance, if a play is strategically inferior, but the player is ahead, and the play cements the win, I would call it a safety play, not a bad play. Similarly, if the player is behind but anticipates that her play may entice the opponent to make an even worse reply, I would call it a psychological ploy, not a bad play.

I also consider the level of the player. If an amateur dan makes a DDK play, I would normally consider that a bad play, although I could be convinced that it was a desperate attempt. Looking for a place to resign, as they say.

For me, generally speaking, I would consider a play to be bad when, given what is known, it is likely to affect the result of the game.

Gérard TAILLE wrote:
`[go]\$\$W\$\$ --------------\$\$ | X 1 X O X . .\$\$ | . X O O X . .\$\$ | X . O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

Let's try to understand the remaining misunderstanding : is white move a bad (or whatever you want) move?

Strictly speaking the answer is quite simple : we cannot say white move is a bad move because a "bad" move has simply never been defined!

Can we go further and try to define what kind of move can be said as "bad"?

Considering the position P above, we can probably try the approach used by CGT by using various environments. Lets imagine a very large ensemble of environment E = {E1, E2, E3 ...}.

Strictly speaking, CGT does not consider environments at all in difference games. Berlekamp introduced the idea of a universal enriched environment, and I based my redefinition of thermography upon it. I have come up with the idea of an ideal environment, but have not written any academic paper making use of it. In terms of an ideal environment or Berlekamp's universal enriched environment, taking the ko is at worst an inaccuracy. If the temperature of the environment is low, that is not an argument for playing in the environment.

Gérard TAILLE wrote:
a strong definition of a bad move may be the following:
If it does not exist an environment in which is one of the best move, then is a bad move for this set of environment E.

I think you have to compare the play with the alternatives.

Gérard TAILLE wrote:
Unless E is restricted to a small set of environment (for example you may decide to put in E only the ideal environments for every t values) this defintion seems too strong:

Let's for example take as position P the following position
`[go]\$\$W\$\$ --------------\$\$ | X a X O O . .\$\$ | X O O O O . .\$\$ | X b X X O . .\$\$ | X O O O O . .\$\$ | X X . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

We certainly would like to say that a black move at "a" is bad (seeing the possibility to play at "b") but if E contains all environments made of pure {u|-u} gote points then, by taking the environment {2|-2} a black move at "a" becomes correct! With such defintion it becomes difficult to have a bad move and for me such defintion becomes completly unuseful.

In CGT, with a non-ko environment, a play at b plainly dominates a play at a. And, even though nobody has proved it, AFAIK, experience says that it is dominated in ko environments, as well. A play at a may be correct in some circumstances, but it is still dominated. An amateur SDK who plays it instead of b is, at best, careless. For an amateur dan player I would say that it is a bad play.

By contrast,

`[go]\$\$W\$\$ --------------------------\$\$ | X a . . . . . . . O O . .\$\$ | X O O O O O O O O O O . .\$\$ | X b . . . . . . . . O . .\$\$ | X O O O O O O O O O O . .\$\$ | X X . . . . . . . . . . .\$\$ | . . . . . . . . . . . . .\$\$ | . . . . . . . . . . . . .[/go]`

For an amateur I think I would call a play at a an inaccuracy.

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 Post subject: Re: This 'n' that #953 Posted: Fri Jun 18, 2021 1:47 pm
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Bill Spight wrote:
Gérard TAILLE wrote:
Bill, do you have any idea for defining an interesting "bad" move notion or do we have to give up and forget this notion?

Well, to borrow from the chess literature, there are a range of errors from inaccuracies to blunders. I am not exactly sure what an inaccuracy is, but I expect that blunders are very bad and inaccuracies are not bad. Where the threshold to bad is, I don't know. There are tactical, strategical, and psychological aspects to the question.

For instance, if a play is strategically inferior, but the player is ahead, and the play cements the win, I would call it a safety play, not a bad play. Similarly, if the player is behind but anticipates that her play may entice the opponent to make an even worse reply, I would call it a psychological ploy, not a bad play.

I also consider the level of the player. If an amateur dan makes a DDK play, I would normally consider that a bad play, although I could be convinced that it was a desperate attempt. Looking for a place to resign, as they say.

For me, generally speaking, I would consider a play to be bad when, given what is known, it is likely to affect the result of the game.

Gérard TAILLE wrote:
`[go]\$\$W\$\$ --------------\$\$ | X 1 X O X . .\$\$ | . X O O X . .\$\$ | X . O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

Let's try to understand the remaining misunderstanding : is white move a bad (or whatever you want) move?

Strictly speaking the answer is quite simple : we cannot say white move is a bad move because a "bad" move has simply never been defined!

Can we go further and try to define what kind of move can be said as "bad"?

Considering the position P above, we can probably try the approach used by CGT by using various environments. Lets imagine a very large ensemble of environment E = {E1, E2, E3 ...}.

Strictly speaking, CGT does not consider environments at all in difference games. Berlekamp introduced the idea of a universal enriched environment, and I based my redefinition of thermography upon it. I have come up with the idea of an ideal environment, but have not written any academic paper making use of it. In terms of an ideal environment or Berlekamp's universal enriched environment, taking the ko is at worst an inaccuracy. If the temperature of the environment is low, that is not an argument for playing in the environment.

Gérard TAILLE wrote:
a strong definition of a bad move may be the following:
If it does not exist an environment in which is one of the best move, then is a bad move for this set of environment E.

I think you have to compare the play with the alternatives.

Gérard TAILLE wrote:
Unless E is restricted to a small set of environment (for example you may decide to put in E only the ideal environments for every t values) this defintion seems too strong:

Let's for example take as position P the following position
`[go]\$\$W\$\$ --------------\$\$ | X a X O O . .\$\$ | X O O O O . .\$\$ | X b X X O . .\$\$ | X O O O O . .\$\$ | X X . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

We certainly would like to say that a black move at "a" is bad (seeing the possibility to play at "b") but if E contains all environments made of pure {u|-u} gote points then, by taking the environment {2|-2} a black move at "a" becomes correct! With such defintion it becomes difficult to have a bad move and for me such defintion becomes completly unuseful.

In CGT, with a non-ko environment, a play at b plainly dominates a play at a. And, even though nobody has proved it, AFAIK, experience says that it is dominated in ko environments, as well. A play at a may be correct in some circumstances, but it is still dominated. An amateur SDK who plays it instead of b is, at best, careless. For an amateur dan player I would say that it is a bad play.

By contrast,

`[go]\$\$W\$\$ --------------------------\$\$ | X a . . . . . . . O O . .\$\$ | X O O O O O O O O O O . .\$\$ | X b . . . . . . . . O . .\$\$ | X O O O O O O O O O O . .\$\$ | X X . . . . . . . . . . .\$\$ | . . . . . . . . . . . . .\$\$ | . . . . . . . . . . . . .[/go]`

For an amateur I think I would call a play at a an inaccuracy.

When reading your post it is clear that it is too difficult to define what could be a bad move.
Even if "a" dominates "b" sure you will hesitate to declare "b" as bad move because it may happen that "b" is the only best move in certain ko environment.

As an example let's take the following position:

`[go]\$\$\$\$ ---------------------------------------\$\$ | . . X X O O O b O a O O O O O X . . . |\$\$ | X X . X X X X X O X X X X X X X . . . |\$\$ | O X X O O O O O O . . . . . . . . . . |\$\$ | O O O O . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ ---------------------------------------[/go]`

In CGT context the environment is a non-ko one, and the point a and b are analysed as pure gote points {5|-5} and {3|-3}. No doubt that "a" dominates "b".
But even in such a simple case a ko environment may completly change the result : in the following position (the environment is at the bottom of the board)

`[go]\$\$W White to play\$\$ ---------------------------------------\$\$ | . . X X O O O b O a O O O O O X . . . |\$\$ | X X . X X X X X O X X X X X X X . . . |\$\$ | O X X O O O O O O . . . . . . . . . . |\$\$ | O O O O . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . O X X X . . . . . . . . . . |\$\$ | . . . . . O X O X X X . . . . . . . . |\$\$ | . . . . . O . . O . X . . . . . . . . |\$\$ ---------------------------------------[/go]`

White to play : the best move is now "b" !
You see the point? In a non-ko context a simple {3|-3} gote point is only a simple {3|-3} gote point and nothing else, full stop.
In a ko environment a simple {3|-3} gote point may create a ko threat with a completly different result!

Obviously trying to define what a bad move is, seems not a good idea!

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 Post subject: Re: This 'n' that #954 Posted: Fri Jun 18, 2021 4:44 pm
 Honinbo

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Gérard TAILLE wrote:
Bill Spight wrote:
Gérard TAILLE wrote:
Bill, do you have any idea for defining an interesting "bad" move notion or do we have to give up and forget this notion?

Well, to borrow from the chess literature, there are a range of errors from inaccuracies to blunders. I am not exactly sure what an inaccuracy is, but I expect that blunders are very bad and inaccuracies are not bad. Where the threshold to bad is, I don't know. There are tactical, strategical, and psychological aspects to the question.

For instance, if a play is strategically inferior, but the player is ahead, and the play cements the win, I would call it a safety play, not a bad play. Similarly, if the player is behind but anticipates that her play may entice the opponent to make an even worse reply, I would call it a psychological ploy, not a bad play.

I also consider the level of the player. If an amateur dan makes a DDK play, I would normally consider that a bad play, although I could be convinced that it was a desperate attempt. Looking for a place to resign, as they say.

For me, generally speaking, I would consider a play to be bad when, given what is known, it is likely to affect the result of the game.

Gérard TAILLE wrote:
`[go]\$\$W\$\$ --------------\$\$ | X 1 X O X . .\$\$ | . X O O X . .\$\$ | X . O X X . .\$\$ | . O O X . . .\$\$ | . O X X . . .\$\$ | . X . . . . .\$\$ | . X . . . . .\$\$ | X X . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

Let's try to understand the remaining misunderstanding : is white move a bad (or whatever you want) move?

Strictly speaking the answer is quite simple : we cannot say white move is a bad move because a "bad" move has simply never been defined!

Can we go further and try to define what kind of move can be said as "bad"?

Considering the position P above, we can probably try the approach used by CGT by using various environments. Lets imagine a very large ensemble of environment E = {E1, E2, E3 ...}.

Strictly speaking, CGT does not consider environments at all in difference games. Berlekamp introduced the idea of a universal enriched environment, and I based my redefinition of thermography upon it. I have come up with the idea of an ideal environment, but have not written any academic paper making use of it. In terms of an ideal environment or Berlekamp's universal enriched environment, taking the ko is at worst an inaccuracy. If the temperature of the environment is low, that is not an argument for playing in the environment.

Gérard TAILLE wrote:
a strong definition of a bad move may be the following:
If it does not exist an environment in which is one of the best move, then is a bad move for this set of environment E.

I think you have to compare the play with the alternatives.

Gérard TAILLE wrote:
Unless E is restricted to a small set of environment (for example you may decide to put in E only the ideal environments for every t values) this defintion seems too strong:

Let's for example take as position P the following position
`[go]\$\$W\$\$ --------------\$\$ | X a X O O . .\$\$ | X O O O O . .\$\$ | X b X X O . .\$\$ | X O O O O . .\$\$ | X X . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

We certainly would like to say that a black move at "a" is bad (seeing the possibility to play at "b") but if E contains all environments made of pure {u|-u} gote points then, by taking the environment {2|-2} a black move at "a" becomes correct! With such defintion it becomes difficult to have a bad move and for me such defintion becomes completly unuseful.

In CGT, with a non-ko environment, a play at b plainly dominates a play at a. And, even though nobody has proved it, AFAIK, experience says that it is dominated in ko environments, as well. A play at a may be correct in some circumstances, but it is still dominated. An amateur SDK who plays it instead of b is, at best, careless. For an amateur dan player I would say that it is a bad play.

By contrast,

`[go]\$\$W\$\$ --------------------------\$\$ | X a . . . . . . . O O . .\$\$ | X O O O O O O O O O O . .\$\$ | X b . . . . . . . . O . .\$\$ | X O O O O O O O O O O . .\$\$ | X X . . . . . . . . . . .\$\$ | . . . . . . . . . . . . .\$\$ | . . . . . . . . . . . . .[/go]`

For an amateur I think I would call a play at a an inaccuracy.

When reading your post it is clear that it is too difficult to define what could be a bad move.
Even if "a" dominates "b" sure you will hesitate to declare "b" as bad move because it may happen that "b" is the only best move in certain ko environment.

In the last example it is because a play at a gains only 127/128 of a point, while a play at b gains 255/256 of a point, an almost always negligible difference.

Gérard TAILLE wrote:
As an example let's take the following position:

`[go]\$\$\$\$ ---------------------------------------\$\$ | . . X X O O O b O a O O O O O X . . . |\$\$ | X X . X X X X X O X X X X X X X . . . |\$\$ | O X X O O O O O O . . . . . . . . . . |\$\$ | O O O O . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ ---------------------------------------[/go]`

In CGT context the environment is a non-ko one, and the point a and b are analysed as pure gote points {5|-5} and {3|-3}. No doubt that "a" dominates "b".
But even in such a simple case a ko environment may completly change the result : in the following position (the environment is at the bottom of the board)

`[go]\$\$W White to play\$\$ ---------------------------------------\$\$ | . . X X O O O b O a O O O O O X . . . |\$\$ | X X . X X X X X O X X X X X X X . . . |\$\$ | O X X O O O O O O . . . . . . . . . . |\$\$ | O O O O . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . O X X X . . . . . . . . . . |\$\$ | . . . . . O X O X X X . . . . . . . . |\$\$ | . . . . . O . . O . X . . . . . . . . |\$\$ ---------------------------------------[/go]`

White to play : the best move is now "b" !
You see the point? In a non-ko context a simple {3|-3} gote point is only a simple {3|-3} gote point and nothing else, full stop.
In a ko environment a simple {3|-3} gote point may create a ko threat with a completly different result!

Obviously trying to define what a bad move is, seems not a good idea!

Thermography explains why.

`[go]\$\$W White to play\$\$ ---------------------------------------\$\$ | . . X X O O O b O a O O O O O X . . . |\$\$ | X X . X X X X X O X X X X X X X . . . |\$\$ | O X X O O O O O O . . . . . . . . . . |\$\$ | O O O O . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . O X X X . . . . . . . . . . |\$\$ | . . . . . O X O X X X . . . . . . . . |\$\$ | . . . . . O . . O . X . . . . . . . . |\$\$ ---------------------------------------[/go]`

A play at a gains 5 points, a play at b gains 3 points, but makes a difference of a large White ko threat.

A play in the bottom by Black yields 6 points for Black. What about a play by White? If Black is komaster Black can reply by taking and winning the ko for a local score of 4 - t. If White is komaster White can win the ko for a local territory score of 0. Since the ko is fought at temperature 0, the ko threat makes a difference of 4 points in the bottom score when White plays first. That is, a play in the bottom gains either 3 points or 1 point. Since 3 - 1 = 5 - 3, it does not matter which play White chooses on the top side.

`[go]\$\$W White to play, White takes the hottest play\$\$ ---------------------------------------\$\$ | . . X X O O O 2 O 1 O O O O O X . . . |\$\$ | X X . X X X X X O X X X X X X X . . . |\$\$ | O X X O O O O O O . . . . . . . . . . |\$\$ | O O O O . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . O X X X . . . . . . . . . . |\$\$ | . . . . . O X O X X X . . . . . . . . |\$\$ | . . . . . O 3 4 O 6 X . . . . . . . . |\$\$ ---------------------------------------[/go]`

= dame or pass

s = 9 + 4 = 13

`[go]\$\$W White to play, White creates ko threat\$\$ ---------------------------------------\$\$ | 6 5 X X O O O 1 O 2 O O O O O X . . . |\$\$ | X X . X X X X X O X X X X X X X . . . |\$\$ | O X X O O O O O O . . . . . . . . . . |\$\$ | O O O O . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . O X X X . . . . . . . . . . |\$\$ | . . . . . O X O X X X . . . . . . . . |\$\$ | . . . . . O 3 4 O 8 X . . . . . . . . |\$\$ ---------------------------------------[/go]`

takes ko, fills ko, = dame or pass

N.B. is correct at t = 0.

s = 10 + 3 = 13

Perhaps you meant to put only 4 White stones next to a, in which case White b is the undisputed champion.

_________________
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 Post subject: Re: This 'n' that #955 Posted: Sat Jun 19, 2021 3:16 am
 Lives in gote

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Yes Bill I prefer to correct my diagram with only four white stones but that is not really the point. BTW I switch "a" and "b" to fit what you said here after

`[go]\$\$\$\$ ---------------------------------------\$\$ | . . X X O O O a O b O O O O X . . . . |\$\$ | X X . X X X X X O X X X X X X . . . . |\$\$ | O X X O O O O O O . . . . . . . . . . |\$\$ | O O O O . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ ---------------------------------------[/go]`

Bill Spight wrote:
In CGT, with a non-ko environment, a play at b plainly dominates a play at a. And, even though nobody has proved it, AFAIK, experience says that it is dominated in ko environments, as well.

In fact I was reacting to your above sentence where you questionned a domination in CGT with non-ko environement and a domination in ko environment in the case of very simple, pure {u|-u} gote points.
In my example the point is not to know if "a" is better than "b" in a ko environment (it is obvious for most go players and I have no doubt that thermography, in the ko context, explains that). The point is rather to understand what means CGT with non-ko environment.
IOW, in my exapmle, and considering we are in "CGT with non-ko environment", is it correct to say that the point "b" is a pure simple {4|-4} point, the point "a" is a pure {3|-3} point and as a consequence "b" dominates "a"?
If it is true then my example answer your question isn't it?

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 Post subject: Re: This 'n' that #956 Posted: Sat Jun 19, 2021 5:34 am
 Honinbo

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Gérard TAILLE wrote:
Yes Bill I prefer to correct my diagram with only four white stones but that is not really the point. BTW I switch "a" and "b" to fit what you said here after

`[go]\$\$\$\$ ---------------------------------------\$\$ | . . X X O O O a O b O O O O X . . . . |\$\$ | X X . X X X X X O X X X X X X . . . . |\$\$ | O X X O O O O O O . . . . . . . . . . |\$\$ | O O O O . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ ---------------------------------------[/go]`

Bill Spight wrote:
In CGT, with a non-ko environment, a play at b plainly dominates a play at a. And, even though nobody has proved it, AFAIK, experience says that it is dominated in ko environments, as well.

That statement was referring to a different diagram, which is why the letters are different.

`[go]\$\$W\$\$ --------------\$\$ | X a X O O . .\$\$ | X O O O O . .\$\$ | X b X X O . .\$\$ | X O O O O . .\$\$ | X X . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . . . . .\$\$ | . . . , . . .[/go]`

This is the diagram, in which a play at b dominates a play at a in CGT. Neither play involves creating or destroying a ko threat. My remark had nothing to do with anything like the other diagram in this post.
.
Gérard TAILLE wrote:
In fact I was reacting to your above sentence where you questionned a domination in CGT with non-ko environement and a domination in ko environment in the case of very simple, pure {u|-u} gote points.

Beside the point, as I was referring to something else.

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 Post subject: Re: This 'n' that #957 Posted: Sat Jun 19, 2021 5:58 am
 Honinbo

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Rather than edit my previous note, let me continue here.

Gérard TAILLE wrote:
The point is rather to understand what means CGT with non-ko environment.

My interpretation of a non ko environment in CGT is one in which there is no ko fight or potential ko fight.

Gérard TAILLE wrote:
IOW, in my exapmle, and considering we are in "CGT with non-ko environment", is it correct to say that the point "b" is a pure simple {4|-4} point, the point "a" is a pure {3|-3} point and as a consequence "b" dominates "a"?

I do not understand the relevance of your diagram, in which there is a ko fight. If the idea is to understand CGT with a non-ko environment, then don't use a ko environment to do so.

_________________
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 Post subject: Re: This 'n' that #958 Posted: Sat Jun 19, 2021 6:29 am
 Lives in gote

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Bill Spight wrote:
Rather than edit my previous note, let me continue here.

Gérard TAILLE wrote:
The point is rather to understand what means CGT with non-ko environment.

My interpretation of a non ko environment in CGT is one in which there is no ko fight or potential ko fight.

Gérard TAILLE wrote:
IOW, in my exapmle, and considering we are in "CGT with non-ko environment", is it correct to say that the point "b" is a pure simple {4|-4} point, the point "a" is a pure {3|-3} point and as a consequence "b" dominates "a"?

I do not understand the relevance of your diagram, in which there is a ko fight. If the idea is to understand CGT with a non-ko environment, then don't use a ko environment to do so.

`[go]\$\$\$\$ ---------------------------------------\$\$ | . . X X O O O a O b O O O O X . . . . |\$\$ | X X . X X X X X O X X X X X X . . . . |\$\$ | O X X O O O O O O . . . . . . . . . . |\$\$ | O O O O . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ ---------------------------------------[/go]`

There is a misunderstanding Bill. In my previous post I was talking on the diagram above where I did not draw any specific environment. On contrary I assume we are in a "CGT with non-ko environment".
In this context my understanding is the following:
1)the point a is a simple {3|-3} area
2) the point b is a simple {4|-4} area
3) because I assume we are in the context of "CGT with non-ko environment" I can simply ignore that a white move at "a" create a ko threat and I am allowed to play a difference game and prove white "b" dominates "a".

Clearly the point 3) is critical and my question is the following:
in the specific context of "CGT with non-ko environment" can (must!) I ignore the creation of a ko threat and can I play a difference game showing "b" dominates "a" ?

If not, does that mean that we are not allowed to play a difference game (I am still in a "CGT with non-ko environment" context!) if a ko threat appears somewhere in a sequence?

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 Post subject: Re: This 'n' that #959 Posted: Sat Jun 19, 2021 6:41 am
 Honinbo

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Gérard TAILLE wrote:
Bill Spight wrote:
Rather than edit my previous note, let me continue here.

Gérard TAILLE wrote:
The point is rather to understand what means CGT with non-ko environment.

My interpretation of a non ko environment in CGT is one in which there is no ko fight or potential ko fight.

Gérard TAILLE wrote:
IOW, in my exapmle, and considering we are in "CGT with non-ko environment", is it correct to say that the point "b" is a pure simple {4|-4} point, the point "a" is a pure {3|-3} point and as a consequence "b" dominates "a"?

I do not understand the relevance of your diagram, in which there is a ko fight. If the idea is to understand CGT with a non-ko environment, then don't use a ko environment to do so.

`[go]\$\$\$\$ ---------------------------------------\$\$ | . . X X O O O a O b O O O O X . . . . |\$\$ | X X . X X X X X O X X X X X X . . . . |\$\$ | O X X O O O O O O . . . . . . . . . . |\$\$ | O O O O . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ ---------------------------------------[/go]`

There is a misunderstanding Bill. In my previous post I was talking on the diagram above where I did not draw any specific environment. On contrary I assume we are in a "CGT with non-ko environment".
In this context my understanding is the following:
1)the point a is a simple {3|-3} area
2) the point b is a simple {4|-4} area
3) because I assume we are in the context of "CGT with non-ko environment" I can simply ignore that a white move at "a" create a ko threat and I am allowed to play a difference game and prove white "b" dominates "a".

Clearly the point 3) is critical and my question is the following:
in the specific context of "CGT with non-ko environment" can (must!) I ignore the creation of a ko threat and can I play a difference game showing "b" dominates "a" ?

Well, ko threats are, as a rule, sente. So, no, even in a non-ko environment a move which creates a sente is not the same as a move which does not. Edit: This ko threat costs one point to eliminate, so I think that CGT would not recognize its significance, except as a ko threat.

Edit2: I feel that I need to reiterate that I was not talking about positions with moves that create ko threats.

Gérard TAILLE wrote:
If not, does that mean that we are not allowed to play a difference game (I am still in a "CGT with non-ko environment" context!) if a ko threat appears somewhere in a sequence?

What has been proven about difference games has been without ko fights. Difference games can still provide useful heuristics, in general.

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 Post subject: Re: This 'n' that #960 Posted: Sat Jun 19, 2021 7:26 am
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`[go]\$\$\$\$ ---------------------------------------\$\$ | . . X X O O O a O b O O O O X . . . . |\$\$ | X X . X X X X X O X X X X X X . . . . |\$\$ | O X X O O O O O O . . . . . . . . . . |\$\$ | O O O O . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |[/go]`

Bill Spight wrote:
Well, ko threats are, as a rule, sente. So, no, even in a non-ko environment a move which creates a sente is not the same as a move which does not. Edit: This ko threat costs one point to eliminate, so I think that CGT would not recognize its significance, except as a ko threat.

Gérard TAILLE wrote:
If not, does that mean that we are not allowed to play a difference game (I am still in a "CGT with non-ko environment" context!) if a ko threat appears somewhere in a sequence?

What has been proven about difference games has been without ko fights. Difference games can still provide useful heuristics, in general.

OK Bill. In general difference games can still provide useful heuristics but if a ko threat is created somewhere in a sequecnce then your are not allowed to claim that a move "b" dominates really a move "a". In my exemple above because white "a" creates a ko threat we are not allowed to play a difference game and claim "b" dominates "a".

It is not good news.

`[go]\$\$B\$\$ -----------------\$\$ | . . . . . . O |\$\$ | X X . . . . O |\$\$ | . X O O O O O |\$\$ | . X X X O . . |\$\$ | . . . X O O O |\$\$ | . . . X X X X |\$\$ | . . . . . . . |\$\$ -----------------[/go]`

When I remember all difference games we played to prove a move dominates another in the diagram above, I realise now that most of the results of these difference games were not reliable because we forgot to look at the ko threats created against the white group. At least in the future we will be more careful.

dhu163 is right when claiming "I'm not too clear what "non-ko" environment means myself".

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