Part 10/12
White Starts in the Initial Position
White only has non-ko options. Dia. 23 shows White's best non-ko option. White's scaffold is derived from Black's wall of the remaining ko in the position after move 1 by adding the tax T for White 1.
White scaffold:
1/3 + T if T ≥ 1/3
1 - 2T + T = 1 - T if T ≤ 1/3
Dias. 31 - 33: We interpret White's scaffold.
Dia. 31: This is the case T ≥ 1/3 of White's scaffold. After move 1, the players play elsewhere. The marked remaining basic endgame ko has the count 1/3. White's 1 excess play incurs the tax T. Therefore, the result is 1/3 + T.
Dias. 32 + 33: This is the case T ≤ 1/3 of White's scaffold. The players play locally if they have a legal local play. Due to the prisoner difference 1, we have the count 1. Black's 1 excess play incurs the tax -T. Therefore, the result is 1 - T.
Engame value of ko
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RobertJasiek
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Re: Engame value of ko
Last edited by RobertJasiek on Tue Feb 21, 2023 3:51 pm, edited 1 time in total.
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Re: Engame value of ko
Part 11/12
Thermograph
We have
Black scaffold:
3 - 3T if T ≥ 7/9
2/3 if 1/3 ≤ T ≤ 7/9
1 - T if T ≤ 1/3
White scaffold:
1/3 + T if T ≥ 1/3
1 - T if T ≤ 1/3
Case T ≤ 1/3:
Black wall: 1 - T
White wall: 1 - T
Case 1/3 ≤ T ≤ 7/9:
Compare 2/3 ? 1/3 + T <=> 1/3 ? T
so 2/3 ≤ 1/3 + T <=> 1/3 ≤ T.
Black's scaffold is at most White's scaffold. We are in a cave.
Black wall: 2/3
White wall: 2/3
Case T ≥ 7/9:
Compare 3 - 3T ? 1/3 + T <=> 2 2/3 ? 4T <=> 2/3 ? T
so 3 - 3T < 1/3 + T <=> 2 2/3 < 4T <=> 2/3 < T
Black's scaffold is smaller than White's scaffold. The cave continues.
Set the walls as in the previous, lower case.
Black wall: 2/3
White wall: 2/3
Thermograph for All Cases:
Black wall:
2/3 if T ≥ 1/3. Mast.
1 - T if T ≤ 1/3.
White wall:
2/3 if T ≥ 1/3. Mast.
1 - T if T ≤ 1/3.
Move value and count:
We find equality of the walls for the smallest temperature of the mast. This occurs for the case 1/3 ≤ T ≤ 7/9.
Equality: 2/3 = 1/3 + T <=> 1/3 = T
Move value: M := T = 1/3
Count: 2/3 = 1/3 + 1/3 = 2/3
Thermograph
We have
Black scaffold:
3 - 3T if T ≥ 7/9
2/3 if 1/3 ≤ T ≤ 7/9
1 - T if T ≤ 1/3
White scaffold:
1/3 + T if T ≥ 1/3
1 - T if T ≤ 1/3
Case T ≤ 1/3:
Black wall: 1 - T
White wall: 1 - T
Case 1/3 ≤ T ≤ 7/9:
Compare 2/3 ? 1/3 + T <=> 1/3 ? T
so 2/3 ≤ 1/3 + T <=> 1/3 ≤ T.
Black's scaffold is at most White's scaffold. We are in a cave.
Black wall: 2/3
White wall: 2/3
Case T ≥ 7/9:
Compare 3 - 3T ? 1/3 + T <=> 2 2/3 ? 4T <=> 2/3 ? T
so 3 - 3T < 1/3 + T <=> 2 2/3 < 4T <=> 2/3 < T
Black's scaffold is smaller than White's scaffold. The cave continues.
Set the walls as in the previous, lower case.
Black wall: 2/3
White wall: 2/3
Thermograph for All Cases:
Black wall:
2/3 if T ≥ 1/3. Mast.
1 - T if T ≤ 1/3.
White wall:
2/3 if T ≥ 1/3. Mast.
1 - T if T ≤ 1/3.
Move value and count:
We find equality of the walls for the smallest temperature of the mast. This occurs for the case 1/3 ≤ T ≤ 7/9.
Equality: 2/3 = 1/3 + T <=> 1/3 = T
Move value: M := T = 1/3
Count: 2/3 = 1/3 + 1/3 = 2/3
Last edited by RobertJasiek on Tue Feb 21, 2023 3:53 pm, edited 1 time in total.
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Re: Engame value of ko
Part 12/12
Epilogue
Exercise: Correct all my mistakes! If there are none, prove it!
Exercise: Draw the scaffolds and thermographs!
In message #12,
https://www.lifein19x19.com/viewtopic.p ... 29#p262229
Bill Spight has proclaimed the thermograph that now I have calculated for the colour-reversed case.
I have hardly studied strategy yet. The cases T >= 7/9, T = 7/9, 1/3 <= T <= 7/9, T = 1/3, T <= 1/3 must be distingushed for Black's or White's start. Gerard Taille, Bill Spight and Schachus have made first suggestions but a rewriting and completion would be useful. However, maybe this task should be delayed until the other initial positions will have received their accurate calculations so we confirm their counts, can derive the gains and thereby better understand correct strategy.
Epilogue
Exercise: Correct all my mistakes! If there are none, prove it!
Exercise: Draw the scaffolds and thermographs!
In message #12,
https://www.lifein19x19.com/viewtopic.p ... 29#p262229
Bill Spight has proclaimed the thermograph that now I have calculated for the colour-reversed case.
I have hardly studied strategy yet. The cases T >= 7/9, T = 7/9, 1/3 <= T <= 7/9, T = 1/3, T <= 1/3 must be distingushed for Black's or White's start. Gerard Taille, Bill Spight and Schachus have made first suggestions but a rewriting and completion would be useful. However, maybe this task should be delayed until the other initial positions will have received their accurate calculations so we confirm their counts, can derive the gains and thereby better understand correct strategy.
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Gérard TAILLE
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Re: Engame value of ko
How can you reach the above position taking into account the rules you explained in your pervious post ?RobertJasiek wrote:Part 2/12
Positions
The unsettled initial positions and followers are:
...
Each of these positions is 'simple' because each possible alternating cycle of plays is a 2-play cycle and each player has at most one play that is a basic ko capture. In the local endgame, long cycles involve successive black plays and successive white plays while either opponent must play elsewhere or pass.
On a player's move, choose his best among the available of these options:
- best local non-ko play
- his ko option followed by the opponent's local play that does not recapture this ko
- his ko option followed by the opponent's play elsewhere and the player's local play
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RobertJasiek
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Re: Engame value of ko
The three options do not come as rules but as definition of thermographic calculus for generalised thermography. On his turn, the player considers all up to three options - those available. Each option has T is a parameter. Therefore, for every T, the second-moving opponent, if White, minimises among options 2 and 3, then the player, if Black, maximises among this and option 1. For reversed colours, swap min-max.
In part 2 of my messages, only a preliminary study is done to find out whether the initial position is 'simple' (a simple loopy game). In the preliminary study, the choices due to thermographic calculus are not considered yet. Only the basic rules of the game and, I think, the basic ko rule apply. So let Black ko capture, White elsewhere, Black ko capture, White elsewhere, Black connect and you have this subposition, which meets the 'simple' definition's criteria.
In part 2 of my messages, only a preliminary study is done to find out whether the initial position is 'simple' (a simple loopy game). In the preliminary study, the choices due to thermographic calculus are not considered yet. Only the basic rules of the game and, I think, the basic ko rule apply. So let Black ko capture, White elsewhere, Black ko capture, White elsewhere, Black connect and you have this subposition, which meets the 'simple' definition's criteria.
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Gérard TAILLE
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Re: Engame value of ko
Surely I must have missed a point.RobertJasiek wrote:The three options do not come as rules but as definition of thermographic calculus for generalised thermography. On his turn, the player considers all up to three options - those available. Each option has T is a parameter. Therefore, for every T, the second-moving opponent, if White, minimises among options 2 and 3, then the player, if Black, maximises among this and option 1. For reversed colours, swap min-max.
In part 2 of my messages, only a preliminary study is done to find out whether the initial position is 'simple' (a simple loopy game). In the preliminary study, the choices due to thermographic calculus are not considered yet. Only the basic rules of the game and, I think, the basic ko rule apply. So let Black ko capture, White elsewhere, Black ko capture, White elsewhere, Black connect and you have this subposition, which meets the 'simple' definition's criteria.
How can you accept to consider a stupid move like black move
in the diagram above?If so then don't you have also to consider all the positions in the following very provocative stupid sequence:
White to play from the initial position: My view is that when building the tree needed to analyse a position you MUST simply delete all obvious stupid moves in order to reduce drastically the nodes you have to evaluate.
as a consquence the position cannot appear in the tree.
Do you really want to put this position in the tree as a follower of the initial position?
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RobertJasiek
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Re: Engame value of ko
The definition of simple (loopy game) does not depend on any classification of strong or weak moves.
Therefore, the position must also be considered.
You have correctly identified my mistake of (besides settled followers) not listing all those positions occurring after what go players call weak (stupid, if you prefer) moves. Also they must be listed and checked for the definition in principle.
In practice, you are right. A strict application of the (almost) formal definitions is impractical in most cases and we must (or want to) simplify analysis by skipping the obviously immaterial, like we do in tactical reading.
Similarly, in Dia. 14, I made the lazy mistake of not considering the non-ko play besides the ko option and letting Black maximise among both. You would say, it has been well done by me because the non-ko play is obviously inferior to the ko option:)
With some experience, one might look for pragmatic simplifications, such as combining the evalation of Dias. 13-15 or not dissecting every remaining basic endgame ko. However, I have wanted to stick application as close to the definitions as I could. First, one needs to understand what one is doing before one max simplify.
Simplification is often tempting but not always valid, especially not in accurate endgame evaluation or CGT. We must always be careful.
Therefore, the position must also be considered.
You have correctly identified my mistake of (besides settled followers) not listing all those positions occurring after what go players call weak (stupid, if you prefer) moves. Also they must be listed and checked for the definition in principle.
In practice, you are right. A strict application of the (almost) formal definitions is impractical in most cases and we must (or want to) simplify analysis by skipping the obviously immaterial, like we do in tactical reading.
Similarly, in Dia. 14, I made the lazy mistake of not considering the non-ko play besides the ko option and letting Black maximise among both. You would say, it has been well done by me because the non-ko play is obviously inferior to the ko option:)
With some experience, one might look for pragmatic simplifications, such as combining the evalation of Dias. 13-15 or not dissecting every remaining basic endgame ko. However, I have wanted to stick application as close to the definitions as I could. First, one needs to understand what one is doing before one max simplify.
Simplification is often tempting but not always valid, especially not in accurate endgame evaluation or CGT. We must always be careful.
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RobertJasiek
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Re: Engame value of ko
The more frightening thing about your White 1 is the possibility of a long cycle without basic ko recapture or suicide. So, strictly speaking, the initial position is not 'simple'. Only after pruning the obviously inferior moves and variations, the remaining game is 'simple'. It will become an exercise to evaluate corner stage ko with advanced ko thermography...;(
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Gérard TAILLE
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Re: Engame value of ko
You are right Robert, strictly speaking the two stage ko you proposed is not 'simple'. If the goal for you is that you can take into account such position you have to modify something: Firstly by changing some defintions and maybe secondly by adding a procedure to avoid disturbing weak moves. Not that easy to formalize is it?RobertJasiek wrote:The more frightening thing about your White 1 is the possibility of a long cycle without basic ko recapture or suicide. So, strictly speaking, the initial position is not 'simple'. Only after pruning the obviously inferior moves and variations, the remaining game is 'simple'. It will become an exercise to evaluate corner stage ko with advanced ko thermography...;(
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Gérard TAILLE
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Re: Engame value of ko
Obviously all the process you defined in your post allows to draw a thermograph but I do not see clearly what work do you do at each node.RobertJasiek wrote:Part 7/12
...
Dias. 19 + 20: This is the case 1/3 ≤ T ≤ 7/9 of Black's scaffold. After move 2, the players play elsewhere. The marked remaining basic endgame ko has the on-board count -1/3. Together with the prisoner difference 1, we have the count -1/3 + 1 = 2/3. The equal numbers of black and white plays incur the net tax 0. Therefore, the result is 2/3 + 0 = 2/3.
Especially what about the count? In the diagram above you have shown how you derived the count to reach the value count = 2/3. Does that mean that you make this derivation for each node?
If yes can you telll us what is the count of the following position
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Re: Engame value of ko
Berlekamp invented this and Siegel tried his best to express it consistently in more accessible mathematical annotation but did not quite succeed. However, what we might want, easier practical application by pruning, does not fit easily into such mathematical theory but would require demanding extensions for it. Currently, we are better off by doing such pruning informally.Gérard TAILLE wrote: You are right Robert, strictly speaking the two stage ko you proposed is not 'simple'. If the goal for you is that you can take into account such position you have to modify something: Firstly by changing some defintions and maybe secondly by adding a procedure to avoid disturbing weak moves. Not that easy to formalize is it?
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Re: Engame value of ko
Ah, I see. The problem is that I have used "count" in two different manners:Gérard TAILLE wrote:I do not see clearly what work do you do at each node.
Especially what about the count?
1) A diagram's on-board count incl. the prisoners. Especially in a settled position.
2) A count derived during an algebraic calculation from the scaffolds. This presumes you are familiar with basic thermography. I have not explained this algebraic procedure here but presumed that the reader knows it. If not, you know where you can find it:) However, you might try to learn it by looking at my calculations.
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Re: Engame value of ko
Good joke. This will require about the same amount of work as I have applied to the other initial position! The count pops out while calculating the intial positon's thermograh after having analysed the followers and options towards them.Gérard TAILLE wrote:what is the count of the following position
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Re: Engame value of ko
Why, of course. Thermography is a doubly recursive, iterative evaluation on the move options! Similar to tactical reading, except that emphasis is on value calculations.Gérard TAILLE wrote:Does that mean that you make this derivation for each node?
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Re: Engame value of ko
It is not clear because in the post viewtopic.php?p=276760#p276760 you analyse the position as a follower and you only derive the black scaffold: 3 - 3TRobertJasiek wrote:Why, of course. Thermography is a doubly recursive, iterative evaluation on the move options! Similar to tactical reading, except that emphasis is on value calculations.Gérard TAILLE wrote:Does that mean that you make this derivation for each node?
With my (poor) knowledge of thermography I cannot deduce the count with only this information.
Then it seems you do not need to derive the count for this position and you confirm that in your previous post viewtopic.php?p=276787#p276787 where you claimed it would require a great deal of new work
My basic question is the following : do you use count only on the leaves or also through the nodes of the tree by iteration?