Cassandra, read the paper to its end, or put stones on a board and test each "basic-ko" of a standard triple-ko (no outside liberties). You CANNOT recapture a single stone in each of the three.

With a "standard" "Triple-Ko":

B1 captures W1.
W3 captures B3. This is a Ko-threat, related to B1 / W1.
B2 captures W2. This is a Ko-threat, related to W3 / B3. This answers White's Ko-threat, related to B1 / W1.
W1 captures B1. This is a Ko-threat, related to B2 / W2. This answers White's Ko-threat, related to W3 / B3.
B3 captures W3. This is a Ko-threat, related to W1 / B1. This answers White's Ko-threat, related to B2 / W2.
W2 captures B2. This is a Ko-threat, related to B3 / W3. This answers White's Ko-threat, related to W1 / B1.

The "Triple-Ko-"cycle is completed.

+ + + + + +

With regard to Ko #1:

B1 captures W1.
This is a Ko-threat, related to B1 / W1.
This answers White's Ko-threat, related to B1 / W1.
W1 captures B1.
This is a Ko-threat, related to W1 / B1.
This answers White's Ko-threat, related to W1 / B1.

+ + + + + +

With regard to Ko #2:

W3 captures B3.
This is a Ko-threat, related to W3 / B3.
This answers White's Ko-threat, related to W3 / B3.
B3 captures W3.
This is a Ko-threat, related to B3 / W3.

+ + + + + +

With regard to Ko #3:

B2 captures W2.
This is a Ko-threat, related to B2 / W2.
This answers White's Ko-threat, related to B2 / W2.
W2 captures B2.

+ + + + + +

Each of B1, W3, B2 has become re-captured !!!

+ + + + + +

What do I have overlooked ?

_________________
The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)

Good God people, we have diagrams and eidogo!

_________________
Occupy Babel!

Click Here To Show Diagram Code

[go]$$B
$$ +-----------+
$$ | X X X X X |
$$ | . X . X O |
$$ | X O X O . |
$$ | O O O O O |
$$ +-----------+[/go]

Click Here To Show Diagram Code

[go]$$B
$$ +-----------+
$$ | X X X X X |
$$ | . X . X O |
$$ | X O X O 1 |
$$ | O O O O O |
$$ +-----------+[/go]

Click Here To Show Diagram Code

[go]$$B
$$ +-----------+
$$ | X X X X X |
$$ | . X . X . |
$$ | X . X . X |
$$ | . . . . . |
$$ +-----------+[/go]

Now a recapture is impossible;)

This fake posting is far, far below your level.

Is this your "new" way, when you have run out of arguments with regard to contents ?

+ + + + +

E O T

_________________
The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)

Cassandra, the topic was: does a triple-ko consist only of basic-ko-intersections / of basic-kos. It does not, because an immediate recapture is impossible for one of the single stones. There are, however, six local-ko-intersections; there are three kos; so the name triple-ko is fine.

Your other problem with understanding has been your attempt to play ko threats. Ko threats [i.e., unrelated plays elsewhere on the board] are immaterial for the sake of my ko definition.

Just a last reply, before the battery actually gives up the ghost.

In general, a "Triple-Ko" consists of three "Basic-Ko", and you know this very well.

What you call the "standard shape" of a "Triple-Ko" is nothing more than a simplifying example, which main design motivation was to minimize the size of the diagram. And which -- for the "usual" reader -- is wholly sufficient to give an idea of the process of the mutually capturing of the various stones in question.

It is not only far below your level -- but also reflects poorly on the quality of your research -- to fade out the dependencies, and restrictions, which follow from reducing the shape, and combining all of the "Basic-Ko" with only one group each.

As far as your fake posting is concerned, as a matter of course there will never be a Ko in a game of Go after the Self-Atari of one of one side's groups.

Or, in another context, if one followed your fake posting in earnest, this would imply that any "Basic-Ko", wherin a Ko-capture simultaneously is an Atari, would lose its property, and become a "what then ???"-Ko.

_________________
The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)

Cassandra, kos can occur and vanish. In particular, a basic-ko can occur and vanish. More specifically, during a 6 plays cycle of a standard triple-ko without outside liberties, at any moment, there are exactly four basic-ko-intersections, but there are six local-ko-intersections.

By your own characterisation of a basic-ko, "a cycle, wherin each player captures the just played single stone of the opponent, which has captured a single stone of the player", there must be the possibility of an immediate recapture (expressed differently: a principle possibility of recurring capture and recapture). Even your own characterisation does not say "a single stone capture followed by exactly one ko threat followed by exactly one ko threat answer followed by a single stone recapture".

Such a triple-ko consists, according to my definition, of three kos, but does not consist of three basic-kos.

After move 1 of a long cycle, the two intersections that were not a basic-ko have become a basic-ko. After, e.g., 6 moves of a long cycle, again they are not a basic-ko.

What might have hapened beforx ?

Click Here To Show Diagram Code

[go]$$B
$$ +-----------+
$$ | X X X X X |
$$ | . X . X O |
$$ | X O X O . |
$$ | O O O O O |
$$ +-----------+[/go]

White pased >>> But why on earth did shi not recognizz that her stoney were in Atari ???

Click Here To Show Diagram Code

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

Wite did not caxture one of Black'x stones, but hopex for a Black pasz. >>> But wyy on earth did shi not recognizz that her stonyy were in Atari ???

Click Here To Show Diagram Code

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

Selx-Atari

Click Here To Show Diagram Code

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

Selx-Atyri

Click Here To Show Diagram Code

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

Sxlx-Atyry

Click Here To Show Diagram Code

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

Zxlx-Atyry

Click Here To Show Diagram Code

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

Zxlx-Ytyry

Click Here To Show Diagram Code

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

Zxlx-Ytyxy

Click Here To Show Diagram Code

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

Zxyx-Yzyxy

+ + + + + +

Conclusion:

Either White never intended to avoid losing the game or the outcome of the game never depended on a Ko-fight.

_________________
The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)

It is an interesting (even research) topic to study whether possibly creation of particular positions could have been the result of good strategy, but this is immaterial for my ko definition.

With regard to "Ko", an evaluation of the position in your fake posting does not make any sense under the assumption that it was Black to move, just because there would be no Ko. You seem to forget that "reducing" a position's type to minimum size does not have benefits only, but also drawbacks.

Trying to include or exclude something within your theory on the basis of a mistaken assumption cannot provide to good and viable results.

Either there is no Ko at all (Black to move), or all Ko-shapes are "Basic-Ko"-shapes (White to move), because any played stone can be re-captured.

Even if we assume that your point of view was correct, then it would follow that your fake posting's position cannot be called a "standard shape" for evaluating "Triple-Ko".

_________________
The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)

'Local-ko-intersection' demands "a set of cycles exists" and "a player". This player can be Black or White. For the sake of applying the definition to a position, either colour is tested for "a player". In our particular position, White can start and answer-force a cycle if Black ("the opponent") does prevent local-area-improvement of White on the cycle-set of the 6 intersections.

Dear Robert,

Letting our discussion here pass review, I remain with a very sad feeling. It is your thread, so I will not judge too hard, but I do not see any tiny sign that you are interested in some transmitting support for bridging from “your world” into the “common world”.
However, I think that the general quality, and importance, of your work deserves attempts to make clear, where the lack of understanding for your discussion style comes from. And I think that I am not alone in this forum with this attitude.

So I will try to give you some explanations, seen from my point of view. And I ask for the kind understanding of the other readers for this somewhat very longish post.

My main complains:

-- You are not responsive to what the other one has written.
-- This is true, even if the other one made large efforts, and tried to give detailed explanations in a “language”, which the other one thought was spoken “inside your world”.
-- At the slightest impression that you could be put on the defensive, you deviate from the previous course of the discussion, and try to establish a new, mainly unrelated, issue for further discussion.

I had this kind of course of discussion with you several times in the past and I am very tired now about it. This post took my about two hours, and is my last attempt to give you some help, so please think about it in earnest.

(( My personal point of view is formatted in italics below. ))


+ + + + + + + + + + +

My post:
viewtopic.php?p=151005#p151005

Core contents:

-- Starting point is your message “... allows better insight to ko strategy”.
-- My understanding of your classification of “Ko”.
-- I think that your classification of “Ko” follows the general strategy considerations.
-- My understanding of “Triple-Ko”.
-- I doubt your usage of the term “Triple-Ko” in your paper (giving a concrete example).

+ + + + + + + + + + +

Your post:
viewtopic.php?p=151014#p151014

Core contents:

-- Repetition (( may be clarification from your point of view )) of what was my understanding of your classification of “Ko”.
(( This is OK. But please note that my “I want to do A, because I want to reach B.” is the same as your “I assume reaching B to be mandatory, so doing A is forced.” ))
-- Your demand for presenting recent papers on “Ko-strategy”.
(( This is typical “your world”. Thoughts like “If I wanted to pre-empt this local loss, I would have to choose this line of play.” / “If I wanted to pre-empt that global loss, I would have to choose that line of play.” are not restricted to “Ko”, nor are they restricted to “Endgame”, nor to “Life & Death”, nor to any other aspect of the game. So I assume it to be very unlikely that there exists any paper that applies these thoughts to only one special aspect of the game. In my eyes, it is more part of a general understanding of the game’s (winning) strategy, as well as of “common sense”. ))
-- You claim your understanding of “Triple-Ko” ((, very obviously with having one very special application case in mind. )).
(( This is typical “your world”. There is no hint apparent that you would have really considered what I have written as my understanding, and what is true for the general case of a “Triple-Ko”. On the contrary, you tried to restrict the discussion of “Triple-Ko” to one very special application case inside “your world”. You were not responsive at all to the concrete example that I had mentioned from your paper. In addition, you seem to mix up the “general” usage of the term with your usage, and to do so on purpose. ))

+ + + + + + + + + + +

My post:
viewtopic.php?p=151015#p151015

Core contents:

-- Reference to your own definition of “Basic-Ko.”
-- Based on this, detailed explanation of my understanding.

+ + + + + + + + + + +

Your post:
viewtopic.php?p=151016#p151016

Core contents:

-- Demand to read your paper in total.
(( This is typical “your world”. I have some doubts of what I have read already. You will know the contents of your paper best, so why not giving some explanation ? ))
-- Your reference to your understanding of a “standard triple-ko”.
(( This is typical “your world”. Not only do you refer to a very special application case of “Triple-Ko”, but also claim this very special case to be “standard”. ))

+ + + + + + + + + + +

My post:
viewtopic.php?p=151018#p151018

Core contents:

-- My understanding of the detailed move sequence in a “standard” “Triple-Ko”.
(( Obviously I made a mistake by also using the term “standard”, because my reference was a more general one than yours. However, not drawing any diagrams was done on purpose, to allow a discussion of the “general” (in my understanding) case. In addition, this move sequence also applies for the very special application case that you obviously had in mind, but this was a secondary effect only. ))

+ + + + + + + + + + +

Your post:
viewtopic.php?p=151039#p151039

Core contents:

-- Demonstration of what you thought to be one valid move sequence starting from your “standard triple-Ko”.
(( In my understanding, not even valid in “your world”. ))

+ + + + + + + + + + +

My post:
viewtopic.php?p=151049#p151049

Core contents:

-- My way of demonstrating my lack of understanding for your discussion style.
(( By the way: The discussion has left its starting point. This is because you were very engaged to restrict the discussion issue, and to pull the discussion inside “your world”, to become able to “fight” on a “battle ground” that you are very familiar with. ))

+ + + + + + + + + + +

Your post:
viewtopic.php?p=151051#p151051

Core contents:

-- Explanation of what you thought was the discussion’s topic.
(( This is “your world”. Again, you refer to what you assume to be the “standard” case of a “Triple-Ko”. ))

+ + + + + + + + + + +

My post:
viewtopic.php?p=151054#p151054

Core contents:

-- My understanding of your “standard” case of a “Triple-Ko”.
-- A detailed explanation, why you were mistaken with your choice of a “standard”.
(( I am trying to enlarge the discussion issue, but in vain. ))

+ + + + + + + + + + +

Your post:
viewtopic.php?p=151060#p151060

Core contents:

-- Repetition of your point of view, concerning “your” “standard” case.
-- Claiming that my “definition” of “Basic-Ko” is mistaken (( when applied to “your” “standard” case )).
-- Claiming your definition of “triple-ko”, but which cannot be found in your paper.
(( This is “your world”. Obviously, your mutual change between “Basic-Ko” and “Non-Basic-Ko” does make sense only inside “your world”, but does not help the general understanding of what you want to explain. ))

+ + + + + + + + + + +

My post:
viewtopic.php?p=151067#p151067

Core contents:

-- Explanation, why I think that your mutual change between “Basic-Ko” and “Non-Basic-Ko” does not make any sense, even with “your” “standard” case.
(( I am again trying to enlarge the discussion issue, but in vain again. ))

+ + + + + + + + + + +

Your post:
viewtopic.php?p=151072#p151072

Core contents:

-- Opening a secondary issue.

+ + + + + + + + + + +

My post:
viewtopic.php?p=151075#p151075

Core contents:

-- Another explanation, why I think your move-sequence post to be a fake one.
(( I am again trying to enlarge the discussion issue, but in vain again. ))

+ + + + + + + + + + +

Your post:
viewtopic.php?p=151090#p151090

Core contents:

-- Explaining one of your definitions again.
(( This is “your world”. Again, you were not responsive to what I had written before. Finally, you refer to “White can start”, in contrary to what you did in your fake posting, where you started with Black. ))

+ + + + + + + + + + +

_________________
The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)

You need to pick a better topic than RJ's research to accomplish that. Within this topic, per definition and per implied agreement, you are moving within the confines of "his world". And for these purposes (of his research) - "his world" is good, proper, and appropriate. No reason to try to change that... yet - until there are more practical results, and then we might need to start translating.

The problem with Robert's "world" is *not* within his researtch - its when he tries to impose it on ours.
Here - you are doing the opposite, which is almost equally bad - impose our "world" on his.
The world's don't readily mix, this is the problem you both need to understand.


As much as I can sympathize with any attempts to poke holes in RJ's armor, in this case I think you might be.

_________________
- Bantari
______________________________________________
WARNING: This post might contain Opinions!!

I think that you are mistaken here.

Roberts discussion style is as destructive "inside his world" as it is "on the bridge" to the "common world". And I think that several of my posts here were written from a standpoint "inside his world".

In my experience, Robert does not like at all (and this is one issue, when he feels to be in danger to get in the defensive) to consider potential INTERNAL contradictions.

A concrete example:
It seems to me that Robert uses the term "triple-ko" for every cycle, wherein three discrete "spots" are involved, e.g. one "Two-Stage-Ko" combined with a "Round-Robin-Shape", as I referenced from his paper. Another application case would be a "Double-Ko" combined with a "Basic-Ko".

Even if we assume this to be valid "inside his world", the shape that Robert calls "standard" "inside his world" does have properties, which are not shared by those of the two other cases mentioned above. Therefore, "standard" cannot be the appropriate term, because the majority of all "Triple-Ko" cases does not share all of its properties.
This is true as long as Robert does not define "standard" as having a meaning "inside his world" that is similar to the meaning of "exceptional case" "outside his world".

Other examples of what I mean with "INTERNAL" contradiction are included in the referenced discussion with Robert.

_________________
The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)

Cassandra,

- Basic-ko, basic-ko-intersection and ko are terms in my paper, while triple-ko is not a term in that paper. Since the phrase triple-ko creates only confusion, we might as well drop the phrase from our discussion.

- That an immediate recapture is not possible I put as an excercise for you. It is a trivial exercise, but it is necessary to understand the answer. Apparently I expected too much from you. When an immediate recapture is impossible, you must accept this fact and must not try to prove the opposite. As long as you cannot understand the simplest fact (what is or is not a basic-ko), you cannot understand more complicated facts.

- With meta-discussion about "my world", you achieve exactly nothing. A definition is not understood by meta-discussion or by insisting on "your world", but must be understood by thinking about the definition itself.

- There is a principle alternative possibility to get an approximative understanding, if you can accept what I say and do not insist on remaining in "your world". You need to accept that ko threat strategy is immaterial. If you do not want to accept this, then you do not have access to the alternative possibility. Instead, you need to think about the definition itself, or give up trying to understand it at all.

- If you really want to understand at least the core ideas, I suggest that you start again. Start with understanding what is a basic-ko. Apply it to this position

Click Here To Show Diagram Code

[go]$$B not a basic-ko
$$ +-----------+
$$ | X X X X X |
$$ | . X . X W |
$$ | X O X O C |
$$ | O O O O O |
$$ +-----------+[/go]

by understanding why, in this position, the marked intersections are not a basic-ko.

This left me speechless. Didn't expect such kind of side attack, less on someone that was actually trying to understand. Robert, as a mathematician (I got the impression somewhere that you finished your degree) you should be pretty aware of what trivial really means, in the context of proofs and mathematics, or at least maths related language. Just to give a trivial example: in a class of numerical analysis I can assume linear algebra operations (matrix computations, essentially) are trivial and thus known or easy to remember. I can't go on and assume the implicit function theorem: I have to explain, clarify and understand that people may not get it anyway. Trivial exercises should be understood on the basis of a proper text used for learning (in Conway's Complex Analysis I and II there are many trivial exercises, which can easily amount to 3 pages of math notation and need the previous 300 pages of knowledge, which is structured in a proper way).

I'm not sure (I'm not diving into this thread to see what exactly is) which paper Cassandra is reading. I just opened ko.pdf from your site, which is the one I assume she is reading. If this is it, kudos to Cassandra for trying to read it. Myself, after having read more than 200 (just a guesstimate from the number of papers I've had at one point in my folders) papers in advanced mathematics, find this text discouraging from any kind of further reading.

_________________
Geek of all trades, master of none: the motto for my blog mostlymaths.net

Dear Robert,

I will not discuss this issue any longer with you, just because it is useless, and because there is no common basis for understanding.

Please refer to Herman's assumption in
viewtopic.php?p=150692#p150692
for an deeper explanation of "useless".

Your behaviour here is like the one described by Herman, with regard to your playing Go.
Even those who might be interested in entering "your world", will be bitten away by you.

As far as the position is concerned, which is named "standard triple-ko" by you, I have already explained in detail, why this cannot be "standard", and what the (surely unwanted) consequences (even "within your world") would be, if you were right. But -- this is one of my complains -- you are absolutely not responsive to what I have written in this context.
Therefore, it is absolute nonsens to try to be suggesstive of that I needed several private lessons, concerning the issue in question.

I will try to give you an explanation, why -- even "within your world" -- the position in question is no valid basis for discussion (neither for research, when aiming at generating general principles).

I admit that this (reduced, and simplified) position is generally used e.g. in dictionaries as ONE (!!!) example case to explain "Triple-Ko". However -- as I have written already -- with the motivation of reducing the DIAGRAM'S SIZE, as well as showing a MOVE SEQUENCE that leads to a cycle.

It is self-evident that there will be a cycle, only in the case that neither side gives way (e.g. in the case of a "Double-Ko" combined with a "Basic-Ko" connected the "Basic-Ko", leaving a "Double-Ko" behind on the board). With a more "general" application case of "Triple-Ko" (or a position from a real professional game), it might be more difficult to explain / to understand this effect than with the reduced, and simplified, position in question.

Even for a "weaker" player it will be clear -- please refer to my diagrams -- that neither option 1 "Pass" for deverting from the cycle will really work (the player's group will be captured), nor will option 2 "Connect a Ko" (the player's group will be captured).

Concluding, one has to reason that the reduced, and simplified, position in question will have been choosen as example of "Triple-Ko" for PEDAGOGICAL PURPOSES only. No one will have had the intention to make this the BASIC CASE for any theoretical research on "Triple-Ko".

Your deduction that there were only two "Basic-Ko" in a "Triple-Ko" at any time, is simply wrong -- with regard to the general application case.

_________________
The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)

Cassandra, I cannot be as reponsive to each of your statements as you wish, because such detailed discussion is not tolerated here. If you think that you need detailed replies for an understanding, we need to discuss at a place of (almost) free speech, such as rec.games.go or dgob.de. There, I can answer 20 times as detailed as I can answer here.

Cassandra, RBerenguel: understanding the paper requires effort. I encourage effort by leaving somebody, who tries to understand, something to think and find out by himself.

Triple-ko: see my previous reply. - (There are other triple-ko-like ko shapes with three basic-kos.)

Reading a paper, writing a paper are two endeavours which require a lot of effort. In this paper I can't see any editorial effort to clear the exposition: it can be seen in many phrases, construction and the overall structure. In some sense, the effort the writer puts to make the exposition clear saves a lot of hassle to enough people to make for a more efficient world. But its a function of complexity vs. readership.

Even in the relatively small world of entire transcendental functions dynamics I was required to write very clear introductions, preliminaries (6 pages of introduction, 6 pages of preliminary results, in a paper with 34 pages not counting bibliography, so almost 50% of the content is preparations) and extremely detailed proofs, even though the readership would be more than familiar with most of the concepts used afterwards.

Also, OpenOffice¹? Really?


1: Not to dismiss OpenOffice as a capable office suite, but there are solutions specially crafted for math research, go typesetting and which are required in all math and most compsci journals in the world

_________________
Geek of all trades, master of none: the motto for my blog mostlymaths.net