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 Post subject: Ultimate theory of weiqi rules
Post #1 Posted: Sat May 13, 2023 4:11 am 
Oza

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I am posting the following text on behalf of Chen Zuyuan, who is probably the world's foremost expert on go rules. Apart from the fact that he is a scientist by training and vocation, he is thoroughly conversant with the texts of ancient Chinese go and so has a panoply of texts from two millennia to guide him in his thoughts. He also devised the ruleset used for world mind sports events in China.

There is no ulterior motive behind this posting beyond the fact that Chen felt he would like to have his thoughts recorded somewhere on an English-speaking site. He decided that L19 was the best forum.

I have obviously had a hand in the English version. I am neither a mathematician nor a scientist and certainly not a rules aficionado. If you care to suggest corrections to the English translation, I will happily take them on board. But if you have comments on the underlying ideas, do note that the Chinese text is the definitive version.

围棋规则理论的终极化研究

Study on the Ultimate Theory of Weiqi Rules


陈祖源 Chen Zuyuan

前言
终极理论一词(Ultimate Theory)出自物理学,即是以特殊的数学形式完备的描述物理规律。从其发端的麦克斯韦方程组就可以看出它与从牛顿到法拉第的以实验为基础的经典物理学在方法论上的差异。当物理学被数学完美的表达时物理学也完美了,终极了,这就是终极理论的意义。我们以往对围棋规则的研究出发点是围棋已经存在,我们是去尽可能准确合理的规范它。而仿照物理学的终极化的研究方法,是用数学和逻辑的方式自动的生成围棋。也就是说我们把围棋当作是在一块棋盘上演绎的一个独特的数学体系。数学的基础是公理,数学体系是在公理化演绎下产生的。围棋规则的终极化研究就是在围棋公理下产生围棋,那么这样表达的围棋规则理论就是完美的、终极的。

PREFACE
The term “Ultimate Theory” comes from physics, being a complete description of the laws of physics in a special mathematical form. From Maxwell’s equations at its origin, we can see the methodological differences between it and the classical physics that were based on experiments from Newton to Faraday. When physics is perfectly expressed by mathematics, physics is also perfect and ultimate, which is the meaning of the Ultimate Theory. The starting point of our previous research on weiqi rules has been that weiqi [go] already exists, and we have tried to describe it as accurately and as reasonably as possible. The research approach instead, imitating the ‘ultimate’ in physics, is to generate weiqi automatically in a mathematical and logical way. That is, we treat weiqi as a unique mathematical system interpreted on a board. Mathematics is based on axioms, and mathematical systems arise under axiomatic deduction. The ultimate study of weiqi rules is to the formation of weiqi via the weiqi axioms, so that the theory of weiqi rules expressed in this way will be perfect and ultimate.

一、围棋规则的公理化构架
公理就是不证自明的,依据理性而公认的,最基本的命题。围棋规则由两类3条公理和1条规则推理,1个目标推理组成:
设定性公理2条;围棋公理1条;规则推理1条;目标推理1条。

I. AXIOMATIC FRAMING OF THE RULES OF WEIQI
An axiom is a self-evident, rationally based and accepted, most fundamental proposition. The rules of weiqi consist of three axioms, one rule corollary, and one example of goal reasoning, in two categories: two established axioms, one weiqi axiom, the one rule corollary, and the one case of goal-reasoning.

(一)设定性公理:
1、基本描述:围棋由棋盘和黑白两色棋子组成,棋盘为纵横各n道(标准棋盘取n=19)。标准棋盘19×19,有361个交叉点;棋子数量应该充足。

2、程序规则:双方各执一色棋子。轮流着于棋盘上的空点。着于棋盘上的棋子不再移动。围棋的设定是少而基本的,是自然的,也是必然的,因此可以叫公设。

(i) Established axioms:
1. Basic description: Weiqi consists of a weiqi board and black and white stones, the board is n lines vertically and horizontally (the standard board is taken to be n = 19). The standard board is 19 x 19 and has 361 intersections; there must be a sufficient number of stones.

2. Rules of procedure: Each player has one colour of stones. They take turns to place one stone on an empty point on the board. Stones placed on the board do not move again. These established elements of weiqi are few and basic, natural and inevitable, and can therefore be called established axioms.

(二)围棋公理,即围棋的核心规则或特性规则:
3、吃子规则:无气之子从棋盘上提出。这是围棋的核心,是围棋之为围棋的所在。

(ii) Weiqi axiom, i.e. the core rules or characteristic rules of weiqi:
3. Rule of capture: the stones with no liberties must be removed from the board. This is the core of weiqi and is what makes it the surrounding game it is.

公理即人为的规定到此为止,规则的其余均应从此三条公理推出,不应再有别的人为的规定。
The axioms, that is, the artificial stipulations, end here, and all the rest of the rules must be inferred from the three axioms, so that no other artificial stipulation need be made.

(三)推理
4、规则推理:打劫规则
本条是2、和3、的推论。此条后面再解释。

(iii) Corollary
4. Rule corollary: the ko rule
This item is an inference from 2. and 3. This item will be discussed later.

5、目标推理:围棋的目标
游戏的目标是规则的产物。围棋规则以公理体系构建,则目标就必须出自公理之中,即是公理的推理。因此公理化的目标只能是公理中的两种实物:公理1中的棋盘或棋子。

5. Goal corollary: the goal of weiqi
The goal of the game is a product of the rules. Since the rules of weiqi are constructed in a system of axioms, the goal must come out of the axioms, i.e. be an axiomatic corollary. Thus the goal of an axiomatisation can only be one of two physical objects in the axioms: the board or the stones in Axiom 1.

(1)以公理1中的棋子为目标,则公理1中的棋盘就是实现目标的工具:在棋盘上依照第2、3、4条公理,生存更多的棋子。这就是古代围棋规则,一般称之为数子规则。 当然也可以是吃对方更多的棋子。死子多就是活子少,活子规则和死子规则其实是一回事。

(1) With the stones in Axiom 1 as the goal, the board in Axiom 1 is the vehicle for achieving the goal: more stones survive on the board in accordance with Axioms 2, 3 and 4. This is the ancient rule of weiqi, generally known as ‘stone scoring.’ Of course, it can also mean capturing more of the opponent's stones. More dead stones means fewer live stones, and the scoring with live stones or dead stones is actually the same thing.

(2)以公理1中的棋盘为目标,则公理1中的棋子就是实现目标的工具。用棋子依照第2、3、4条公理,去占取棋盘上的点(19×19=361),以占取更大的部分为目标。

(2) Taking the board in Axiom 1 as the goal, the stones in Axiom 1 are the tools to achieve the goal. That means using the stones to occupy the points on the board (19 x 19 = 361) in accordance with Axioms 2, 3 and 4, with the goal of taking the larger share.

但什么是占取?最公理化的占取就是用棋子直接的占领,那结果就等同(1)。但(1)中的眼位是不能算的,因为那里不能生存棋子。但如果目标是棋盘上的点,眼位这个点是占取的棋子的组成部分。因此包含眼位的占取可以认为也是公理化的,虽然程度可能稍弱一些。这就是现在的中国规则,一般称之为计地规则。

But what is ‘occupation’? The most axiomatic occupation is to occupy with stones directly, and the result is equivalent to (1). But the eye-positions in (1) do not count, because no stones can live there. But if the goal is points on the board, the points of the eye-positions are part of the occupying stones’ group. Therefore, containing the eye-positions can be considered axiomatic as well, although perhaps to a slightly weaker extent. This is now the Chinese rules, generally known as ‘area scoring.’

解释:那么能不能还有其他目标呢?公理1围棋组成只有棋盘和棋子,再没有三,因此公理化推论下的必然目标就只有这两个。再设其他目标就是在公理体系以外的人为的加入了。一个公理性的围棋规则体系,如果目标设定在公理体系之外,就很可能会导致体系的不和谐。日本的计空规则,包括死子回填,是一个与公理性无关的人为的设计。不是从公理推导出来的人为设计的规定,很可能会与其他公理冲突,导致自我逻辑悖理。

Discussion: Can there be other goals then? Axiom 1 is that weiqi consists of only the board and the stones, and there is no third element, so that the only two natural goals under axiomatic inference are these. To set other goals would be an artificial addition outside the axiomatic system. An axiomatic system of weiqi rules with goals set outside the axiomatic system is likely to lead to discord in the system. Japanese territory scoring, including backfill of dead stones, is an artificial design that has nothing to do with axiomatics. Artificially designed rules that are not deduced from axioms are likely to conflict with other axioms and lead to self-logical paradoxes.

空的价值意义只有在作为(1)或(2)目标体系中的推理时才成立。使用空这个推理是因为它可以简化(1)或(2)的目标的计算。本质的的公理化的目标只能是(1)或(2)。日本规则的失误就在于被表面所模糊,失落了本质。

The value meaning of the territory only holds when it is used as an inference in the system of goals of (1) or (2). The inference of territory is used because it simplifies calculation of the goal of (1) or (2). The intrinsic axiomatisation of a goal can only be (1) or (2). The failure of Japanese rules is that they are obscured by appearances and so lose their essence.

二、关于打劫规则
打劫规则是围棋中一个特殊的规则,一直来都以“在反复回提一个子之前需在棋盘其他地方下一着棋”的方式规定。这个规定显然不够自然。因此人们提出了禁止全局同形,这是一个很好的理论,但这也是新设了一条公理,而严格地说它的公理性偏弱。事实上我们无须特别规定打劫规则,它已经逻辑性的隐含在规则2和3之中。

II. About the ko rule
The ko rule is a special rule in weiqi and has always been laid down in the form of "Before a stone can be repeatedly removed, a move must be made elsewhere on the board." This rule is obviously not natural enough. Thus, the idea of prohibiting repetition of the game position has been proposed, which is a good theory, but this is also a new axiom,and strictly speaking its axiology is weak. The fact is that we do not need to specify the ko rule, which is already logically implicit in Rules 2 and 3.

劫子立即回提这个动作:把对方刚下的棋拿走,把对方刚吃的棋子重新放回去,因此它在本质上是否定了对方上一着棋,即否定了公理2赋予的对方的一次着子权利。虽然它有公理3的依据,但有了着子规则才有吃子规则,公理3是在公理2以后产生的,因此公理2优先。因此打劫规则可以认为是公理2和3的推理。

Immediate removal of a ko-stone by this action:
Taking away a stone the opponent has just played and putting back a stone the opponent has just captured, essentially negates the opponent's previous move, i.e. negates the right of the opponent to place a stone under Axiom 2. Although it has a basis in Axiom 3, there is a rule of placing before there is a rule of capturing, and Axiom 3 arises after Axiom 2, so Axiom 2 takes precedence. Thus the ko-rule can be considered as a corollary of Axioms 2 and 3.

如果把禁止否定从上一着引申到之前任一着:
禁止否定对方的棋着,即不得使对方面临其曾面临过的局面。这也就是禁止全局同形。虽然引申会导致逻辑必然性降低,但面对循环劫尤其是假生,采用这个引申还是需要的。

Where the prohibition on negation is extended from the previous move to any of the previous moves:
It is forbidden to negate the opponent's moves, that is, to make the opponent face a game situation he has already faced. This means prohibiting repeated game positions. Although extension leads to a reduction in logical necessities, it is nevertheless still necessary to adopt this extension for multiple kos, and in particular for cases like moonshine life.

三、其他
1、虚着
由于围棋以活子或占地为目标,着一子就是活子也是占地,因此如果不着子就是表示不再下棋了,如果对方也不着子,双方都不下棋了,对局结束。因此不着子视为提议终局和同意终局,可以理解为是规则5的推理。如果对方继续下棋,棋局当然继续,这时刚才那未着子的行为称为虚着或弃着。

III. MISCELLANEOUS
1. Void move (pass)
Since weiqi aims at living or occupying space, a move is a move to live or to occupy space, so if no move is made it implies that the game is no longer to be played, and if the opponent does not make a move either, both players are no longer playing and so the game is over. Therefore, making no moves is regarded as proposing the end of the game and then agreeing to the end of the game, which can be understood as a corollary to Rule 5. If the opponent continues to play, the game of course continues, and the unplayed move is then called a void move or a pass.

2、禁着点
在公理化体系中自杀当然是不禁的。

2. Forbidden moves
Suicide is of course not forbidden in the axiomatic system.

3、终局与协议终局
这是协议性程序规则,是在规则1、2、3、4的前提下保证规则5的结果的实现。

3. End of the game and agreement as to end of the game
These are agreed procedural rules that guarantee the achievement of the outcome of Rule 5 subject to Rules 1, 2, 3 and 4.

4、贴先
这属于约定,不是围棋的基本规则。所有现行的围棋规则文本中都没有贴先的规定,而是在比赛规则中设定。因为原则上说贴先只是平衡先行之利的方法之一,是不是用这种方法,贴多少,怎么贴,都是可以有选择性的。

4. Compensation for first move [komi]
This is a convention, not a basic rule of weiqi. There is no provision for compensation in any of the current weiqi rules texts, but rather it is set out in the competition rules. Because, in principle, komi compensation is only one of the ways to balance the advantage of playing first, the available options are whether to use this method, how much to compensate, and how to compensate.


Last edited by John Fairbairn on Sun May 14, 2023 1:27 am, edited 2 times in total.

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Post #2 Posted: Sat May 13, 2023 4:20 am 
Gosei

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Do you think we could have his permission to add the text to Senseis Library?

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Post #3 Posted: Sat May 13, 2023 4:52 am 
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From the preface I already like the writers as we seem to think in similar ways. Thank you both very much!

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Post #4 Posted: Sat May 13, 2023 5:28 am 
Oza

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Quote:
Do you think we could have his permission to add the text to Senseis Library?


I had already raised the possibility with him but it seems best to wait until we can be sure we have a settled final text. SL is too prone to tampering. And as for other sites beyond SL, it's important not to have different versions floating around that are not maintained centrally.

And L19's discussion mode may come into its own!

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Post #5 Posted: Sat May 13, 2023 6:21 am 
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Dear John,

What follows below is neither a suggestion to correct your English translation nor a comment on the underlying ideas of the Ultimate Theory.

Rather, it is a suggestion to deepen a very specific issue that you could usefully share with Chen Ziyuan when the opportunity arises.

+ + + + + + + + + + + + + + + + + + + +

Chen Ziyuan translated by John Fairbairn wrote:
Study on the Ultimate Theory of Weiqi Rules

In total, an extremely interesting perspective.

However, I have some stomach pains with ...
Chen Ziyuan translated by John Fairbairn wrote:
III. MISCELLANEOUS
...
2. Forbidden moves
Suicide is of course not forbidden in the axiomatic system.

My feeling is that this statement / determination requires a corresponding derivation as found for "ko" and "position repetition".
As the wording currently stands, it is decidedly weak. And the permission of suicide is "of course" NOT self-evident either.

Why?
Multiple and repeated suicides by both players would thus be permissible, but would inevitably lead to an empty board again in the end (with White to move), thus negating the complete game so far (being equivalent to a Black pass in the very beginning of the original game, in principle stating that Black does not really want to play this game). And that is exactly what is not permissible according to the systematics of the other argumentation.

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Post #6 Posted: Sat May 13, 2023 7:06 am 
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John writes: "Chen Zuyuan, who is probably the world's foremost expert on go rules"

I say: As far as I know, Chen Zuyuan is the leading expert on old rules. I am the leading expert on new rules, as far as available to me in a language I understand.

Translation: consistency of "intersection" or "point" would help. When referring to vertices of the board, I suggest the former. Point, however, is a scoring unit.

Title: From the title, I expected very much more. The text is a suggestion of establishing rules from axioms.

Rules: Mathematically, rules can be written as definitions. In non-mathematical texts, we prefer rules as a legal text form, which is less precise than definitions.

Axioms: His so called axioms are a mixture of mathematical axioms and philosophical axioms, which can be derived from well chosen mathematical axioms. There is, however, not a unique set of such mathematical axioms. E.g., one can start with either lines or vertices (to be later perceived as intersections of lines, which can be seen as pairs of vertices having adjacency properties. Stones can be introduced as physical objects or as abstract objects of colouring. The choice has consequences on whether removed stones are understood as natural or unnatural for scoring. More fundamentally, why any scoring? There can also be a winning condition (the opponent has no more move, so that passes need not be introduced from few axioms). We know the consequence: pass fights. However, as long as axioms are not established as mathematical axioms and chosen well, discussion of all consequences in the text is somewhat premature, as is the intention implied by the title. We must distinguish necessary axioms from preferences.


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Post #7 Posted: Sat May 13, 2023 7:14 am 
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I didn't understand the part about the ko rule. Is it the (simple) ko rule or the superko rule which is supposedly a consequence of the axioms?

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Post #8 Posted: Sat May 13, 2023 7:21 am 
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Quote:
4. Compensation for first move [komi]
This is a convention, not a basic rule of weiqi. There is no provision for compensation in any of the current weiqi rules texts, but rather it is set out in the competition rules. Because, in principle, komi compensation is only one of the ways to balance the advantage of play first, the available options are whether to use this method, how much to compensate, and how to compensate.


Could the second sentence be better worded with 'playing first' or 'first play'/'first move' instead of 'play first'

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Post #9 Posted: Sat May 13, 2023 9:01 am 
Oza

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Quote:
Could the second sentence be better worded with 'playing first' or 'first play'/'first move' instead of 'play first'


Yes, thanks. My typo. Change made in OP above.

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Post #10 Posted: Sat May 13, 2023 9:50 am 
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Just returned from walking HONTE ...

My very serious recommendation would be that Mr Chen eliminates ANY reference to Japanese Rules from his text. (I firmly assume that he knows that e.g. "backfill of dead stones" is merely a mechanism to simplify the determination of the final result, but it has absolutely nothing to do with Japanese Rules in the strict sense.)
He also makes the serious mistake of confusing the current concrete formulation of the official Japanese Rules with ("Japanese-style") rules based on "territory".

To my mind, Mr Chen is trying to give the impression that the transition (historical in China) from "stone" to "area" is "natural", and so, in this respect, Chinese Rules are clearly superior to Japanese Rules.
However, it is NOT clear from the axioms why, in addition to STONES, parts of the ORIGINAL BOARD should also be taken into account when determining the final result. Or, the other way round ...

If we take the BOARD as the basis for determining the final result, it is not clear from the axioms why other than the ORIGINAL components of the BOARD (these are EMPTY points) are used for this.
To determine the final result on the basis of the BOARD, we can count the occasions when the ORIGINAL condition of the BOARD was either preserved (i.e. empty points where the opponent cannot live in -- aka "territory") or restored (i.e. taking opponent's stones of the board -- aka "prisoners").
You will not be very surprised that the ("Japanese-style") "territory scoring" results -- but with counting territory in seki.
(And we all know that its current official concrete design in Japan is not really ideal ...)

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Last edited by Cassandra on Sun May 14, 2023 2:46 am, edited 1 time in total.

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Post #11 Posted: Sun May 14, 2023 1:32 am 
Oza

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There was another typo by me: in the byline I put Chen Ziyuan instead of ZUYUAN. I have now changed this. I think I missed this because I am working on a book about the famous player Chen Zixian (currently being proofread).

Mr Chen does read the forum and this was his reaction: "It's good to see a lot of reaction, concern and discussion, but I will stay out of it. My intent in writing this article was that my research into the rules ends here."

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Post #12 Posted: Sun May 14, 2023 6:07 am 
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这里并不否认陈先生的《围棋规则演变史》一书是很好的集成资料,但不得不指出,陈某人关于中国古代围棋规则的“计活子”观点,并非是他的原创,原创作者的笔名是燕来(张云琪),见:https://zhuanlan.zhihu.com/p/536668093

This does not deny that Mr. Chen's book "History of the Evolution of WeiQi Rules" is a good integrated material, but it must be pointed out that Mr. Chen's "Counting Living Stones on the board" viewpoint on ancient Chinese WeiQi rules is not his original creation. The pen name of the original author is Yan Lai (real name: Zhang Yunqi), as shown in: https://zhuanlan.zhihu.com/p/536668093


燕来(张云琪)也不是所谓的“科学家”,而是民间研究者,他曾在中国围棋论坛受到过很多讥讽与嘲笑。

Yan Lai(Zhang Yunqi) is not a so-called "scientist", but a folk researcher. He has received a lot of ridicule and ridicule in the Chinese Go forum.

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Post #13 Posted: Sun May 14, 2023 9:05 am 
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Thank you John, I like this article a lot, and mostly agree with it.

jlt wrote:
I didn't understand the part about the ko rule. Is it the (simple) ko rule or the superko rule which is supposedly a consequence of the axioms?


As far as I understand, Chen Zuyuan suggests that the superko rule must be used. The key sentence is "Although extension [to superko] leads to a reduction in logical necessities, it is nevertheless still necessary to adopt this extension for multiple kos, and in particular for cases like moonshine life."

(I have added [to superko] for clarification)

Chen Zuyuan / John Fairbairn wrote:
It is forbidden to negate the opponent's moves, that is, to make the opponent face a game situation he has already faced. This means prohibiting repeated game positions.


The first sentence seems to me to define Situational superko, while the second sentence defines Positional superko.
As I understand the logic behind it, I would say that situational superko is more appropriate here.
John, are you sure that the original sentence is "This means prohibiting repeated game positions." and not "This means prohibiting repeated game situations / configurations."

Cassandra wrote:
To my mind, Mr Chen is trying to give the impression that the transition (historical in China) from "stone" to "area" is "natural", and so, in this respect, Chinese Rules are clearly superior to Japanese Rules.
However, it is NOT clear from the axioms why, in addition to STONES, parts of the ORIGINAL BOARD should also be taken into account when determining the final result. Or, the other way round ...

If we take the BOARD as the basis for determining the final result, it is not clear from the axioms why other than the ORIGINAL components of the BOARD (these are EMPTY points) are used for this.


I like this part of the text : "Axiom 1 is that weiqi consists of only the board and the stones, and there is no third element, so that the only two natural goals under axiomatic inference are these."

My understanding is that in go, there are a board and some stones. Therefore the two most natural goals are either to have the most stones, or to have the most of the board.

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Post #14 Posted: Sun May 14, 2023 9:42 am 
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OK, after reading again the text I see that the author considers that the ko rule "follows" from the axioms, and then generalizes to the superko rule. However I don't understand the expression "extension leads to a reduction in logical necessities". Perhaps I don't speak English well enough but I can't make sense of the expression "reduction in logical necessities".

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Post #15 Posted: Sun May 14, 2023 10:22 am 
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I understand it this way : it is logical to say that recapturing in a ko is forbidden, because it's exactly like if I was removing my opponent's move.
Thus forbidding this is a "logical necessity".

But if I do so indirectly, after a long cycle, the logic is less clear. I have not prevented my opponent to play.
We may apply a superko rule if we admit that closing a long cycle is equivalent to preventing the first move of the cycle. But that's less logic. Thus it is a "reduction in logical necessities".

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Post #16 Posted: Sun May 14, 2023 10:46 am 
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OK it makes sense, although it feels strange to use a mathematical vocabulary (axioms) and try to deduce everything from the axioms but then say that something is a logical consequence of the axioms, but maybe not...

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Post #17 Posted: Sun May 14, 2023 11:07 am 
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Long cycles result from decisions by BOTH players, not just by one player as in "simple" ko.

Consequently, there is NO logical justification for favouring the player who starts this cycle.

Strictly speaking, applying the rational for "superko", capturing stones is also neglecting opponent's moves, and therefore should also be forbidden.

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Post #18 Posted: Mon May 15, 2023 1:45 am 
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Ian: Mr Chen has said you can post his article on SL. He is familiar with the site and likes it.


This post by John Fairbairn was liked by: pgwq
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Post #19 Posted: Mon May 15, 2023 6:20 am 
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John Fairbairn wrote:
Ian: Mr Chen has said you can post his article on SL. He is familiar with the site and likes it.


感谢忠实的约翰一再转达陈先生的信息。

Thanks to Trusty John for relaying Mr. Chen's message.

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Post #20 Posted: Mon May 15, 2023 8:17 am 
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jlt wrote:
OK, after reading again the text I see that the author considers that the ko rule "follows" from the axioms, and then generalizes to the superko rule.
 
The philosophy behind the ko rule (not to negate opponent's move) seems questionable to me, for example it does not explain recapturing a ko after a pass. Which is truly necessary under territory scoring, but also preferable under area scoring to avoid defects like 1EyeFlaw. OC, recapturing after a pass does not completely negate a move - the seemingly repeated situation actually differs in "hot stone" - but such explanation assumes a ko rule to begin with.

Cassandra wrote:
However, it is NOT clear from the axioms why, in addition to STONES, parts of the ORIGINAL BOARD should also be taken into account when determining the final result. Or, the other way round ...
 
The anti-Japanese tone on the goal does feel arbitrary. Area scoring (w/o some button) is known to be inferior to territory scoring in accuracy (masking small endgame mistakes). Directly giving points for played stones involves a "random" B bias, from his 50% chance to play 1 stone more than W.

The goal or score is less biased if it only includes elements of skill - the utility / effects of played stones (territory, or parts of the board), and not played stones themselves. Go is the surrounding game, not the filling game.

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