At the moment, I do not enter an understanding of the section "Attempt of proof" yet because I first want to understand the presuppositions that you make. So currently I comment only on the section "Additional more restrictive assumption for X":

You use the term "life", but what is "life"? In the context of the WAGCmod proposition, "life" is undefined! So please define it!

What is a "basic set"? Define this term if you want to use it!

A J2003-alive string is not "dead"? Do you mean J2003-dead? What is the relevance for a proof? All we need to do is applying the J2003-alive definition. So we can circumvent consideration of J2003-dead. Therefore I do not understand why you mention it at all.

Likewise WAGC-alive and WAGC-dead. Why do you mention WAGC-dead at all?

Now to the core questions: Do you want to restrict the considered available statuses of string X? Do you want to require it to have "life" (for whichever definition you give for it) before you make some proof attempt? Why? This would substitute the "Proposition on WAGCmod" by a weaker proposition. The "Proposition on WAGCmod" allows X to be WAGC-alive or not WAGC-alive, and Chris Dams has proven it for both.

This is my final attempt, Robert.


The set SET(J2003-alive) contains all strings with the status property "J2003-alive".

The members of this set can be devided into 2 subsets:

• SUBSET1(J2003-alive) contains all strings that are also members of the set SET(2-eye-alive), which contains all strings with the status property "2-eye-alive".
• SUBSET2(J2003-alive) contains all strings that are not member of the set SET(2-eye-alive).

In a position, a string is WAGC-alive-in-seki if it is J2003-alive and not two-eye-alive.

The set SET(WAGC-alive-in-seki) contains all strings that

• are also members of the set SET(J2003-alive)
• but are no members of the set SET(2-eye-alive), which is SUBSET1(J2003-alive).

The complement of SUBSET1(J2003-alive) within the set SET(J2003-alive) is SUBSET2(J2003-alive).
The SET(WAGC-alive-in-seki) is identical to the set SUBSET2(J2003-alive).

In a position, a string is WAGC-alive if it is either two-eye-alive or WAGC-alive-in-seki.

The members of the set SET(WAGC-alive) can be devided into 2 subsets:

• SUBSET1(WAGC-alive) contains all strings that are also members of the set SET(2-eye-alive), which is identical to SUBSET1(J2003-alive).
• SUBSET2(WAGC-alive) contains all strings that are member of the set SET(WAGC-alive-in-seki), which is identical to SUBSET2(J2003-alive)

SET(WAGC-alive) is identical to SET(J2003-alive).

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

Let me bring a clearer structure into your proof attempt. I use your set-naming convention.


1) S := set of all strings in a position.

2) SET(2-eye-alive) := {ALL s in S | s is 2-eye-alive}.

3) SET(J2003-alive) := {ALL s in S | s is J2003-alive}.
4) SUBSET1(J2003-alive) := SET(J2003-alive) AND SET(2-eye-alive).
5) SUBSET2(J2003-alive) := SET(J2003-alive) \ SET(2-eye-alive).

From definitions (3)~(5),
6) SET(J2003-alive) = SUBSET1(J2003-alive) XOR SUBSET2(J2003-alive).

7) SET(WAGC-alive-in-seki) := SET(J2003-alive) \ SET(2-eye-alive).

From definitions (5)+(7),
8) SET(WAGC-alive-in-seki) = SUBSET2(J2003-alive).

9) SET(WAGC-alive) := SET(2-eye-alive) XOR SET(WAGC-alive-in-seki).

10) SET(2-eye-alive) =???= SUBSET1(J2003-alive).

Here is a gap. Neither (4) alone nor (4) and (5) together define SET(2-eye-alive). From these definitions alone, we cannot know yet whether (4) contains all elements of SET(2-eye-alive).


***


I do not understand ", which is SUBSET1(J2003-alive)" but think that it is superfluous. I also think that "The complement of SUBSET1(J2003-alive) within the set SET(J2003-alive) is SUBSET2(J2003-alive).", the term SUBSET1(WAGC-alive) and the term SUBSET2(WAGC-alive) are superfluous. "SUBSET2(WAGC-alive) contains all strings that are member of the set SET(WAGC-alive-in-seki), which is identical to SUBSET2(J2003-alive)" is superfluous because it is already proven as stated in (8).

Please allow that I write in prose further on. It seems that you understand better what I want to say when I use prose. Any symbolic language used by me seems not to be appropriate for this purpose.

You are right, but this is done outside your WAGCmod construction's text. Mind the preconditions !



In my understanding, using colloquial for prose,

The set SET(uncapturable) contains all strings, which

• do not consist of any stone that can be captured and
• (already have two eyes or (exclusively)
• from part of a Seki).

The set SET(capturable-1) contains all strings, which

• consist of at least one stone that can be captured and
• (form part of a Uttegaeshi or (exclusively)
• form part of a Nakade (inside the opponent's non-J2003-alive strings) or (exclusively)
• form part of a Seki and
• (consist of (at least) one stone, which successor will become part of a two-eyed-group or (exclusively)
• consist of no stone, which successor will become part of a two-eyed group (-> Seki))).

The set SET(capturable-2) contains all strings, which

• consist of at least one stone that can be captured and
• are no member of SET(capturable-1) and
• (consist of (at least) one stone, which successor will become part of a two-eyed group, may be as a real eye-point or (exclusively)
• form part of a Seki).

All of the blue coloured subsets are by their definition subsets of the set SET(2-eye-alive), too.

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

What do you mean by this?



The preconditions are: We have a position. We have an arbitrarily chosen but specific string.



We do not get anywhere with informal statements like "already have two eyes". This is what you wish for the uncapturable strings but maths, unlike religion, is not about wishes.



Maybe this list is helpful, maybe not.



A string never consists of no stone.



It does not help for an only moderately complicated proof though.

You refer to a definition of J2003-alive that is done outside WAGCmod.


"Precondition" is everything that is defined outside WAGCmod, e.g. "J2003-alive" with its three subsets.



Let me try an application of my "colloquial for prose" lists on J1989.



In my understanding, using colloquial for prose,

the set SET(J1989-alive) consists of two subsets.

The subset SUBSET1(J1989-alive) contains all strings, which

• do not consist of any stone that can be captured and
• (already have two eyes or (exclusively)
• form part of a Seki and cannot be captured, even if all found "dead" stones would have been removed from the evaluation's starting position).

The subset SUBSET2(J1989-alive) contains all strings, which

• are no member of SUBSET1(J1989-alive) and
• (form part of an Uttegaeshi or (exclusively)
• form part of a Nakade (inside the opponent's non-J1989-alive strings) or (exclusively)
• "capturing them would enable ..." with (at least) one of the stone(s) referred to being a successor of (at least) one of the strings primary points or (exclusively)
• capturing a one-stone-string would enable to turn the stone's primary point into an eye-point or (exclusively)
• "capturing them would" not "enable ..." (-> Seki) or (exclusively)
• "capturing them would enable ...", but none of the stone(s) referred to being a successor of (at least) one of the strings primary points and no captured one-stone-string being succeeded by an eye-point (-> Seki)).

All of the blue coloured subsets are by their definition subsets of the set SET(2-eye-alive), too.


Replacing "J2003" by "J1989" in your WAGCmod construction's text then would result in "WAGC-alive" = "J1989-alive".

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

For the sake of the proposition, the J2003 definition of J2003-alive is presumed. So although it is not on the same webpage, it is not "outside". Also the WAGC-alive definition is a presupposition.

J1989-alive differs from J2003-alive. E.g., the former does not know capturable-2. Therefore "WAGC-alive == J1989-alive" is a falsehood.

Anyway, your informal SET(J1989-alive) is by far too ambiguous.

For the former: see below.

For the latter: have you forgotten about the wide application range of "would enable ..." ?

The GIF below shows an extract of the PDF attached. It contains a further development of the table posted some days ago.



To compare rule sets you have to make yourself free from any dependency on any of the rule sets. In my opinion it seems best to divide the positions referred to by any of the rule sets into subsets. The parameters for dividing come from all of the rule sets. For example, if one new rule set would it make necessary to distinguish what had been one row only, you will have to implement a new parameter, resulting in some new rows. What is the same if I perhaps have forgotten anything.

The symbols give hints to following aspects concerning the comparison of rule sets:

>>> Ellipse:
If "2-eye-alive" is used as a "link" between the rule sets, it is a precondition that both rule sets use the same procedure during evaluation to get this status, if the aim of the comparison is to prove the rule sets as "identical" (or at least their "alive"-status). The procedure to get "2-eye-alive" has to be seen as a parameter of the rule set using it.
Otherwise the comparison makes no sense from the very beginning.

>>> Arrow:
There are rule sets (as J1989) that use a two-step procedure for status evaluation (of "alive"), in which step 2 overrides the results of step 1.
In contrary, both J2003 and Cassandra use a mere combination of exclusive results, there is no overriding.

>>> Rectangle:
It may be a sufficient condition that all subsets of "2-eye-alive" are also subsets of "rule set-alive" (in which case the WAGCmod construction automatically gives "rule set 1-alive" = "rule set 2-alive" - both are linked by the complement of "2-eye-alive"), but it is no required condition for two rule sets providing identical results for scoring.

This is due to the fact that the coexistence of both opponents' "alive" - "alive" strings gives the same concluding result as the coexistence of both opponents' "not alive"- "not alive" strings: None of the strings is taken off the board as prisoners.

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

You should forget about application of "enable" as quickly as possible! "enable" is a wrong model for modern Japanese professional tradition.

I do not understand the subcolumns of your 2-eye-alive column.

What does the back-slash before Seki-collapse mean?

Seki-collapse must not be a subcolumn of J1989 because it has nothing to do with the J1989 Rules. Better you introduce a major column "Official J1989 Commentary". Therefore your arrows become meaningless.

I do not understand "To compare rule sets you have to make yourself free from any dependency on any of the rule sets.".

I strongly criticise the selection of examples in your table. Instead of listing all standard shapes, you jump into the rarities.

It is possible to classify positions but I do not understand your classification scheme.

What do you mean by "If "2-eye-alive" is used as a "link" between the rule sets"?

Why must it be a precondition that both rule sets use the same procedure during evaluation to get this status?

Identical rulesets and identical alive-statuses due to their respective application are two very different things. Which do you mean?

Maybe you want to see sense in a comparison only if "The procedure to get "2-eye-alive" has to be seen as a parameter of the rule set using it.". I do not have such a restrictive view.

As a side-effect, I do not understand Ellipse.

All subsets of 2-eye-alive? Do you even know what you are speaking of?

What is "rule set-alive"? Do you use "rule set" as a parameter here? Which rulesets do you accept, which not?

As a side-effect, I do not understand Rectangle.

One may not say in general "This is due to the fact that the coexistence of both opponents' "alive" - "alive" strings gives the same concluding result as the coexistence of both opponents' "not alive"- "not alive" strings: None of the strings is taken off the board as prisoners." because alive, underlike dead, allows for the definition option of having territory.

"would enable ..." is part of J1989, so one cannot forget about it when discussing J1989. It's another kettle of fish to find / have a well-defined procedure instead, which gives the same (may be wished) results.

I do not understand the subcolumns of your 2-eye-alive column.
"J2003 Ko-Pass" means using your "hypothetical Ko-Pass" during evaluation (this procedure had been referred to be my as "very one-sided discrimination").
"J1989 Ko-Pass" means using the "Pass for every special Ko" during evaluation.
"none; all of" means using no Ko-Pass, i.e. using the same Ko-rule as during "Play". Combined with referring to "all" primary points as a successor of a captured string (J2003 and J1989 refer to "at least one").

What does the back-slash before Seki-collapse mean?
Difference. "X" in column "uncapturable" minus "X" in column "\ Seki-collapse" results in a concluding "empty" in column "uncapturable".

Seki-collapse must not be a subcolumn of J1989 because it has nothing to do with the J1989 Rules. Better you introduce a major column "Official J1989 Commentary". Therefore your arrows become meaningless.
See above, referring to "enable". I suppose that the commentary includes what are the wished results of the rule set.

I do not understand "To compare rule sets you have to make yourself free from any dependency on any of the rule sets.".
Lady Justice holds the weighting scale in her left hand. She is no part of it.

I strongly criticise the selection of examples in your table. Instead of listing all standard shapes, you jump into the rarities.
Sorry, Robert. The cat is put among the pigeons by "rarities", not by "standard shapes". I suppose that the examples are well-known, so extra diagrams are uncalled-for. If you have better examples, let me know.

It is possible to classify positions but I do not understand your classification scheme.
Motivation:

• stones per string: Evaluation of single stones is not identical to evaluation of strings (with at least 2 stones).
• permanent: Evaluation of stones / strings that cannot be captured is not identical to evaluation of stones / strings that can becaptured from the primary position.
• taboo-points: Evaluation depends on the number of taboo-points, which a stone / string under evaluation has already respectively can get.
• successor: Evaluation depends on the type of the successor of the stone / string under evaluation. Successor can be found at
  all primary point(s) / not all, but at least one primary point (of a string) / at least on one point inside the "local area" / at least one primary point in 50% (i.e. caught in a cycle) / nowhere.
• taboo-points of successor: Evaluation depends on the number of taboo-points, the successor of a string under evaluation can get.
• after removal of dead stones: Evaluate all stones / strings. Take all "dead" stones of the board from the primary position. Evaluate strings again, whose first evaluation had been caught in a cycle. Is my simulation of "collapse of the Seki".

What do you mean by "If "2-eye-alive" is used as a "link" between the rule sets"?
Within the WAGCmod construction, "2-eye-alive" is used as bridging element between two rule sets. So you have to be sure that the bridge is the same (i.e. gives identical results) if seen from either end.
You use the term "2-eye-alive" only. But you generally have to distinguish between "2-eye-alive, as found with rule set 1 procedure" and "2-eye-alive, as found with rule set 2 procedure". May be that the procedures are the same, or their result, but this cannot be taken as self-evident.

Why must it be a precondition that both rule sets use the same procedure during evaluation to get this status?
"Same procedure" may be a bit misleading in the wording. As explained above, the procedures of both rule sets must provide identical results.
If you assess a "bridge" as the same, when referring to two photos taken from one end only, you cannot be sure that the island, which can be guessed in the dust of the horizon, will be the same.

Identical rulesets and identical alive-statuses due to their respective application are two very different things. Which do you mean?
The status "alive" is not the primary point of interest. It's just an indermediate step on the way to "Score", a means to the end. For scoring you need one property / status (given to a stone / string) and its complement. "Territory" is decided by the interaction of the strings' properties.

Rule sets that claim to be "identical" must provide the same score.
"Identical alive-statuses ..." might be sufficient, but is not necessary.

Maybe you want to see sense in a comparison only if "The procedure to get "2-eye-alive" has to be seen as a parameter of the rule set using it.". I do not have such a restrictive view.
Perhaps you should have. See the GIF.



As a side-effect, I do not understand Ellipse.
Black's middle string in #16 can be forced into a 2-eye-formation with applying the J2003 hypothetical Ko-Pass only. Using J1989, the Double-Ko-Seki runs into a "Pass for this special Ko" cycle. Using Cassandra, there will be a Triple-Ko cycle.

All subsets of 2-eye-alive? Do you even know what you are speaking of?
What do you mean ?

What is "rule set-alive"? Do you use "rule set" as a parameter here? Which rulesets do you accept, which not?
"rule set" = "J2003", "J1989", ...

As a side-effect, I do not understand Rectangle.
Black's 4-stone string in #1 is "capturable-1" in J2003, so "alive", "would enable ..." in J1989, so "alive", but "removable" in Cassandra.
The fate of Black's string is already apparent in J2003 and J1989 (i.e. will not end as prisoner), in Cassandra it is not (will become finally apparent in "Score"). What is due to "alive" being "active", "not alive" / "removable" being "passive", what I mentioned in a previous posting. The concluding result "Seki" will be the same.

One may not say in general "This is due to the fact that the coexistence of both opponents' "alive" - "alive" strings gives the same concluding result as the coexistence of both opponents' "not alive"- "not alive" strings: None of the strings is taken off the board as prisoners." because alive, underlike dead, allows for the definition option of having territory.
Your reasoning might be true, but is irrelevant in the context we are discussing here.

Remember that my statement refers to the coexistence of strings, which cannot become 2-eyed as a whole (including successor). All the rule sets in my table act in a different manner for the concluding declaration of territory. A "not alive" - "not alive" coexistence is only possible in what is known as Seki. The question whether to count territory in Seki (might be possible in an "alive" - "alive" coexistence) has to be answered in "Count", not in "Evaluate".

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

"enable": If you want to show off a weak understanding of J1989 as it may have existed until ca. 1996, then you can talk about enable without thinking. Otherwise you should respect the research of the last 14 years and its results: The original text is too ambiguous; my pure text interpretation is clearer. The original text's intention as expressed in my pure text interpretation leads to results that contradict the rules authors' intended effect of their rules on examples. Therefore the right thing to do, except for describing historical nonsense in detail, is to substitute J1989 enable by J2003's local-1/2/3 and capturable-1/2/3.

When you want to use J2003 Ko-Pass for 2-eye-alive, you have to state more: a) For determination of whether a string is 2-eye-alive? b) All conditions related to J2003 Ko-Pass? c) How do you want to handle long cycles?

J1989 Ko-Pass for 2-eye-alive: You have to explain how you want to fill the flaws related the J1989's ko-pass rule. Besides see above.

"none" for 2-eye-alive: See above. You also need to explain whether some or all history-bans carry over from Play to Evaluate.

While also I suppose that the commentary includes what are the wished results of the ruleset, I also think that "collapse of the seki" is lazy and careless commenting rather than anything to be taken too serious.

Still I do not understand "To compare rule sets you have to make yourself free from any dependency on any of the rule sets.".

Of course, I have better examples. I also think that I have already shown them to you at dgob.de. - If all rarities are explained but later one finds that one standard example is not explained correctly, then all the effort was in vain.

Your functional classification attempt is somewhat new but I think you should also use more usual classification schemes (like identifying snapback or double ko).

The WAGCmod link is not so much 2-eye-alive but the equivalence statement WAGC-alive equals J2003-alive.

Due to the established proof by Dams, we can be sure that the particular two different procedures result in the same.

With my procedure comparison remark, I have wanted to say that one can compare also non-identical rulesets!:)

All subsets of a set of N strings: Each subset can consist of any selection of at least 0 and at most N different strings of the set. There are VERY MANY possible subsets...

"rule set" = "J2003", "J1989", ... does not help me; what is "..." and where is its end?! Since you appear to use it as a parameter, we need to know exactly HOW flexible you want to be.

So why do you mark Ellipse and Rectangle in your table?

"All the rule sets in my table act in a different manner for the concluding declaration of territory.": So you think, but please define for each column exactly the entire rulesets! Only then can we try to find out whether your table is correct.

Hi, Robert,

please let me take some time to edit my table to better meet your requirements before answering your previous posting.

I think I have got a feeling what one (of perhaps several) cause for some misunderstanding in our discussion is: From time to time I probably trigger some of your "reflexes" accidentally.

May be that you will understand even "Lady Justice's view" better thereafter.

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

- When you claim to provide a mathematical proof, then correct mathematics is what I ask from you.
- You continue discussion that has evolved from the proof topic, so you should expect me to keep mathematical standards high.
- In discussions on rules I tend to be accurate regardless of whether it is a about maths or informal discussion. I dislike discussions on the level of inaccurate mud proceeding as if they were knowledgable. Therefore, whenever you enjoy imprecision, expect my strong methodical opposition. Discussion shall not drop back to the mediaeval age of missing education. Educate yourself better, use acceptable terms and careful reasoning and presentation of contents (contents, not fake contents however colourful or well formatted) and you do not need to fear a great number of questions for clarity on what you write.
- Do not let the reader guess what you write but write so that it is self-explaining or else does contain all the necessary explanations clearly.

Dear Robert, I hope you do not mean "correct" in terms of the accuracy Chris' "proof" has.

As you suggested, I studied Chris' "proof". Result is as is had supposed before. Nothing new to report. Chris had forgotten about the preconditions and that he is working in two rule sets, too.

To make it apperent to you I have added to Chris' text what I refer to as "parameters". I.e. to what rule set a used term does belong to.

--------------------------------------------------------


From: Chris Dams <chr...@wn5.nospamplease.nl>
Newsgroups: rec.games.go
Subject: Re: Model for the World Amateur Go Championship Rules
Date: Wed, 12 Apr 2006 20:56:06 +0000 (UTC)
Message-ID: <e1jph6$13q$1@wnnews.sci.kun.nl>
References: <p67q32p0mo82bm1c2h5gpolhs93hncpu52@4ax.com> <e1j9jn$pla$1@wnnews.sci.kun.nl> <27gq32p0vhdvn6j0e5o1fneud7erkaas4a@4ax.com>


Dear Robert,

Robert Jasiek <jas...@snafu.de> writes:

>On Wed, 12 Apr 2006 16:24:23 +0000 (UTC), Chris Dams
><chr...@wn5.nospamplease.nl> wrote:
>>> In a position, a string of a player is _two-eye-alive_ @ WAGCmod if the
>>>opponent cannot force @ WAGCmod no intersection of the string with a
>>>two-eye-formation on.
>>
>>> _J2003-alive_@ WAGCmod is defined like in J2003 as either uncapturable @ J2003,
>>>capturable-1 @ J2003, or capturable-2 @ J2003.
>>
>>> In a position, a string is _WAGC-alive-in-seki_ @ WAGCmod if it is
>>>J2003-alive @ WAGCmod and not two-eye-alive @ WAGCmod.
>>
>>> In a position, a string is _WAGC-alive_ @ WAGCmod if it is either
>>>two-eye-alive @ WAGCmod or WAGC-alive-in-seki @ WAGCmod.
>>
>>From these definitions it follows that WAGC-alive is identical to
>>J2003-alive. This does not follow from these definitions alone !

>I doubt this. Me too !If you claim it, then please present a formal proof!

I have to admit that I, at first, interpreted "either" as "or" in the
definition of WAGC-alive. However, I think the identity of J2003-alive
and WAGC-alive is still provable. Proof is given below.

Let us denote

> In a position, a string is _WAGC-alive-in-seki_ @ WAGCmod if it is
> J2003-alive @ WAGCmod and not two-eye-alive @ WAGCmod.

as

WAGC-alive-in-seki @ WAGCmod == J2003-alive @ WAGCmod && (!two-eye-alive @ WAGCmod)

In the same notation we also have from

> In a position, a string is _WAGC-alive_ @ WAGCmod if it is either
> two-eye-alive @ WAGCmod or WAGC-alive-in-seki @ WAGCmod.

WAGC-alive @ WAGCmod == two-eye-alive @ WAGCmod ^^ WAGC-alive-in-seki @ WAGCmod.

Substituting the former into the latter expression, we find

WAGC-alive @ WAGCmod == two-eye-alive @ WAGCmod ^^ (J2003-alive @ WAGCmod && (!two-eye-alive @ WAGCmod)).

In propositional calculus this reduces to

WAGC-alive @ WAGCmod == J2003-alive @ WAGCmod || two-eye-alive @ WAGCmod.

If we now also have the implication two-eye-alive @ WAGCmod -> J2003-alive @ WAGCmod if follows
that

WAGC-alive @ WAGCmod == J2003-alive @ WAGCmod.

For the implication two-eye-alive @ WAGCmod->J2003-alive @ WAGCmod, imagine that a string is
two-eye-alive @ supposed J2003. The string can either be uncapturable @ J2003 or not uncapturable @ J2003.
(1) The string is uncapturable @ J2003 -> It is J2003-alive @ J2003
(2) It is not uncapturable @ J2003 -> The string is either capturable-1 @ J2003 or
not capturable-1 @ J2003
(2a) It is capturable-1 @ J2003 -> It is J2003-alive @ J2003
(2b) It is not capturable-1 @ J2003 -> Because the string is two-eye-alive @ J2003 there
is in every hypothetical-strategy @ J2003 of its opponent a
hypothetical-sequence @ J2003 in which
we reach a two-eye-formation that includes one of its intersections.
For every
hypothetical-strategy @ J2003 H of the opponent, we choose a
hypothetical-sequence @ J2003 S(H) in it
where the oponent reaches a two-eye-formation and subsequently only
passes. Because the two-eye-formation cannot be capture by only moves
of its opponent, it consists of permanent stones. In S(H) the
two-eye-formation that is formed on the captured string has either a
stone on local-1 @ J2003 of the string or it does not have a stone on local-1 @ J2003
of the string
(2b1) If it has a stone on local-1 @ J2003 of the string, it is also on
local-2 @ J2003.
(2b2) If it does not have a stone on local-1 @ J2003 of the string,
then local-1 @ J2003 of the string consists of the one or both
of the empty intersections of the two-eye-formation. Actually,
it consists of one of the intersections since if it would consist
of both, these would have to be adjacent to each other which
contradicts the definition of a two-eye-formation. So, local-1 @ J2003
of the string consists of one intersection and during S(H) it
becomes one of the the empty points of a two-eye-formation. This
implies that this two-eye formation includes strings that occupy
the intersections adjacent to local-1 @ J2003. Because local-1 @ J2003 consists of
one intersections these adjacent intersections where empty or
occupied by opposing stones. Hence, these intersections belong to
local-2 @ J2003 of the string.
In both (2b1) and (2b2) we see that the two-eye-formation that is
formed in S(H) has permanent stones on local-2 @ J2003 of the string. Hence,
if every hypothetical-strategy @ J2003 of the opponent of the string there
is a hypothetical-sequence @ J2003 where a permanent-stone is played on
local-2 @ J2003. Hence, the opponent cannot force both caputre of the string
and no local-2 @ J2003 permanent stone. Hence, the string is capturable-2 @ J2003.
Hence, it is J2003-alive @ J2003.
Hence, under the assumption that the string is two-eye-alive @ supposed J2003, we find that
it is J2003-alive @ J2003. QED.

Best,
Chris

-----------------------------------------------------


As I had written in several postings before, there is an immediate and direct reference to the J2003 evaluation procedure.

There is nothing to "prove" that two-eye-alive @ J2003 is a subset of J2003 @ J2003, because this is the result of a circular reference within their definitions.
Both two-eye-alive and (at least) the J2003 subsets capturable-1 and capturable-2 are a function of "two-eye-formation", and - not to be forgotten !!! - of the "forcing" procedure applied. In each case the variables used as parameters within the function are identical. So it's clear cut that the two resulting variables are not independent from another !

Should - what cannot be taken from your WAGCmod text - the WAGCmod "forcing" procedure be identical to the J2003 "forcing" procedure, then there is nothing to prove within the relationship of the two rule sets. Your construction uses the complement of "two-eye-alive" as bridging element; the result is what I have written before, just give the child another name.

Should the WAGCmod "forcing" procedure be not identical to the J2003 one, then it will be quite obvious that there is something missing in Chris' posting that has not been proven yet:
Does two-eye-alive @ WAGCmod as a subset of WAGCmod-alive equal two-eye-alive @ J2003 ?


My classification's table, together with a new set of examples, will last a bit longer. It's not as simple as the refutation of a so called "proof".

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

Do not expect me to teach you mathematical proving. Your request for explicitly stated presuppositions is ok though. So I clarify below.


These terms are defined in the following extract of J2003/35a:

position, string(s), player, (opponent, by "player" and standard context), intersection(s), adjacent, force

"Axioms [...] A player can force something if there is at least one hypothetical-strategy of his so that each compatible hypothetical-sequence fulfils that something."


These terms are defined in the following extract of J2003/35a:

uncapturable, capturable-1, capturable-2

"Axioms [...] A player's final-string is capturable-2 if
* it is neither uncapturable nor capturable-1 and
* the opponent cannot - with the same hypothetical-strategy - force both capture of the string's stones and no local-2 permanent-stone of the player."


J2003's alive is defined by the aforementioned extrat and this citation:

"A final-string is alive if it is either
* uncapturable,
* capturable-1, or
* capturable-2."


The WAGCmod page simply gives this an alias: J2003-alive.


"two-eye-formation" and "two-eye-alive", using more basic terms, are defined on WAGCmod.


"WAGC-alive-in-seki" is defined on WAGCmod, uses "two-eye-alive", which is defined on WAGCmod, and uses J2003-alive, which is defined in J2003/35a under a different alias. See above.


"WAGC-alive" is defined on WAGCmod and uses "two-eye-alive" and "WAGC-alive-in-seki", see above.


Furthermore, as explained earlier, "WAGC-alive equals J2003-alive" relies on a particular, presumed position and a arbitrary, fixed string.


***


The contruction is not a vicious circle but a hierarchical structure of definitions. Only after the proof was made and by means of it, we can apply it in either direction of implication.

In your @-list, you make a few mistakes. Among them is this misleading statement: "two-eye-alive @ supposed J2003".

Your citation of "I doubt this." is put in a misleading context. I doubted before seeing Chris's proof, shortly afterwards I approved it.

Of course, I do mean "correct", but most obvious presuppositions like earlier definitions are not stated usually. Efficient annotation does not mean incorrectness.

Your statement "Both two-eye-alive and (at least) the J2003 subsets capturable-1 and capturable-2 are a function of "two-eye-formation"" is nonsense. To start with, in mathematics, a "function" is something unrelated.

Since your having such great difficulties with maths proofs, let me suggest to you that you use a helping tool: Make two printed copies of J2003/35a. Derive J2003-alive from one copy and WAGC-alive from the other copy. Only then state "WAGC-alive equals J2003-alive". Maybe this inefficient working helps you.

As to your statement "then there is nothing to prove within the relationship of the two rule sets": If you want to prove that you are not a mathematician, then always state that all is trivial;)

You ask: Does two-eye-alive @ WAGCmod as a subset of WAGCmod-alive equal two-eye-alive @ J2003?

The question is badly posed because two-eye-alive is not defined @ J2003. However, your question implies a better formulated

Proposition:

Given a string in a position, the set-intersection of two-eye-alive and WAGC-alive equals the set-intersection of two-eye-alive and J2003-alive.

Proof:

This follows immediately from the definitions and Chris Dams's proof. QED.

Note:

If you don't see it immediately for J2003-alive=>WAGC-alive, write down these two assumption cases: 1) The string is J2003-alive and two-eye-alive. 2) The string is J2003-alive and not two-eye-alive. Prove it for each case.

I have no problems with mathematics, Robert. In my old days, I had been school primus in this subject.

Let me recapitulate, what the problem is:

In a position, a string of a player is two-eye-alive if the opponent cannot force no intersection of the string with a two-eye-formation on.

from WAGCmod contains two parameters:

• "force"
• "two-eye-formation"

• A player's final-string is uncapturable if the opponent cannot force capture of its stones.
• A player's final-string is capturable-1 if it is not uncapturable and the opponent cannot - with the same hypothetical-strategy - force both capture of the string's stones and no local-1 permanent-stone of the player.
• A player's final-string is capturable-2 if it is neither uncapturable nor capturable-1 and the opponent cannot - with the same hypothetical-strategy - force both capture of the string's stones and no local-2 permanent-stone of the player.

from J2003 contains at least the parameters

• "force"
• "two-eye-formation" (implicit within "uncapturable", "capturable-1", and "capturable-2")

None of the "life-subsets" of J2003 can - following their definition - contain anything else than "two-eye-formation" or "Seki".

It is not necessary to explicitely know, which classes of positions will become subsets of "two-eye-alive", "uncapturable", "capturable-1", or "capturable-2", because this is decided by "force".

When you want to compare the two rule sets, it is mandatory that you do not use two different versions of "force", i.e. two different evaluation procedures. Because you use one and only one "force" within both rule sets.

But using the identical procedure for "force" will give identical results for "two-eye-formation".

Because none of the rule sets can contain anything else than "two-eye-formation" or "Seki" within their "alive"-supersets, it goes without saying that the complements, which are "Seki", must be identical, too.

Might be that you will call this a "proof", too, but it is more than trivial.

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

Please find attached the update of my comparison table (= comparison.pdf), now with a supplement of positions' classification (= classification.pdf), in which you will find the examples referred to in the table.

You will recognise that the cat is not put among the "standard" pigeons. The rule sets are mainly distinguished by their specification for "Ko-Pass" during evaluation, what affects the "exotic" positions only.

Please let me know if I should include further columns, row, or examples or might have overlooked anything related to your J2003.

-----------------
2010-06-11 attachments updated >>> version number 3

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

What you call "parameter" is in fact "term defined earlier".

The term alive as defined in J2003 does NOT rely on defining "two-eye-formation"!!!!!!!!!!!!!!!!!!!!!!!!!

You say: "None of the "life-subsets" of J2003 can - following their definition - contain anything else than "two-eye-formation" or "Seki"." Do you make this statement before, during or after Chris's proof? In which context do you assume "seki" to be defined?

Your post is not a proof at all for in particular the above unclarities.