Life In 19x19

Given a final-position, a string in it, and the definitions of local-2\1, capturable-2\1, J2003-alive-2\1, WAGC-alive-in-seki-2\1, WAGC-alive-2/1.

Proposition:

The string is WAGC-alive-2\1 equals the string is J2003-alive-2\1.

Proof:

Part 1 of the proof is analogue to part 1 of Chris Dams's proof.

Part 2 of the proof is as follows:

For the implication two-eye-alive -> J2003-alive-2\1, imagine that a string is two-eye-alive. Either the opponent cannot force capture of the string or he can force capture of the string.

(1) The opponent cannot force capture of the string -> It is uncapturable. -> It is J2003-alive-2\1.

(2) The opponent can force capture of the string. -> Because the string is two-eye-alive there is in every hypothetical-strategy of its opponent a hypothetical-sequence in which we reach the player's two-eye-formation that includes one of its intersections. For every hypothetical-strategy H of the opponent, we choose a hypothetical-sequence S(H) in it where the player reaches such a two-eye-formation and subsequently only passes. Because the two-eye-formation cannot be captured by only moves of its opponent, it consists of permanent-stones. -> In S(H) the two-eye-formation that is formed on at least one intersection of the captured string has either a stone on local-1 of the string or it does not have a stone on local-1 of the string.

(2a) The two-eye-formation has a stone on local-1 of the string -> the string is capturable-1 -> It is J2003-alive-2\1.

(2b) The two-eye-formation has no stone on local-1 of the string -> Local-1 of the string consists of 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 of the string consists of one intersection and during S(H) it becomes one of 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. Because local-1 consists of one intersection these adjacent intersections were empty or occupied by opposing stones. Hence, these intersections belong to local-2\1 of the string. We see that the two-eye-formation that is formed in S(H) has permanent-stones on local-2\1 of the string. Hence, in every hypothetical-strategy of the opponent of the string there is a hypothetical-sequence where a permanent-stone is played on local-2\1. Hence, the opponent cannot force both capture of the string and no local-2\1 permanent-stone. Hence, the string is capturable-2\1. Hence, it is J2003-alive-2\1.

Hence, under the assumption that the string is two-eye-alive, we find that it is J2003-alive-2\1. QED.

Some minor suggestions:



(2) The opponent can force capture of the string. -> Because the string is two-eye-alive there is in every hypothetical-strategy of its opponent a hypothetical-sequence in which we reach the player's two-eye-formation that includes one of its intersections. For every hypothetical-strategy H of the opponent, we choose a hypothetical-sequence S(H) in it where the player reaches such a two-eye-formation and subsequently only passes. Because the two-eye-formation cannot be captured by only moves of its opponent, it consists of permanent-stones. -> In S(H) the two-eye-formation that is formed on at least one intersection of the captured string has either at least one stone on local-1 of the string or it does not have a stone on local-1 of the string.

(2a) The two-eye-formation has at least one stone stone on local-1 of the string -> the string is capturable-1 -> It is J2003-alive-2\1.

It is certainly possible to make lots of further minor improvements in annotation style. E.g., one could add a few "of the player" phrases.

More importantly, I would like to see studies of the differences and relations between capturable-2 and capturable-2\1. Can we classify the set of different positions?

RobertJasiek wrote:It is certainly possible to make lots of further minor improvements in annotation style. E.g., one could add a few "of the player" phrases.

More importantly, I would like to see studies of the differences and relations between capturable-2 and capturable-2\1. Can we classify the set of different positions?

What do you mean with "different positions" ?

Capturable-2 can be seen as extension of capturable-2\1 or capturable-2\1 as part of capturable-2.

Cassandra wrote:What do you mean with "different positions" ?

a) "different positions" in its strict sense.
b) You suggest some classification scheme and count for us the number of classes.

Capturable-2 can be seen as extension of capturable-2\1 or capturable-2\1 as part of capturable-2.

Do you mean this?

Proposition:

The set of capturable-2\1 strings is a subset of the set of capturable-2 strings.

Proof:

"Trivial."

Remark:

Nice. Now where are example positions?:)

RobertJasiek wrote:More importantly, I would like to see studies of the differences and relations between capturable-2 and capturable-2\1. Can we classify the set of different positions?

Your wish to have "classes" defined for the sake of camparison of capturable-2 and capturable-2\1 is a result of the somewhat ineffective, somewhat overdefined multi-step procedure to identify "life" within your J2003. And what is even worse: the steps cannot be seen independent from each other.



Step 1: Identify uncapturable strings.
Is applied to: all strings.
"Identify" depends on: "force".

A player's final-string is uncapturable if the opponent cannot force capture of its stones.
A permanent-stone is a stone that is played during a hypothetical-sequence and then not removed during the rest of the hypothetical-sequence.
For a final-string, local-1 is all the string's intersections.


Step 2: Identify capturable-1 strings.
Is applied to: all not uncapturable strings.
"Applied to" depends on: "uncapturable"
"Identify" depends on: "force", "local-1".

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.

Step 3: Identify local-2(\1).
Is applied to: all not uncapturable strings.
"Applied to" depends on: "uncapturable"
"Identify" depends on: "force", "uncapturable", "capturable-1".

(3a) For a player's final-string, local-2 is local-1 and, recursively, any adjacent intersection without a stone of a string that is of the player and either uncapturable or capturable-1.
(3b) For a player's final string, local-2\1 is any intersection of the string's local-2, which does not belong to the string's local-1.

Step 3 and step 4 are NOT restricted to those strings without status so far, when using local-2.
There is no need to include strings with the known status "capturable-1", what is done by including local-1 in local-2.


Step 4: Identify capturable-2(\1).
Is applied to: all not uncapturable strings (with local-2\1: which are not capturable-1).
"Applied to" depends on: "uncapturable"
"Identify" depends on: "force", "uncapturable", "capturable-1", "local-1" (only if referring to "local-2"), "local-2\1".

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.

A player's final-string is capturable-2\1 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\1 permanent-stone of the player.

Your fear is:
There will be a "forced" hypothetical strategy in step 4, which places a permanent stone in local-1 of a string that had not been identified as "capturable-1" in step 2.

But this permanent stone in local-1 fulfils the condition for "capturable-1" in step 2, what apparently both players overlooked when busy with step 2.

So - late, but not too late - the string has been identified as "capturable-1".

Thereafter step 3 und step 4 (both dependent on the result of step 2) must be done again.

Your very special chain will not take part in the second course !



What is needed to identify "capturable-2" is nothing more than local-2\1.

Cassandra wrote:the somewhat ineffective, somewhat overdefined multi-step procedure to identify "life" within your J2003.

Sure. It is necessary for J2003's purposes. For other purposes, one can approach things much more easily. E.g., "alive" is "being on the board".

Your fear is:
There will be a "forced" hypothetical strategy in step 4, which places a permanent stone in local-1 of a string that had not been identified as "capturable-1" in step 2.

But this permanent stone in local-1 fulfils the condition for "capturable-1" in step 2, what apparently both players overlooked when busy with step 2.

So - late, but not too late - the string has been identified as "capturable-1".

It is not defined that way. My "fear" persists.

Thereafter step 3 und step 4 (both dependent on the result of step 2) must be done again.

Your very special chain will not take part in the second course !

If you can prove that:)

What is needed to identify "capturable-2" is nothing more than local-2\1.

Interesting claim. Can you prove it?

Robert, the need to have local-1 included in local-2 follows from this tiny flaw in your construction, which is not free from feedback loops.

But no further information of the string's status can be gained that is not known already by using local-2\1. This is the decisive point: "no further information".

It is not decisive that the opponent can force "no permanent stone in local 1" when the aim is to answer the question "Is the string capturable-1 ?", and can force either "at least one permanent stone in local-1" or "at least one permanent stone in local-2\1" when the aim is to answer the question "Is the string capturable-2 ?". This is true even for White's single stone in 1989 Nihon Kiin Life & Death example #1. But the "ordinary" use of "force" would end in "local-2\1", an end in "local-1" is just a prestidigitation by the opponent.

When the opponent does not have this choice, there is no possibility to have a permanent stone in local-1 (Nihon Kiin #4, for example).

Cassandra wrote:the need to have local-1 included in local-2 follows from this tiny flaw in your construction, which is not free from feedback loops.

It is not a flaw in my construction and there is no feedback loop. Regardless, as you have pointed out, it is (for other purposes) also possible to use different constructions like such with fewer levels and greater segregation of the local-1.

But no further information of the string's status can be gained that is not known already by using local-2\1.

Prove it! (BTW, "information" is a very mighty word. You might try to start with much more modest claims. Even each such will be very difficult to prove. E.g., "Capturable-2 equals capturable-2\1." or "Given a hypothetical-strategy for capturable-2\1, the hypothetical-strategy together with certain further left-parts would not [do something new] when used for capturable-2".)

It is not decisive that the opponent can force "no permanent stone in local 1" when the aim is to answer the question "Is the string capturable-1 ?", and can force either "at least one permanent stone in local-1" or "at least one permanent stone in local-2\1" when the aim is to answer the question "Is the string capturable-2 ?". This is true even for White's single stone in 1989 Nihon Kiin Life & Death example #1. But the "ordinary" use of "force" would end in "local-2\1", an end in "local-1" is just a prestidigitation by the opponent. When the opponent does not have this choice, there is no possibility to have a permanent stone in local-1 (Nihon Kiin #4, for example).

As long as we do not know much better the relation between capturable-2 and capturable-2\1, we should be careful with such statements.

RobertJasiek wrote:

Cassandra wrote:the need to have local-1 included in local-2 follows from this tiny flaw in your construction, which is not free from feedback loops.

It is not a flaw in my construction and there is no feedback loop. Regardless, as you have pointed out, it is (for other purposes) also possible to use different constructions like such with fewer levels and greater segregation of the local-1.

But no further information of the string's status can be gained that is not known already by using local-2\1.

Prove it! (BTW, "information" is a very mighty word. You might try to start with much more modest claims. Even each such will be very difficult to prove. E.g., "Capturable-2 equals capturable-2\1." or "Given a hypothetical-strategy for capturable-2\1, the hypothetical-strategy together with certain further left-parts would not [do something new] when used for capturable-2".)

It is not decisive that the opponent can force "no permanent stone in local 1" when the aim is to answer the question "Is the string capturable-1 ?", and can force either "at least one permanent stone in local-1" or "at least one permanent stone in local-2\1" when the aim is to answer the question "Is the string capturable-2 ?". This is true even for White's single stone in 1989 Nihon Kiin Life & Death example #1. But the "ordinary" use of "force" would end in "local-2\1", an end in "local-1" is just a prestidigitation by the opponent. When the opponent does not have this choice, there is no possibility to have a permanent stone in local-1 (Nihon Kiin #4, for example).

As long as we do not know much better the relation between capturable-2 and capturable-2\1, we should be careful with such statements.

Let me try to explain my current view on your J2003-world, Robert. Perhaps this will give you a better understanding of what lies behind my quoted statements above.

Starting point of our journey will be your fear that the opponent might misuse the definition of "capturable-2\1".

When player and opponent cooperate, they are able to create each of the following results for the status evaluation of a single string:

(A) The string will remain on the board.
(B) The string will not remain on the board.
(B1) A permanent stone of the player becomes established on every point of the primary string.
(B2) A permanent stone of the player becomes established on at least one point of the primary string.
(B3) There is no point of the primary string, where a permanent stone of the player becomes established.
(B31) At least one permanent stone of the player becomes established in a certain area of the board.
(B32) There in no point in a certain area of the board, where a permantent stone of the player becomes established.

But this sort of cooperation is clearly not meant when using "force".

Instead the instructions for the opponent are as follows:

• Use all your capabilities to gain (B32).
• If this is not possible, use all your capabilities to gain (B31).
• If this is not possible, use all your capabilities to gain (B2).
• (B1 is not relevant for J2003.)
• If this is not possible, the default result is (A).

The instrctions for the player read:

• Use all your capabilities to gain (A).
• (B1 is not relevant for J2003.)
• If this is not possible, use all your capabilities to gain (B2).
• If this is not possible, use all your capabilities to gain (B31).
• If this is not possible, the default result is (B32).

The analysis of the set of "possible" evaluation sequences will end well-defined, with only one (1 !) status for every single string.

If the well-defined status is (A), this corresponds to "uncapturable" of J2003.

If the well-defined status is (B2), this corresponds to "capturable-1" of J2003.
All the points of the primary string are refered to as "local-1" in J2003.

When talking about the well-defined status (B31), the aspect comes to life refered to as "feedback loop" by me. This "feedback loop" has to do with what is "a certain area of the board" in my primary list.

But first I should highlight that in (B31) there will be no permanent stone of the player become established in "local-1". It follows, that the definition of a "certain area", which includes "local-1" is equivalent to the definition of the same (rest of the) "certain area" (defined before), with "local-1" excluded.
In J2003 this "certain area" is defined as "local-2", and this equivalent to "local-2\1" for the purpose of analysis.

The definition of "local-2" in J2003 refers to "capturable-1", what is equivalent to (B2) here. This will make it mandatory to repeat the analysis for all what has been identified as (B2) so far, if "later" in the course a status (B2) is found, which has not been visible to player and opponent before. This is because some "certain area" has changed now.
(This is true for the "later" finding of (A), too, with analogue consequences.)

But even if this happens accidentally, all final evaluation statuses will remain well-defined.

The well-defined status (B31) is refered to as "capturable-2" in J2003 and is equivalent to "capturable-2\1".

Cassandra wrote:When player and opponent cooperate, they are able to create each of the following results for the status evaluation of a single string:

Provided we do not use certain superko rules.

But first I should highlight that in (B31) there will be no permanent stone of the player become established in "local-1".

Proof?

In J2003 this "certain area" is defined as "local-2", and this equivalent to "local-2\1" for the purpose of analysis.

Proof?

"capturable-1", what is equivalent to (B2) here.

Not precisely.

The well-defined status (B31)

Please write down that definition!

"capturable-2" in J2003 [...] is equivalent to "capturable-2\1".

Proof???

RobertJasiek wrote:

Cassandra wrote:When player and opponent cooperate, they are able to create each of the following results for the status evaluation of a single string:

Provided we do not use certain superko rules.

If the number of groups on the board is not too small, even super-Ko should be no obstacle when there is cooperation. And J2003 does not use super-Ko.

But first I should highlight that in (B31) there will be no permanent stone of the player become established in "local-1".

Proof?

Opponent and player meet at (B31).
The opponent could not realize (B32). The player could not realize (A), (B2).
The opponent was not forced to allow a stone on local-1 (= B2). The player was not able to force a stone on local-1 (= B2).

In J2003 this "certain area" is defined as "local-2", and this equivalent to "local-2\1" for the purpose of analysis.

Proof?

See above: There will be no stone on local-1.

"capturable-1", what is equivalent to (B2) here.

Not precisely.

Why not ?

The well-defined status (B31)

Please write down that definition!

It is sufficient what I have written already. It is not necessary to define "certain area" for the purpose of showing that including local-1 in local-2 is overdoing things.

"capturable-2" in J2003 [...] is equivalent to "capturable-2\1".

Proof???

See above: There will be no stone on local-1.

Why not?

Just because of the usual, trivial wording and text embedding details.

Opponent and player meet at (B31).
The opponent could not realize (B32). The player could not realize (A), (B2).
The opponent was not forced to allow a stone on local-1 (= B2). The player was not able to force a stone on local-1 (= B2).

This is not a mathematical proof. In particular because your (A) to (B32) phrases presume the players cooperation while what would have to be proven depends on force.

See above: There will be no stone on local-1.

It is sufficient what I have written already. It is not necessary to define "certain area" for the purpose of showing that including local-1 in local-2 is overdoing things.

See above: There will be no stone on local-1.

Since your mathematical proof is missing, the consequences may not be made (thus far).

Why not?

Just because of the usual, trivial wording and text embedding details.

The string will be captured and have a successor on at least one of its primary points. What more is needed for "capturable-1" ?

Opponent and player meet at (B31).
The opponent could not realize (B32). The player could not realize (A), (B2).
The opponent was not forced to allow a stone on local-1 (= B2). The player was not able to force a stone on local-1 (= B2).

This is not a mathematical proof. In particular because your (A) to (B32) phrases presume the players cooperation while what would have to be proven depends on force.

This is not correct, Robert.

Each of my (A) to (B32) is possible when player and opponent cooperate and some sort of "cooperation" is what you fear for not including local-1 in local-2. (A) to (B32) is nothing less than the set of all theoretically possible evaluation sequences (spoken in J2003-language "all hypothetical sequences of all hypothetical strategies").

But there will remain NO room at all for "cooperation" because of the instructions for player and opponent. These instructions clarify "force" for both. Following these instructions eliminates all "superfluous" sequences not relevant for the string under evaluation.

See above: There will be no stone on local-1.
It is sufficient what I have written already. It is not necessary to define "certain area" for the purpose of showing that including local-1 in local-2 is overdoing things.
See above: There will be no stone on local-1.

Since your mathematical proof is missing, the consequences may not be made (thus far).

Let sequences survive the application of the instructions, which belong to (B31). No player's stone will be found on local-1.

Cassandra wrote:some sort of "cooperation" is what you fear

No, it is not cooperation what I "fear".

Conjecture: "The set of capturable-2 strings is a subset of the set of capturable-2\1 strings."

Now my fear is the possibility of falsehood of this conjecture. Compare Proposition 5. If we could prove the conjecture, then we would know the equality of capturable-2 and capturable-2\1. The current state of the art though is: Neither is the conjecture proven nor has a counter-example been found.

No player's stone will be found on local-1.

Our wish is not a proof.