What is a Korean Ko-threat?

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Robert Jasiek

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14.07.2004, 05:04:1914.07.04
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To recall, the Korean 1992 Rules have the following precedental rule:

>>
# O a . # O . . .
. # # # # O . . .
# # O O O O . . .
O O O . . . . . .
. . . . . . . . .

In [the figure], if Black clearly has [more ko-threats], Black a is not
needed.
<<

Now what is a "ko-threat"?

Let us try to exclude what a Korean ko-threat is not, for sure.

Suppose the prisoner-difference is +BIG. (E.g., BIG = 10,000.) Then even
each pass is a ko-threat that wins the game for Black.

Suppose the prisoner-difference is -BIG. Then none of the
ko-threat-local plays on the board wins the game for Black. [See the
previous thread "Korean Indirect-kos" for already defined terms.]

So obviously, a Korean ko-threat does not refer to winning the game but
to some local value of the ko.

Which local value of a ko? Let us try to answer a more specialized case
first: if the ko is a visible-indirect-ko, like in the figure.

***

Definition: The _visible-indirect-ko-defender_ of a visible-indirect-ko
is the player whose ko-stone is in the visible-indirect-ko.

Notes: It would not make sense to define the
visible-indirect-ko-defender as the player who can force something
according to the definition of visible-indirect-ko since that definition
allows "force or indirect-force". If one wanted such, then a different
definition of visible-indirect-ko would be necessary.

Definition: The _visible-indirect-ko-attacker_ of a visible-indirect-ko
is the opponent of the visible-indirect-ko-defender of the
visible-indirect-ko.

Definition: A _cluster_ is an intersection with a property and,
recursively, any adjacent intersection with that property.

Note: The word of the term has been suggested by Robert Pauli.

Definition: A _weak-defender-cluster_ of a visible-indirect-ko is a
cluster with the property that the defender of the visible-indirect-ko
controls each intersection and each intersection of the
visible-indirect-ko is part of the cluster.

Definition: The _defender-cluster_ of a visible-indirect-ko is a
weak-defender-cluster of the visible-indirect-ko if the
visible-indirect-ko-defender of the visible-indirect-ko can force the
weak-defender-cluster of the visible-indirect-ko and cannot force a
different weak-defender-cluster of the visible-indirect-ko.

Note: The purpose is to ensure that the defined object is given
unequivocally, i.e. there can be one and only one such object.

Definition: If a visible-indirect-ko has a defender-cluster, then the
_defender-value_ of the visible-indirect-ko is - favouring the
visible-indirect-ko-defender of the visible-indirect-ko - the maximal
defender-cluster's score under J2003 for all positions reached when the
visible-indirect-ko-defender of the visible-indirect-ko forces the
defender-cluster of the visible-indirect-ko.

Note: "forces" and not "indirect-forces". I.e. the defender-value exists
only if the visible-indirect-ko-defender of the visible-indirect-ko can
"force".

***

Application to the example:

Black as the visible-indirect-ko-defender forces this position:

# . # . # O . . .
. # # # # O . . .
# # O O O O . . .
O O O . . . . . .
. . . . . . . . .

The defender-cluster D is:

D D D D D O . . .
D D D D D O . . .
D D O O O O . . .
O O O . . . . . .
. . . . . . . . .

The defender-value is 4. (The prisoner-difference is 1.)

If Black forces the following position,

# . # # # O . . .
. # # # # O . . .
# # O O O O . . .
O O O . . . . . .
. . . . . . . . .

then the defender-cluster's score under J2003 is not maximal; it is only
3.

***

Definition: A _weak-attacker-cluster_ is [to be defined.]

Definition: The _attacker-cluster_ of a visible-indirect-ko is [to be
defined.]

Definition: If a visible-indirect-ko has an attacker-cluster, then the
_attacker-value_ of the visible-indirect-ko is [to be defined.]

Note: "indirect-force" has a significant role for existence of an
unequivocal attacker-value. In some sense, the attacker must be allowed
to try any indirect-force to create a visible-direct-ko on the
visible-indirect-ko before further actions happen.

Definition: If a visible-indirect-ko has an attacker-cluster, a
defender-cluster, an attacker-value, and a defender-value, then the
_difference-value_ of the visible-indirect-ko is its defender-value
minus its attacker-value.

***

Now principally it will be possible to define "ko-threat" for a
visible-indirect-ko that has the required characteristics as a
ko-threat-local play of a size that is at least the difference-value of
the visible-indirect-ko. However, other reference sizes could be
imagined for different definitions of "ko-threat".

--
robert jasiek

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