1996-10-5 last update, 1996-10-5 first day
Robert Jasiek,
jasiek@berlin.snafu.de
Copyright: All rights of the author are preserved according to the
international law.
Examples 3
Introduction
This page gives examples with comments referring to
super ko. If not explicitly
differently stated, comments are valid for positional super ko
and for situational super ko. The Tromp-Taylor rules with either
super ko rule shall be the presupposed rule set.
Besides SK for super ko further
Conventions are defined.
If a whole string should have marks on it, just ignore them;
they are immaterial.
Examples
Basic ko
- fight, W+-[att..]
- In any basic ko after one player has captured in it, the
other player must not recapture immediately, because the
resulting whole board position would be the same as before
the capture of the one player. (Note: Also the same player
moves, what is important under situational SK.)
- For example, W[attatt] is possible.
Double ko
- coexistence, B+[pp],
- The position arises shortly before the game end. Let us
assume B to play. B[p] or B[a] are conceivable.
- B[pp]# ends the game with both sides remaining on the board.
- B[pbapp]# would end the game.
- B[pbapb]# forces W to get another liberty, but B[pbapba]#
is prohibited, because it would repeat the starting position.
W must pass and looses all. So B[pb]# has been a mistake and
B[pp]# would have been correct.
- Under positional SK B[abpab]# is forbidden, because it would
repeat the whole board. B must pass and would loose all. Thus
B[a] is a mistake. Under positional SK only B[pp] is possible.
This means coexistence.
- Under situational SK B[abpab]# is allowed, since it is W's
turn then. After B[abpabp] B must pass, because B[abpabpa]#
would repeat the situation after B[a]. So B[abpabpp]# ends
the game with coexistence after some superfluous moves; B[pp]
would have been enough to end the game with the same outcome.
- Now let us assume that the double ko exists together
with another basic ko c with a W ko stone in it on the board,
Both players might try to use the double ko for ko threats.
SK does not allow B[cbacba]#. If W does not want to sacrifice
the double ko, he should have better not played B[cb]#. This
means that ko threats for c should be searched elsewhere only.
The double ko behaves neutral as to c.
Triple ko
- W[attatt..] is a possible ko fight.
- W[acbacttbacbatt..] is another possibility of equal character.
- The players may choose between move-sequences of 1 or 5 moves
in the triple ko, but they need to search their threats
alternatively in each case .
Three kos
- B[cttctt..] is a possible ko fight.
- B[cbacbttacbactt..] is another possibility of similar
character.
- Play closely resembles triple ko.
Triple ko
- W can be removed before the end of the game: B starts a
sequence of 5 moves, then no W ko threat - even on a bigger
board - would alter the character of the position and
finally W is left with nothing else than a pass play.
- Under positional SK any try by W to capture B, e. g. W[cbp],
can be countered, e. g. W[cbpcb]# is prohibited.
- Under situational SK any try by W to capture B, e. g. W[cbp],
can be countered, e. g. W[cbpcbacba]# is prohibited. When
analysing this MS you will notice the difficulty of keeping
track of a situation including the right to move in
comparison with the position only. More variety as to
possible MSs is provided, but the outcome is the same.
- In a similar position with the ko mouth a being external, but
still adjacent to the big W string the analyses is similar.
Moonshine life
- If W tries to prevent removal, B[abcabc]#
is prohibited. W cannot avoid removal.
Forgotten removal
- Assume W to have had the last move for a capture at a.
- B gets no more chance to capture at a. So he also cannot play
at b. B will not play at c or d.
- If B would not already have the marked stone on the board, he
could vary the board position and remove W: W[aMpapbpp].
- If W should have brilliantly played yose, so that B had no
chance to remove at a and b, W deserves life in the corner.
Sending-2-returning-1 and bent-4
- B[pp] ends the game with coexistence.
- Under positional SK B[aca]# or B[bcb]# are disallowed. So
just B[pp] might be played.
- Under situational SK B[aca]# or B[bcb]# are allowed. E.g.
B[acapp] ends the game with coexistence. B[acapb]# is unwise,
since B[acapbcb]# would repeat a situation and be prohibited.
- B[cpp] ends the game with coexistence.
- B[cpabac] gives a bent-4 ko. With SK B[cpabacbp]# allows B to
capture all.
Four kos
- B should try to kill something. B[abcdbadc]# is prohibited.
B then gets the right side and W the left.
- In history strong players interpreted positions by seeing
"eyes" in them. This resulted in rules with attempts of
definitions for shape. SK does not have problems of locality.
At b shall be a B stone, at d shall be a W stone
- The traditional name for the position is eternal life.
Without restriction as to repetition B[acbd]* would be
possible.
- A proper MS with ko threats is B[acbttdacttbdattcbdtt..].
Molasses ko
- Molasses ko seems to be the most interesting ko
configuration. The reason for the name will be obvious from
the following MSs: In general every fifth move of a game
before its last stage is a tenuki.
- Til today Molasses ko is the only known example with
different outcomes under positional SK respectively
situational SK.
- The special feature of Molasses ko is that not both players
want to pass successively, because the other player's strings
can be threatened. However, after four moves a pass play is
required. Hence a halfcycle within the ko configuration
consists of an odd move number.
- Under positional SK
B[cadbecbdapcadb]# is prohibited. W looses all, because B
has got the last tenuki. The previous fight on the rest of
the board was about getting this last tenuki. Skill was
needed to get more tenukis. Molasses ko under positional SK
produces similar strategy to play under the primitive ko
rule.
- Under situational SK
B[cadbpcbda]# is not prohibited, but B[cadbpcbdaecadbpcbda]#
neither. In general Molasses ko remains in coexistence til
the end of a game. The game will end with B[cadbecbdapp].
- For Molasses ko neither a removal of one colour nor a
coexistence is surprising; a removal can considered to be
caused by the continuous availability of a threatening move,
a coexistence can considered to be caused by a balance of
liberties. One variety of SK is not chosen to fit a special
desire for shape interpretation; special examples with
unfamiliar behaviour ought not to cause preference for a
rule set. Otherwise one will have to change the rules each
time when new spectacular examples are discovered.
Exercises
Quadruple ko
- W[c] gives a sufficient number of liberties for the time
being.
- A complete move-tree-analyses for quadruple ko is difficult
under SK. This is true for humans, not for computers. If you
should be able to solve the position, you might as well try
the general n-tuple ko...
- Single pass plays make the analysis more difficult.
- A typical length of MSs with maximal length is about
exponential in the number of involved kos, i. e. 2^n for
n kos.
- If both players should be able to read out the whole
move-tree and conclude that all MSs would result in
coexistence, they might as well pass as soon as possible.
If their reading is incomplete and they want to kill, then
an extremely long MS follows.
- Maybe the outcome of a n-tuple ko can be given by an explicit
mathematical analysis.
- Concerning future research:
One would like to see a proof that for any position
respectively situation during a part of a cyclical MS a
reasonably defined outcome of all involved stones could be
explicitly given. Then for any kind of repetitive shape
reading would be superfluous. (This is wishful thinking.
Ing ko rules have tought us just
how arduous a general shape analysis would be.)
Ring of 4 basic kos
Two double ko stones
- With ten involved points the position is rather intriguing.
At b shall be a B stone, at d shall be a W stone
- This is not too difficult.
[to be continued]
Links