Hi,
I've just played the following game in kgs, in japanese rule.

I guess that the black group on the right side is dead, but how do I prove it if Black doesn't connect the ko ?

Think about it similarly to the corner bent-four: one side can attempt to kill, but the other side cannot attempt to live.

Hi Pio,

Click Here To Show Diagram Code

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$$ -----------
$$ | O . O O |
$$ | . O O X |
$$ | O O X . |
$$ | O X X X |
$$ | O X . . |
$$ | O X O . |
$$ | O X . X |
$$ | O X X X |
$$ -----------[/go]

I don't know if this is helpful or more confusing --
for your particular board, you can think of it as reduced to the above situation.

White can, at any time she wishes, eventually remove all the Black stones off the board;
there is nothing Black can do to stop it from happening.

( If Black disagrees, you can easily play it out.
Of course, in the process to reduce Black's liberties,
White will have to sacrifice a few White stones --
but the local status of this Black group is dead. )

For your particular board, even if there are any ko threats elsewhere,
White can reply to every single one of them; even if Black wins the 0.5 ko and connects,
it doesn't change the status -- Black is still dead as a group.

Verify the death of the black stones. Verify the life of the white stone. For each imagined verification, study the game tree and decicion-making. Necessarily, White can establish a white two-eye-formation on the intersections of and around the black group and the initial single white stone. Simply speaking, the basic ko rule applies. Whenever Black captures the ko, White plays an arbitrary move as a ko threat and next rcaptures the ko. Whenever Black plays a move elsewhere on the board, White recaptures the ko if possible. Black's plays elsewhere do not affect the local life verification because of not preventing White from establishing a local two-eye-formation.

See also my rules webpage. KGS-Japanese Rules are ambiguous so they require other Japanese rulesets as context rulesets. This is the real complication when assessing the life an death because it must be done for each existing Japanese ruleset and commentary ruleset. It turns out that each existing relevant ruleset assesses the statuses equally. However, for some of them, passes as ko threats and definitions of locality are relevant. For the details see my webpage.

Naive conclusion: treat it like a book life and death problem: the black group is dead. Detailed conclusion: KGS-Japanese Rules are ambiguous but many hours of rules expert analysis confirm the death of the black stones and the life of the white stone.

Thank you all for the answers !