Quote:
As far as I am concerned I consider the point "a" as a real eye and I conclude we are in meari-menashi configuration.
What is meari-menashi?
Following Wolfe's seki graph theory, I think the calculation principle is iterating that a group needs 2 eye-like liberties to survive, namely places where the opponent can't play without giving you at least 1 permanent liberty there.
This occurs usually (always?) only when at that point the opponent dies themselves if they play there, generally by losing a semeai. Hence they at least need 2 other liberties (except for snapback, under the stones etc.), and must reduce your liberties. Playing on a mutual liberty reduces the liberties of both sides by 1 at the cost of a move. This is also true for connecting a weak point such as N19. Because B can get a liberty in sente by capturing, this is a follow up.
I would call internal liberties those which even after capturing you don't live. Are there any others than those found in eyes? Not sure what the official definition (s) is.
External liberties are those where you cannot expect to capture a group there. Note that it is optional whether or not to include potential small eyes in this.
Counting liberties instead of points using CGT, we have that normal external liberties are of the form
{1|-1} to the marginal liberties. Chilling by a tax of 1 per move equates this to 0, i.e. not interesting. Normal internal liberties are the same.j
Normal mutual liberties have marginals of
{0|0}. Chilling makes them negative {-1|1}. They are worse than playing external liberties. However sometimes there are 1st line kosumi tesuji which may be critical, though normally it only works in combination with another weak point.
What does the eye shape look like above?
{{{{2|0}|0}|-1}|0}
The way the eye comes into play is that W starts off with a bonus 3 liberties that B must fill to survive. K18 becomes one of these as an approach liberty. These act equivalently to internal liberties though they appear mutually. At the end of it, B can get an extra liberty in sente which makes these more urgent than normal external/internal liberties.
I don't know if CGT has a convenient way of representing this. Because the number of liberties at every node matters, not just at the leaf nodes because the semeai may end after any move if a group is captured. The marginal liberties isn't completely sufficient because the exact liberties matters.
It is like the local situation is
-3
| \
-2 0
| \
-1 0
| \
2 0
\
0
If B is down to 1 liberties after filling the first 2 approach liberties, W's next move can play that liberty to kill. Then B needs 3 external liberties to survive. In that case, W has 3 approach liberties.
Tsumego problems are difficult and interesting when comparing close marginal gains. Basically if the number of approach liberties is slightly different, a different result boundary appears. I don't know what sort of situations exist let alone their results.
For example what if W has 3 approach liberties from 2 different directions? Then perhaps W get a sneaky extra liberty from this by defending the side which is stronger. i.e. if B has 3 external liberties on both sides, W wins even if B plays first. If one side is 3 and the other is 4, I think we have a golden chicken standing on one leg situation and it is seki. If both sides are 4, then I think it is ko.
nb, if the example situation is copied, it doesn't work as then W has 2 eyes under optimal play. I'm not quite sure how to set up approach liberties in a different way though. I assume a semeai which B can win is required.