Gérard TAILLE wrote:
Oops a approach move seems a very big [constraint].
Dame is empty intersections that is adjacent to stones that are alive. How about using this definition to permit dame to be filled out of order?
The dame-filling rule is that one or more empty intersections connected to only alive stones of one color and either alive or dead stones of the other color, can be filled during confirmation, and this can be done out of normal playing order.
Using this position.
$$B
$$ | . . . . . . .
$$ | O O . . . . .
$$ | . O . . . . .
$$ | X O . . . . .
$$ | . X O O O . .
$$ | X X X X O . .
$$ | O X . X O . .
$$ | . O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .
- Click Here To Show Diagram Code
[go]$$B
$$ | . . . . . . .
$$ | O O . . . . .
$$ | . O . . . . .
$$ | X O . . . . .
$$ | . X O O O . .
$$ | X X X X O . .
$$ | O X . X O . .
$$ | . O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .[/go]
We start by noting that the marked stones are alive, therefore white is allowed to fill the dame without interruption from black.
$$B
$$ | . . . . . . .
$$ | W W . . . . .
$$ | V W . . . . .
$$ | X W . . . . .
$$ | . X O O O . .
$$ | X X X X O . .
$$ | O X . X O . .
$$ | . O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .
- Click Here To Show Diagram Code
[go]$$B
$$ | . . . . . . .
$$ | W W . . . . .
$$ | V W . . . . .
$$ | X W . . . . .
$$ | . X O O O . .
$$ | X X X X O . .
$$ | O X . X O . .
$$ | . O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .[/go]
Now we can show that the marked stones are dead like this:
$$W :b4: pass
$$ | . . . . . . .
$$ | O O . . . . .
$$ | V O . . . . .
$$ | X O . . . . .
$$ | 1 X O O O . .
$$ | X X X X O . .
$$ | 5 X 3 X O . .
$$ | 2 O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .
- Click Here To Show Diagram Code
[go]$$W :b4: pass
$$ | . . . . . . .
$$ | O O . . . . .
$$ | V O . . . . .
$$ | X O . . . . .
$$ | 1 X O O O . .
$$ | X X X X O . .
$$ | 5 X 3 X O . .
$$ | 2 O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .[/go]
Now for your other position we need to apply the dame-filling rule twice, because the tail must be captured twice.
$$B
$$ | . . . . . . .
$$ | . . . . . . .
$$ | O O O O O O .
$$ | . . . . . O .
$$ | X X X X . O .
$$ | X . . . . O .
$$ | X O O O O O .
$$ | . X O O O . .
$$ | X X X X O . .
$$ | O X . X O . .
$$ | . O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .
- Click Here To Show Diagram Code
[go]$$B
$$ | . . . . . . .
$$ | . . . . . . .
$$ | O O O O O O .
$$ | . . . . . O .
$$ | X X X X . O .
$$ | X . . . . O .
$$ | X O O O O O .
$$ | . X O O O . .
$$ | X X X X O . .
$$ | O X . X O . .
$$ | . O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .[/go]
White can fill dame because the marked white stones are alive, and play

.
$$W
$$ | . . . . . . .
$$ | . . . . . . .
$$ | W W W W W W .
$$ | V V V V V W .
$$ | X X X X V W .
$$ | X V V V V W .
$$ | X W W W W W .
$$ | 1 X W W W . .
$$ | X X X X W . .
$$ | O X . X W . .
$$ | . O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .
- Click Here To Show Diagram Code
[go]$$W
$$ | . . . . . . .
$$ | . . . . . . .
$$ | W W W W W W .
$$ | V V V V V W .
$$ | X X X X V W .
$$ | X V V V V W .
$$ | X W W W W W .
$$ | 1 X W W W . .
$$ | X X X X W . .
$$ | O X . X W . .
$$ | . O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .[/go]
Black plays

.
$$W
$$ | . . . . . . .
$$ | . . . . . . .
$$ | O O O O O O .
$$ | V V V V V O .
$$ | . . . . V O .
$$ | . V V V V O .
$$ | 2 O O O O O .
$$ | . X O O O . .
$$ | X X X X O . .
$$ | O X . X O . .
$$ | . O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .
- Click Here To Show Diagram Code
[go]$$W
$$ | . . . . . . .
$$ | . . . . . . .
$$ | O O O O O O .
$$ | V V V V V O .
$$ | . . . . V O .
$$ | . V V V V O .
$$ | 2 O O O O O .
$$ | . X O O O . .
$$ | X X X X O . .
$$ | O X . X O . .
$$ | . O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .[/go]
Now white can use the dame-filling rule again to capture with
$$W
$$ | . . . . . . .
$$ | . . . . . . .
$$ | O O O O O O .
$$ | V V V V V O .
$$ | V V V V V O .
$$ | V V V V V O .
$$ | X O O O O O .
$$ | 3 X O O O . .
$$ | X X X X O . .
$$ | O X . X O . .
$$ | . O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .
- Click Here To Show Diagram Code
[go]$$W
$$ | . . . . . . .
$$ | . . . . . . .
$$ | O O O O O O .
$$ | V V V V V O .
$$ | V V V V V O .
$$ | V V V V V O .
$$ | X O O O O O .
$$ | 3 X O O O . .
$$ | X X X X O . .
$$ | O X . X O . .
$$ | . O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .[/go]
Now the life and death can be determined in same way as before.
$$W :b6: pass
$$ | . . . . . . .
$$ | . . . . . . .
$$ | O O O O O O .
$$ | V V V V V O .
$$ | V V V V V O .
$$ | V V V V V O .
$$ | . O O O O O .
$$ | 3 X O O O . .
$$ | X X X X O . .
$$ | 7 X 5 X O . .
$$ | 4 O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .
- Click Here To Show Diagram Code
[go]$$W :b6: pass
$$ | . . . . . . .
$$ | . . . . . . .
$$ | O O O O O O .
$$ | V V V V V O .
$$ | V V V V V O .
$$ | V V V V V O .
$$ | . O O O O O .
$$ | 3 X O O O . .
$$ | X X X X O . .
$$ | 7 X 5 X O . .
$$ | 4 O X O X . .
$$ | O O O O X . .
$$ | . O X X X . .
$$ | O O X . . . .
$$ | X X X . . . .
$$ | . . . . . . .[/go]
Maybe this is workable to solve the dame filling problem in general?
If so then I am thinking about how to make an iterative process actually work, that is to allow one to make the claim that the outside stones are alive first and then use that to invoke dame filling.
Edit =====
Actually, one may say that

is capturing a ko stone but I think that could be fixed by defining it away by saying something about ko-shapes.