Likely it is almost impossible for neural nets of "moderate" size
to identify life/death stati of a groups.

One difficult example (in the spirit of old Minsky/Papert who asked perceptrons
to identify the connectivity status of a "picture") might be the following 
X-"around the board"-group which is unconditionally alive (due to the two
eyes in upper left and lower right corner). Here I show it for 9x9-board size:

. X X X X X X X X
X . . . . . . . X
X . . . . . . . X
X . . . . . . . X
X . . . . . . . X
X . . . . . . . X
X . . . . . . . X
X . . . . . . . X
X X X X X X X X .

Ingo.
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