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21. Closure
Just as b imfam cm produces the immediate family
of b, so does the
phrase cm imfam cm produce the immediate families
of each of the rows of cm. We will,
however, use a new sparser connection matrix that will be
more instructive, and will use powers of imfam to
produce families of further generations, including an
infinite power to give the closure of the connection matrix;
that is, the connection matrix for all points reachable by a path
of any length:
cm=: (i. =/ <:@i.) 8
<"2 cm imfam^:0 1 2 _ cm
+---------------+---------------+---------------+---------------+
|0 1 0 0 0 0 0 0|0 1 1 0 0 0 0 0|0 1 1 1 0 0 0 0|0 1 1 1 1 1 1 1|
|0 0 1 0 0 0 0 0|0 0 1 1 0 0 0 0|0 0 1 1 1 0 0 0|0 0 1 1 1 1 1 1|
|0 0 0 1 0 0 0 0|0 0 0 1 1 0 0 0|0 0 0 1 1 1 0 0|0 0 0 1 1 1 1 1|
|0 0 0 0 1 0 0 0|0 0 0 0 1 1 0 0|0 0 0 0 1 1 1 0|0 0 0 0 1 1 1 1|
|0 0 0 0 0 1 0 0|0 0 0 0 0 1 1 0|0 0 0 0 0 1 1 1|0 0 0 0 0 1 1 1|
|0 0 0 0 0 0 1 0|0 0 0 0 0 0 1 1|0 0 0 0 0 0 1 1|0 0 0 0 0 0 1 1|
|0 0 0 0 0 0 0 1|0 0 0 0 0 0 0 1|0 0 0 0 0 0 0 1|0 0 0 0 0 0 0 1|
|0 0 0 0 0 0 0 0|0 0 0 0 0 0 0 0|0 0 0 0 0 0 0 0|0 0 0 0 0 0 0 0|
+---------------+---------------+---------------+---------------+
The closure of cm can therefore be expressed
as cm imfam^:_ cm, and a monadic closure function
can be defined as follows:
(closure=: imfam^:_ ~) cm
0 1 1 1 1 1 1 1
0 0 1 1 1 1 1 1
0 0 0 1 1 1 1 1
0 0 0 0 1 1 1 1
0 0 0 0 0 1 1 1
0 0 0 0 0 0 1 1
0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0
The complete definition of the closure function may now be
displayed as follows:
closure f.
([ +. +./ .*.)^:_~
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