The way J handles inverses to functions is quite unique and, while not
always very useful, can lead to compact but very descriptive code:
+/&.:*: NB. distance
+&.^. NB. multiplication
u&.(a.&i.) NB. u applied to y’s indices in a., then indexed out of a.
etc.
However, some less easily invertible functions, such as , (ravel), {: and
related,
A., m&{ , and others, don’t have a monadic inverse because they are “lossy”
functions,
in that there is no function u such that
y -: u@v y NB. where v is a lossy function
for an arbitrary y.
However, while there are other lossy functions which I have not cited, the ones
I
did are particular in that they are invertible with trains of related dyadic
functions:
u&., y <—> ($ $ u@,) y
u&.(m&{.) y <—> (m&}. , u@(m&{.)) y NB. extends to others like }.
etc.
NB.
although {. already has an inverse
u&.A. y <—> (A.~ u@A.) y
These are all simply “cosmetic”, in that you can obviously write them out
by hand, but then again, &. is basically cosmetic too. Nevertheless, it leads
to more elegant and concise code (and sometimes leads to new ways of
thinking about a problem). However, one inverse that could possibly
lead to (possibly big?) gains in speed with special code would be:
u&.(m&{) y <—> (m&}~ u@(m&{)) y
This might already be implemented for the fork case.
I’ve checked, and m&{ already has an inverse, although I don’t see what it’s
useful for.
On a side note, I’ve noticed that <;.1 has an inverse according to b._1, but
using it yields a nonce error (which means it hasn’t been implemented yet,
right?). <;._1@(m&,) could be recognised as a whole as well, so that
u&.(<;._1@(m&,)) y <—> ({. ;@, m ,&.> }.)@u@(<;._1@(m&,))
Or it could be added as an obverse to the cut standard library verb.
Again, this can all be done by hand, of course.
Louis
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