The Universe of Discourse

Thu, 03 Jan 2008

Note on point-free programming style
This old comp.lang.functional article by Albert Y. C. Lai, makes the point that Unix shell pipeline programming is done in an essentially "point-free" style, using the shell example:

    grep '^X-Spam-Level' | sort | uniq | wc -l

and the analogous Haskell code:

    length . nub . sort . filter (isPrefixOf "X-Spam-Level")

Neither one explicitly mentions its argument, which is why this is "point-free". In "point-free" programming, instead of defining a function in terms of its effect on its arguments, one defines it by composing the component functions themselves, directly, with higher-order operators. For example, instead of:

  foo x y = 2 * x + y

one has, in point-free style:

  foo = (+) . (2 *)

where (2 *) is the function that doubles its argument, and (+) is the (curried) addition function. The two definitions of foo are entirely equivalent.

As the two examples should make clear, point-free style is sometimes natural, and sometimes not, and the example chosen by M. Lai was carefully selected to bias the argument in favor of point-free style.

Often, after writing a function in pointful style, I get the computer to convert it automatically to point-free style, just to see what it looks like. This is usually educational, and sometimes I use the computed point-free definition instead. As I get better at understanding point-free programming style in Haskell, I am more and more likely to write certain functions point-free in the first place. For example, I recently wrote:

        soln = int 1 (srt (add one (neg (sqr soln))))

and then scratched my head, erased it, and replaced it with the equivalent:

        soln = int 1 ((srt . (add one) . neg . sqr) soln)

I could have factored out the int 1 too:
        soln = (int 1 . srt . add one . neg . sqr) soln

I could even have removed soln from the right-hand side:

        soln = fix (int 1 . srt . add one . neg . sqr)

but I am not yet a perfect sage.

Sometimes I opt for an intermediate form, one in which some of the arguments are explicit and some are implicit. For example, as an exercise I wrote a function numOccurrences which takes a value and a list and counts the number of times the value occurs in the list. A straightforward and conventional implementation is:

        numOccurrences x []     = 0
numOccurrences x (y:ys) =
if (x == y) then 1 + rest
else                 rest
where rest = numOccurrences x ys

but the partially point-free version I wrote was much better:

        numOccurrences x = length . filter (== x)

Once you see this, it's easy to go back to a fully pointful version:

        numOccurrences x y = length (filter (== x) y)

Or you can go the other way, to a point-free version:

        numOccurrences = (length .) . filter . (==)

which I find confusing.

Anyway, the point of this note is not to argue that the point-free style is better or worse than the pointful style. Sometimes I use the one, and sometimes the other. I just want to point out that the argument made by M. Lai is deceptive, because of the choice of examples. As an equally biased counterexample, consider:

        bar x = x*x + 2*x + 1

which the automatic converter informs me can be written in point-free style as:

        bar = (1 +) . ap ((+) . join (*)) (2 *)

Perusal of this example will reveal much to the attentive reader, including the definitions of join and ap. But I don't think many people would argue that it is an improvement on the original. (Maybe I'm wrong, and people would argue that it was an improvement. I won't know for sure until I have more experience.)

For some sort of balance, here is another example where I think the point-free version is at least as good as the pointful version: a recent comment on Reddit suggested a >>> operator that composes functions just like the . operator, but in the other order, so that:

        f >>> g = g . f

or, if you prefer:

        >>> f g x = g(f(x))

The point-free definition of >>> is:

        (>>>) = flip (.)

where the flip operator takes a function of two arguments and makes a new function that does the same thing, but with the arguments in the opposite order. Whatever your feelings about point-free style, it is undeniable that the point-free definition makes perfectly clear that >>> is nothing but . with its arguments in reverse order.