9-Dec

Schönfinkelisation and Partial Application

3 min read

By Kjetil Valle

December 9, 2019

As we discussed in a previous article, one of the things we can do with a function is to call it with fewer arguments than it is expecting. This will result in a new function where the arguments we did provide are bound to values, and the remaining arguments are still expected as parameters. Since we apply the function to only some of its arguments, we call this technique partial application.

Let's see how this works in Elm:

-- A simple function, adding together two arguments.
-- If you find the type signature confusing, don't worry
-- about that for now.
add : number -> number -> number
add a b =
	a + b

-- A relatively useless function, used to illustrate
-- partial application…
incrementByFive : number -> number
incrementByFife =
	-- Here we partially apply the `add` function. By
	-- only providing the first argument, we get a new
	-- function in return, which will accept another
	-- number, which it will always add 5 to!
	add 5

-- Yes, this would return 42
incrementByFive 37

Currying

It might surprise you that we are able to call the function add with only one argument. But, in fact, it is possible to turn a function of any number of arguments into a function of only one by using a process called currying.

Say we have this little JavaScript function:

function add(a, b) {
  return a + b;
}

We can't just simply partially apply this function. In JavaScript you need to provide all the arguments when applying a function. However, we might define this function instead:

function curriedAdd(a) {
  return function (b) {
    return a + b;
  };
}

By using currying, we can convert add into curriedAdd. And this process will work with functions of any number of arguments. And with this new function, we're free to do things like this again:

incrementByFive = curriedAdd(5);

But why didn't we have to do this in Elm? That's because in Elm, like many other functional programming languages, functions are curried by default. This means that all functions can easily be partially applied! And that's also why the type signature we saw in the first example looks like this:

add : number -> number -> number

A different way to read this type is like this:

add : number -> number -> number

So, as you see, add is in fact a function which takes only one argument (a number) and returns a function (with the type number -> number).

Why is this useful?

The incrementByFive example above is obviously quite contrived. But you'd be surprised how often partial application turns out to be useful when writing code in a functional language.

This often occurs when we're using functions like map or filter. These functions expects an argument which is a function that can be applied to every element of a list, and which must thus take exactly one argument. It is often convenient to create this function by using partial application.

Here are a couple of examples, again using Elm.

-- Convert a list of `Maybe String` into something that
-- can be displayed to the user, using "n/a" where we
-- don't have a value.
displayNames = List.map (Maybe.withDefault "n/a") [ Just "NBN", Nothing, Just "Jinteki", Just "Wayland"]
-- Result: [ "NBN", "n/a", "Jinteki", "Wayland" ]

-- Filter a list of names to include only the people
-- named "John".
johns = List.filter (String.startsWith "John") [ "John Snow", "John Rambo", "James Bond", "John McClane", "Jack Bauer"]
-- Result: [ "John Snow", "John Rambo", "John McClane" ]

Both Maybe.withDefault and String.startsWith expect to get two arguments, but by only applying them partially we get exactly what we need.

As another example, say we have a function with the following type signature in our Elm application.

log : LogConfig -> Message -> Cmd msg

The function requires a configuration type and a message. Depending on the configuration, the message might be logged to a backend service or maybe to the browser console. To specify the configuration everywhere we need to log something would be rather cumbersome. Instead we can just partially apply log to the configuration once, and use the resulting function in the rest of the application!

The more used you get to functional programming, the more places you will find where a partially applied function is exactly what you need.

Schönfinkeli-what-now?

The name currying is a reference to the American logician Haskell Brooks Curry. (If the name sounds familiar, that's no coincidence: Mr. Curry is quite popular among computer scientists, and has in fact three languages named after him: Haskell, Brook and Curry…)

However, he was not the one to first discover the technique of currying. The Russian logician and mathematician Moses Ilyich Schönfinkel had already described the concept previously, and was in fact also attributed by Curry for doing so. So, maybe we should start to refer to it as schönfinkelisation instead?