What is 'Currying'?

I've seen references to curried functions in several articles and blogs but I can't find a good explanation (or at least one that makes sense!)

Currying is when you break down a function that takes multiple arguments into a series of functions that each take only one argument. Here's an example in JavaScript:

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

add(3, 4); // returns 7

This is a function that takes two arguments, a and b, and returns their sum. We will now curry this function:

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

This is a function that takes one argument, a, and returns a function that takes another argument, b, and that function returns their sum.

add(3)(4);

var add3 = add(3);

add3(4);

The first statement returns 7, like the add(3, 4) statement. The second statement defines a new function called add3 that will add 3 to its argument. (This is what some may call a closure.) The third statement uses the add3 operation to add 3 to 4, again producing 7 as a result.

In an algebra of functions, dealing with functions that take multiple arguments (or equivalent one argument that's an N-tuple) is somewhat inelegant -- but, as Moses Schönfinkel (and, independently, Haskell Curry) proved, it's not needed: all you need are functions that take one argument.

So how do you deal with something you'd naturally express as, say, f(x,y)? Well, you take that as equivalent to f(x)(y) -- f(x), call it g, is a function, and you apply that function to y. In other words, you only have functions that take one argument -- but some of those functions return other functions (which ALSO take one argument;-).

As usual, wikipedia has a nice summary entry about this, with many useful pointers (probably including ones regarding your favorite languages;-) as well as slightly more rigorous mathematical treatment.

Here's a concrete example:

Suppose you have a function that calculates the gravitational force acting on an object. If you don't know the formula, you can find it here. This function takes in the three necessary parameters as arguments.

Now, being on the earth, you only want to calculate forces for objects on this planet. In a functional language, you could pass in the mass of the earth to the function and then partially evaluate it. What you'd get back is another function that takes only two arguments and calculates the gravitational force of objects on earth. This is called currying.

It can be a way to use functions to make other functions.

In javascript:

let add = function(x){
  return function(y){ 
   return x + y
  };
};

Would allow us to call it like so:

let addTen = add(10);

When this runs the 10 is passed in as x;

let add = function(10){
  return function(y){
    return 10 + y 
  };
};

which means we are returned this function:

function(y) { return 10 + y };

So when you call

 addTen();

you are really calling:

 function(y) { return 10 + y };

So if you do this:

 addTen(4)

it's the same as:

function(4) { return 10 + 4} // 14

So our addTen() always adds ten to whatever we pass in. We can make similar functions in the same way:

let addTwo = add(2)       // addTwo(); will add two to whatever you pass in
let addSeventy = add(70)  // ... and so on...

Now the obvious follow up question is why on earth would you ever want to do that? It turns what was an eager operation x + y into one that can be stepped through lazily, meaning we can do at least two things 1. cache expensive operations 2. achieve abstractions in the functional paradigm.

Imagine our curried function looked like this:

let doTheHardStuff = function(x) {
  let z = doSomethingComputationallyExpensive(x)
  return function (y){
    z + y
  }
}

We could call this function once, then pass around the result to be used in lots of places, meaning we only do the computationally expensive stuff once:

let finishTheJob = doTheHardStuff(10)
finishTheJob(20)
finishTheJob(30)

We can get abstractions in a similar way.

Currying is a transformation that can be applied to functions to allow them to take one less argument than previously.

For example, in F# you can define a function thus:-

let f x y z = x + y + z

Here function f takes parameters x, y and z and sums them together so:-

f 1 2 3

Returns 6.

From our definition we can can therefore define the curry function for f:-

let curry f = fun x -> f x

Where 'fun x -> f x' is a lambda function equivilent to x => f(x) in C#. This function inputs the function you wish to curry and returns a function which takes a single argument and returns the specified function with the first argument set to the input argument.

Using our previous example we can obtain a curry of f thus:-

let curryf = curry f

We can then do the following:-

let f1 = curryf 1

Which provides us with a function f1 which is equivilent to f1 y z = 1 + y + z. This means we can do the following:-

f1 2 3

Which returns 6.

This process is often confused with 'partial function application' which can be defined thus:-

let papply f x = f x

Though we can extend it to more than one parameter, i.e.:-

let papply2 f x y = f x y
let papply3 f x y z = f x y z
etc.

A partial application will take the function and parameter(s) and return a function that requires one or more less parameters, and as the previous two examples show is implemented directly in the standard F# function definition so we could achieve the previous result thus:-

let f1 = f 1
f1 2 3

Which will return a result of 6.

In conclusion:-

The difference between currying and partial function application is that:-

Currying takes a function and provides a new function accepting a single argument, and returning the specified function with its first argument set to that argument. This allows us to represent functions with multiple parameters as a series of single argument functions. Example:-

let f x y z = x + y + z
let curryf = curry f
let f1 = curryf 1
let f2 = curryf 2
f1 2 3
6
f2 1 3
6

Partial function application is more direct - it takes a function and one or more arguments and returns a function with the first n arguments set to the n arguments specified. Example:-

let f x y z = x + y + z
let f1 = f 1
let f2 = f 2
f1 2 3
6
f2 1 3
6