How are closures used in functional languages

For some reason, I tend to associate closures with functional languages. I believe this is mostly because the discussions I've seen concerning closures is almost always in an environment that is focused around functional programming. That being said, the actual practical uses of closures that I can think are are all non-functional in nature.

Are there practical uses of closures in functional languages, or is the association in my mind mostly because closures are used to program in a style that's also common to functional programming languages (first class functions, currying, etc)?

Edit: I should clarify that I refering to actual functional languages, meaning I was looking for uses that preserve referential transparency (for the same input you get the same output).

Edit: Adding a summary of what's been posted so far:

1. Closures are used to implement partial evaluation. Specifically, for a function that takes two arguments, it can be called with one argument which results in it returning a function that takes one argument. Generally, the method by which this second function "stores" the first value passed into it is a closure.
2. Objects can be implemented using closu开发者_运维技巧res. A function is returned that has closes around a number of variables, and can then use them like object attributes. The function itself may return more methods, which act as object methods, which also have access to these variables. Assuming the variables aren't modified, referential transparency is maintained.

I use lots of closures in Javascript code (which is a pretty functional language -- I joke that it is Scheme with C clothing). They provide encapsulation of data that is private to a function.

The most ubiquitous example:

var generateId = function() {
    var id = 0;
    return function() {
        return id++;
    }
}();
window.alert(generateId());
window.alert(generateId());

But that's the hello, world of Javascript closures. However there are many more practical uses.

Recently, in my job, I needed to code a simple photo gallery with sliders. It does something like:

var slide = function() {
    var photoSize = ...
    var ... // lots of calculations of sizes, distances to scroll, etc
    var scroll = function(direction, amout) {
        // here we use some of the variables defined just above
        // (it will be returned, therefore it is a closure)
    };
    return {
        up: function() { scroll(1, photoSize); },
        down: function() { scroll(-1, photoSize); }
    }
}();

slide.up();
// actually the line above would have to be associated to some
// event handler to be useful

In this case I've used closures to hide all the up and down scrolling logic, and have a code which is very semantic: in Javascript, "slide up" you will write slide.up().

One nice use for closures is building things like decision trees. You return a classify() function that tests whether to go down the left or right tree, and then calls either its leftClassify() or rightClassify() function depending on the input data. The leaf functions simply return a class label. I've actually implemented decision trees in Python and D this way before.

They're used for a lot of things. Take, for example, function composition:

let compose f g = fun x -> f (g x)

This returns a closure that uses the arguments from the function environment where it was created. Functional languages like OCaml and Haskell actually use closures implicitly all over the place. For example:

let flip f a b = f b a

Usually, this will be called as something like let minusOne = flip (-) 1 to create a function that will subtract 1 from its argument. This "partially applied" function is effectively the same as doing this:

let flip f a = fun b -> f b a

It returns a closure that remembers the two arguments you passed in and takes another argument of its own.

Closures can be used to simulate objects that can respond to messages and maintain their own local state. Here is a simple counter object in Scheme:

;; counter.ss
;; A simple counter that can respond to the messages
;; 'next and 'reset.  

(define (create-counter start-from)
  (let ((value start-from))
    (lambda (message)
      (case message
    ((next) (set! value (add1 value)) value)
    ((reset) (set! value start-from))
    (else (error "Invalid message!"))))))

Sample usage:

> (load "counter.ss")
> (define count-from-5 (create-counter 5))
> (define count-from-0 (create-counter 0))
> (count-from-5 'next)
6
> (count-from-5 'next)
7
> (count-from-0 'next)
1
> (count-from-0 'next)
2
> (count-from-0 'reset)
> (count-from-0 'next)
1