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What Is Lambda?

Since the idea might not be familiar to everybody, I thought it would be nice to write a short explanation of what lambda expressions are all about. If you want a longer explanation, see any book on Scheme or Lisp, or, of course, the lambda calculus.

I think of lambda expressions as functions. Here are some examples.

(lambda (x) x)
(lambda (x) (+ x 1))
(lambda (x y) (+ x y))
(lambda (x y) (if (> x y) x y))

These are, respectively, the identity function, a function that increments a number, a function that adds two numbers, and a function that takes the maximum of two numbers. The addition function, for example, accepts two arguments, as indicated in the list (x y), and returns the value of the expression (+ x y).

The correct statement, by the way, is not that lambda expressions are functions, but that they evaluate to functions or rather to closures.

Although closures are in themselves nameless, it's certainly possible to assign names to them, like so.

(define increment (lambda (x) (+ x 1)) )

(increment 17) = 18

However, it's not necessary to assign names; one can use closures directly.

( (lambda (x) (+ x 1)) 17 ) = 18

That's the basic idea, now let's look at a few of the fun parts.

Suppose we want to make some functions that increment not by 1 but by a variable amount, like so.

(lambda (x) (+ x delta))

Since we want a lot of these, it would be nice if we didn't have to define them individually, if we could just supply a value for delta and get back a function in other words, if we had a function that returned functions. But we can do exactly that!

(define add
   (lambda (delta)
      (lambda (x) (+ x delta)) ))

( (add 3) 17 ) = 20

But that's not all we can do. Not only can closures be returned from functions, they can be passed in as arguments. Here's one of the many possibilities.

(define apply-twice (lambda (f x) (f (f x))) )

(apply-twice (add 3) 17) = 23

Amazing, no?

 

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@ January (2001)