Notes (TILC 04/05): Recursion

4 Recursion

recursion: A pattern where a function calls itself.

The basic idea behind recursion in lambda calculus is that we want to build an expression that regenerates itself as we reduce it. What I mean by that is if a function F is going to apply itself inside itself, it’s reduction needs to somehow build a new copy of F. We’re going to want to end up with some function that transforms F into a sequence of repeated applications of F:

F (F (F ....)))

Imagine we want to apply an expression F to itself once. If F is the identity function id:

id id => (\x.x) (\x.x) => \x.x

Let’s say we want a function that does this self-application, that we’ll call M:

M = \f. f f

Now supppose we apply M to M:

M(M) = (\f. f f) (\f. f f) =>
(\f. f f) (\f. f f) => \dots

This is an infinite loop! M(M) regenerates itself perfectly, so that any reduction just goes M(M) => M(M) ... . M(M) is in fact the classic case of a non-terminating lambda expression, and is usually called Ω).

Infinite loops are cool, but what we really want is to modify M(M) so that we add an application of a function R at every loop:

M (\f. R (f f)) = (\f. f f) (\f. R (f f)) =>
(\f. R (f f)) (\f. R (f f)) =>
R ((\f. R (f f)) (\f. R (f f)) =>
R ( R ((\f. R (f f)) (\f. R (f f)))) =>

When we abstract over R, this becomes the famous Y combinator:

Y = \r. (\f. f f) (\f. r (f f)) => \r. (\f. r (f f)) (\f. r (f f))

combinator: A lambda abstraction with no free variables.

Another way to think about the Y combinator is as a fixed-point combinator.

fixed-point. If x = (f x), x is a fixed-point of the function f.

Suppose we have a function fix such that(fix f) that returns a fixed-point of f. Then, by the definition of a fixed-point,

(fix f) = f (fix f)

This can be expanded indefinitely as

fix f => f (fix f) => f (f (fix f)) => ...

Which is precisely the same as the Y combinator:

Y f => f (Y f) = f (f (Y f))

so we can say that Y is the lambda expression that implements the function fix for lambda functions.

We know know how to recurse a function R an infinite number of times:

Y R => R (Y R) => R (R (Y R)) => ...

But suppose we want to only recurse R a finite number of times:

Y R => R (Y R) => R (R (Y R)) => R ( R ( ... (R B)) ...)

We’re going to have to bottom-out at some base-case B.

But how do we structure our function R so that it generates a base case, if YR => R(Y R)? Won’t that generate more copies of R no matter what we do?

Watch what happens if we apply Y to 0_:

Y 0_ => Y (\s z. z) -> (\s z. z) Y (\s z. z) -> (\s z. z) -> 0_

0_ throws away its first argument, and because it throws away the Y, the recursion stops. So we if build an R that at some point throws away the Y combinator, we’ll lose our means of producing new copies of R.

Let’s look at the example in the text, which is supposed to sum all the integers between n and 0_:

R = (\r n. isZero n 0_ (n succ (r (pred n))) )
sum n = Y R n

Let’s do some reductions on this, starting with sum 0_:

sum n = Y R 0_ =>
Y (\r n. isZero n 0_ (n succ (r (pred n)))) 0_ =>
(\rn.isZero n 0_ (n succ (r (pred n)))) (Y R) 0_ =>
(isZero 0_ 0_ (0_ succ ((Y R) (pred 0_)))) =>
true 0_ (0_ succ ((Y R) (pred 0_)))) =>

Now sum 1:

Y R 1 =>
Y (\rn.isZero n 0_ (n succ (r (pred n)))) 1_ =>
(\rn.isZero n 0_ (n succ (r (pred n)))) (Y R) 1_ =>
(isZero 1_ 0_ (1_ succ ((Y R) (pred 1)))) =>
false 0_ (1_ succ ((Y R) (pred 1)))) =>
1_ succ ((Y R) (pred 1))) =>
1_ succ (Y R 0) =>
1_ succ 0 =>

And sum 2_ for good measure:

Y R 2_ =>
Y (\rn.isZero n 0_ (n succ (r (pred n)))) 2_ =>
(\rn.isZero n 0_ (n succ (r (pred n)))) (Y R) 2_ =>
(isZero 2_ 0_ (2_ succ ((Y R) (pred 2_)))) =>
false 0_ (2_ succ ((Y R) (pred 2_)))) =>
2_ succ ((Y R) (pred 2_))) =>
2_ succ (Y R 1_) =>
2_ succ (1_ succ 0_) =>

Let’s break this down into the general case:

loop = \ test f next start. Y (\r s. test s (f (r (next s)))) start

First we test our state s and return either true or false. In the example in the text, test is isZero and s is a number n.

If our test returns true we terminate by returning our base, (0_ in the above example) otherwise we recurse and return f (r (next s)). The function f is what we’re actually interested in evaluating over our recursion. In the above the function f is add n or (n succ m). The r is the stand-in for the rest of the recursion that will be generated with a Y R application. And the next is pred in the above example, because we want to count down from n to 0_ by 1s.

Supposing we want to loop a function f over the numbers from m_ to 0_ (technically the term is fold f over the numbers m_ to 0_, but I don’t want to get into folds just now).

All we have to do is fill apply to loop to the corresponding variables:

loop (\n. isZero n base) f pred m_ =>
  Y (\r n. isZero n base
          (f (r (pred n)))
  (\r n. isZero n base (f (r (pred n))))
  ( Y (\r n. isZero base
            (f (r (pred n)))
(\r n. isZero base
  ( f
    ( (Y (\r n. isZero base (f (r (pred n)))))
      (pred m_))

In this way we can build a wide range of finite recursive structures.