This is an introductory article to the Y combinator from lambda-calculus. But we won’t mention the Y combinator in this article.

This article is the first one of a serie about the Y combinator in ruby:

1. Recursions without names: Introduction to the Y combinator in ruby
2. The Y combinator in ruby
3. Y combinator real life application: recursive memoization in ruby

In this article, we are going to show how to write recursive functions in ruby without giving names to any function.

(If you are curious to see it in other languages, there is a version of the code in clojure and javascript.)

At first glance, it seems impossible: how could we refer to something that we are currently defining without using its name?

All the code snippets of this page are live and interactive powered by the klipse plugin:

1. Live: The code is executed in your browser
2. Interactive: You can modify the code and it is evaluated as you type

Begin with the end in mind

The end result of this article is the recursive implementation of the factorial function without using neither names nor loops.

Here is the code:

->(f){
  f[f]
  }[->(func){
    ->(n) { n == 0 ? 1 : n * func[func][n-1]}
    }][19]

As you can check, no mention of any names.

At first, it feels like magic.

Now, we are going to show the 4 step process that leads to this wonderful piece of code.

(We were inspired by this long but awesome article by Mike Vanier.)

The process

Step 0: recursive function

Let’s start with the recursive implementation of factorial:

$factorial = ->(n) {
  (n === 0) ? 1 : n * $factorial[n - 1]
    }
$factorial[10]

Step 1: simple generator

Let’s write a function that is a generator of the factorial function:

$factorial_gen = ->(f) {
  ->(n) {(n === 0) ? 1 : n * f[n - 1]}
    }
nil

One one hand, factorial_gen is not recursive.

On the other hand, factorial_gen is not the factorial function.

But the interesting thing is that if we pass factorial to factorial_gen it returns the factorial function:

$factorial_gen[$factorial][19]

Before going on reading make sure you understand why it is true that:

$factorial_gen[factorial] is equivalent to $factorial

Step 2: weird generator

Now, we are going to do something very weird: instead of using func, we are going to use (func func). Like this:

$factorial_weird = ->(f) {
  ->(n) {(n === 0) ? 1 : n * f[f][n - 1]}
    }
nil

The funny thing now is that if we apply factorial_weird to itself we get the factorial function:

$factorial_weird[$factorial_weird][19]

Before going on reading make sure you understand why it is true that:

$factorial_weird[factorial_weird] is equivalent to factorial

Step 3: Recursion without names

Now, let’s write down the application of factorial_weird to itself, using the body of factorial_weird instead of its name:

$factorial_no_names = ->(f) {
      ->(n) {(n === 0) ? 1 : n * f[f][n - 1]}
        }[->(f) {
      ->(n) {(n === 0) ? 1 : n * f[f][n - 1]}
        }]
$factorial_no_names[19]

And we got a recursive implementation of factorial without using any names!

We gave it a name just for the convenience of using it.

As you can check, this is a completely valid implementation of factorial:

(1..11).map(&$factorial_no_names).to_a

Do you understand why this is equivalent to the code we shown in the beginning of the article?

->(f){
  f[f]
  }[->(func){
    ->(n) { n == 0 ? 1 : n * func[func][n-1]}
    }][19]

Can you write your own implementation of other recursive functions without names?

Share in the comments your implementation for:

• Fibonacci
• Quicksort
• max
• min
• …

In our next article, we are going to show the Y combinator in action in ruby.