Lambda Calculus 2 - Church Numbers

Counting without numbers?

In the first post I wrote about the lambda calculus we looked at the basic syntax and a simple function that took two numbers and added them together:

$$ \lambda x.\lambda y.x + y $$

This might all look OK until I tell you that, in the untyped lambda calculus, the only primitive data type is a function.

A function. Not a bit, a byte, a string; not a number - a function.

So we should be a little suspicious of $\,\lambda x.\lambda y.x + y$ as this $+$ symbol needs to be defined as a function. Fair enough - addition feels like a the sort of thing that could easily be a function.

But what would we apply to it? We need a number - like one or two. But we need to make them out of functions.

Wait, what? We need to make numbers out of functions?!

And this is where things start to get weird.

So what's a number anyway?

You will now be inducted into a sacred mystery that will allow you to make and understand Lisp jokes on the Internet. Be brave.

In a universe which only has functions - how would we count? Well, we'd have to do it with functions, obviously.

OK, sure - but that's not really getting us anywhere - let's take $2$ as a concrete example. How do I write a function that represents $2$?

Simple - we could just give it a name - like it was JavaScript:

const two = () => {}

Now you're obviously shouting "But that's cheating! What are these 'names' of which you speak? Are they made of functions too?"

And you'd be right. The thing is, we don't just want a symbol for $2$ - the numeral. What we want is a function that represents, in some way, the very essence of two-ness.

What I'm trying to get across here (without jumping to the solution immediately) is that the representation of numbers in the lambda calculus are not mere symbols; they capture a key behaviour that we associate with the idea of 'number'.

And that behaviour is repetition. When we say 'look at those two apples', we're expecting there to be an apple, and then another apple. In Church arithmetic a number is represented by a function that takes two values, and then applies the first value to the second value $n$ times, where $n$ is the number being represented.

Church numbers

So much for the theory, let's take a look at some real numbers.1 First up, the number one:

$$ \lambda f.\lambda x. f(x) $$

We accept a variable called $f$, we take another one called $x$, and we call $f$ with $x$ once. We're kinda hoping that $f$ turns out to be a function that takes a single argument, but as this is the lambda calculus and everything is a function that takes a single argument, we can be fairly absolutely certain it is.

In JavaScript:

f => x => f(x)

And Scheme:

(lambda (f) (lambda (x) (f x)))

So if that's one, we can probably work out what two is, right?

$$ \lambda fx. f\ (f\ x) $$

And three?

$$ \lambda fx. f\ (f\ (f\ x)) $$

OK, so no peeking now. What's zero?






Did you work it out?

$$ \lambda fx. x $$

It's just ignoring the original function and returning the value it would've been applied to. The function $f$ has been applied to $x$ zero times.

Function composition and functional powers

There's a neater way of talking about this, with just a litte more notation. We can abstract the idea of "first apply $f$, then apply $f$ to that result, then apply $f$..." out to the idea of function composition, which you might have seen in some languages. This is just building a new function out of two functions. The new function does exactly the same thing as calling one function after the other (in right-to-left order). This is usually written using the $\circ$ symbol:

$$ (f \circ f)\ x = f\ (f\ x) $$

And so three could become the (slightly) easier to read:

$$ \lambda f.\lambda x.\ (f \circ f \circ f)\ x $$

But we can go further. The idea of composing a function with itself so-many times can be written as a functional power - a 'function to the $n$'. Just like we can say $2^3 = 8$, so we can say:

$$ \lambda f.\lambda x.\ (f\ (f\ (f\ x)))\quad=\quad \lambda fx. (f\circ f\circ f)\ x\quad=\quad \lambda fx. f^3\ x $$

Implementing composition and powers of functions is left as an exercise for the reader in their favourite language.2

Playing around with the computer

I find there to be two productive interesting ways to play around with the lambda calculus and Church numbers when I'm trying to understand what's going on.

Firstly, and probably more obviously, try plugging around with them in your favourite language (as long as it has some sort of anonymous function type). Say Python - if we were to write three as a Church numeral we'd have:

three = lambda f: lambda x: f(f(f(x)))

If we want to test this - to see if it does what we think it does - we just need a function to be $f$:

increment = lambda x: x + 1

and some value to play the role of $x$

zero = 0

So then we just plug in those values:

three(increment)(0) #=> 3

We used three variables to hold the values above, but we could just inline them to get to something that looks a little more lambda-y:

(lambda f: lambda x: f(f(f(x))))(lambda x: x + 1)(0) #=> 3

Which translates to:

$$ (\lambda f.\lambda x.\ f\ (f\ (f\ x)))\ (\lambda x. x + 1)\ 0 = 3 $$

Both zero and increment use Python's built in number system. But we don't have to use numbers to test Church numbers. All we need are values that behave in the required way.3 For instance:

(define inc (lambda (x) (cons '() x)))

(define zero '())

(((lambda (f) (lambda (x) (f x))) inc) zero) ;;=> (())
(((lambda (f) (lambda (x) (f (f x)))) inc) zero) ;;=> (() ())
(((lambda (f) (lambda (x) (f (f (f x))))) inc) zero) ;;=> (() () ())

Lots of things will work - experiment!

Playing around with pen and paper

The second way I like to play with lambdas is with pen and paper. The simplicity of the syntax, and the very few transformations allowed on an expression,4 mean that it's possible to do the evaluation yourself. Let's try it with the above. I'm going to go through it step by step, and I'm going to note which of the reductions steps being performed at each line.

$$ \begin{array}{ c | l | c} & Expression & Reduction \\ \hline 0 & (\lambda fx.\ f\ (f\ (f\ x)))\ (\lambda x. x + 1)\ 0 \\ 1 & (\lambda x.\ (\lambda x.\ x + 1)\ ((\lambda x.\ x + 1)\ ((\lambda x.\ x + 1)\ x)))\ 0 & \beta \\ 2 & (\lambda x. (\lambda a.\ a + 1)\ ((\lambda b.\ b + 1)\ ((\lambda c.\ c + 1)\ x)))\ 0 & \alpha \\ 3 & (\lambda a.\ a + 1)\ ((\lambda b.\ b + 1)\ ((\lambda c.\ c + 1)\ 0)) & \beta \\ 4 & (\lambda a.\ a + 1)\ ((\lambda b.\ b + 1)\ 1) & \beta \\ 5 & (\lambda a.\ a + 1)\ 2 & \beta \\ 6 & 3 & \beta \end{array} $$

This is fun to try out, and while it's not much help when the expression is relatively simple as the one above, it gets pretty vital for me when I want to discover how more complicated expressions work.

In summary, the computer is great for checking that a lambda expression works, but I prefer to do get the pen and paper out if I want to get a feel for what's going on - for what makes it work.

But ...

But what about the $+$ and $1$ and $0$ above? I said that there were only functions in the lambda calculus, aren't we still cheating a little bit?

We are. So in the next post let's define increment, add, multiply and maybe even exponentiation in terms of lambdas. Things are certain to get weirder.

  1. I mean, actually these are the natural numbers including zero, not actually the real numbers. Yay, maths joke! 

  2. I've always wanted to say that. But, seriously - implement them. It's fun! 

  3. I am thoroughly in debt to the amazing book The Little Schemer for the this example. 

  4. $\alpha$-conversion and $\alpha$-reduction - see the first post