Today we’re going to talk about one of

my favorite topics in Computer Science, which is the Lambda Calculus. And in

particular, we’re going to talk about three things: we’re going to think what

actually is it, why is it useful, and where did it actually come from? So

we’re going to start with the last question here – where did it actually come

from? This is Alonzo Church, who was a mathematician at Princeton University

in the United States, and he was the person who invented the Lambda

Calculus. And what he was interested in is what is the notion of a function from

a computational perspective. And his answer to this question is what we now

know as the Lambda Calculus. And there’s an interesting piece of history here,

which many people don’t know. So, it turns out that Alonzo Church was

the PhD supervisor of someone very famous in computer science — Alan

Turing. And of course Alan Turing, amongst many other things which he did,

he invented Turing machines — which Computerphile has done a number of videos

on — and Turing machines capture a basic state-based model of computation. It’s

interesting that his PhD supervisor, Alonzo Church, he captured a basic

functional notion of computation with his Lambda Calculus. And it turns out

that these two, quite different notions, one functional and one state-based, turn

out to be equivalent — and this is what’s called the Church-Turing hypothesis, or

part of the Church-Turing hypothesis. So for Church, a function was a black box,

but you’re not allowed to look inside. And what it does is it takes some input, so

maybe it takes a number like x, and it’s going to process it in some way, and it’s

going to produce an output. So maybe it produces the output x + 1. So this would

be a function that takes a single input, a number called x, processes in

some way, and then produces a single output, which is the number x + 1.

And we could have a slightly more interesting example. Maybe we

have a box with two inputs, x and y, and we process them in some way,

and maybe we produce their sum as the output. So this would be a function which

takes two inputs, x and y, processes them in some way, and then produces their

sum, x + y. And there’s two important things about functions in this sense. The

first is that they’re black boxes; you’re not allowed to look inside, and you can’t

see the mechanics of what’s going on inside that box, all you can do it put

something in and observe what comes out the other side. And the second important

thing is that these functions are pure, they have no internal state; so all that

happens when you map x across to x + 1, is the magic goes on inside the box, and

there’s no internal state, there’s no hidden information that we can use. And

this is quite different from the notion of computation that Alan Turing was interested

in with his Turing machines — he had internal state — there’s no internal state,

these are pure mathematical functions. Now we can think how do you actually define

functions in the Lambda Calculus. And there is a very, very simple syntax for

this, which I’ll introduce to you now. So let’s think about the increment function

in the Lambda Calculus. What you do is you write down a lambda symbol — so this

is the Greek lower-case letter lambda, and that says we’re introducing a function

at this point. And then you just write down the name of the input, so that was x.

And then you have a dot, and then you say how the output is calculated in

terms of the input, that’s x + 1. So we could do the same with addition, you just

need two lambdas, and write lambda x, dot, lambda y, dot, x + y. So this is the

function that takes two inputs, x and y, and then delivers the result x + y.

And this is written down formally in Church’s Lambda Calculus exactly like

this. So, when you’ve got a function, what can you do with it? Well, all you can

do is give it some input, let it do its thing, and it will give you some output.

So let’s have a simple example of this. If we take a function like increment,

which was lambda x, x + 1, and we apply it to a number like 5, what actually

happens? It’s a basic process of substitution; we’re essentially substituting

the number 5 here into the body of this lambda expression and then x

becomes 5, so we get 5 + 1, and then we get the result 6 on the

other side. And this is basically all there is to the Lambda Calculus. It’s

only got three things: it’s got variables, like x, y and z; and it’s got a way of

building functions — this lambda notation; and it’s got a way of applying functions.

This is the only three things that you have in the setting. What is

actually the point of the Lambda Calculus? We’ve introduced this very

simple notation, why should you be interested in learning about it? I think

there’s three answers which I would give to this. The first point I’d make is that

the Lambda Calculus can encode any computation. If you write a program in

any programming language, which has ever been invented, or ever will be invented,

or really any sequential programming language, it can in some way be encoded

in the Lambda Calculus. And of course it may be extremely inefficient when you do

that, but that’s not the point — this is a basic idea of computation, and we want to

think how many and what kind of programs can we encode this, and actually you can

encode anything. And this is really the kind of Church-Turing hypothesis which

I mentioned. Alan Turing, you can encode anything in his Turing machines, and in

Church’s Lambda Calculus, you can encode anything. And actually these two systems

are formally equivalent — any Turing machine program can be translated into an

equivalent Lambda Calculus program, and vice versa. They are

formally equivalent. The second point I would make is that Lambda

Calculus can also be regarded as the basis for functional programming

languages like Haskell. So these are becoming increasingly popular these days.

And actually a very sophisticated language like Haskell is compiled down to

a very small core language, which is essentially a glorified form of Lambda

Calculus. So if you’re interested in functional programming languages

like Haskell, or the ML family, these are all fundamentally based on the

Lambda Calculus — it’s just kind of a glorified syntax on top of that. The

third point which I would make, is that the Lambda Calculus is actually now

present in most major programming languages. So this wasn’t the case 10 or 15

years ago, but it is the case today. So if you look at languages like Java, like

C#, even Visual Basic, F#, and so on, all of these languages now encode

Lambda Calculus, or include Lambda Calculus, as a fundamental component. So

every computer scientist today needs to know about Lambda Calculus. What

I’d like to end up with is a couple of little examples of what you

can do with it. So, the Lambda Calculus has basically got nothing in it: it’s got variables,

it’s got a way of building functions, and it’s got a way of applying functions. It

doesn’t have any built-in data types like numbers, logical values, recursion

and things like that. So if you want to do these things in the Lambda Calculus,

you need to encode them. So I’ll end up showing you a simple encoding, and the

encoding which I’m going to show you is the logical values TRUE and FALSE. And

the key to this is to think what do you do with logical values in a programming

language? And the basic observation is that you use them to make a choice

between doing two things — you say if something is TRUE do one thing, if

something is FALSE do another thing, and we’re going to use this idea of making a

choice between two things to actually encode TRUE and FALSE. So the trick is

for TRUE, you write down this lambda expression. So what it does is it takes

two things, x and y, and then it chooses the first. And FALSE does the opposite.

It’s going to take two things, and it’s going to choose the second. So we’ve got

two lambda expressions here, both of which take two inputs, x and y, and one

chooses the first one, x, and one chooses the second one, y. So fair enough,

what can we actually do with this? Well, let’s think how we

could define a little logical operator. So, NOT is the most simple logical

operator which I could think of. It’s going to flip TRUE to FALSE, and FALSE to TRUE.

It’s logical negation. Based upon this encoding, how could I actually define the

NOT operator or the NOT function. It’s very easy to do. I will take in a logical

value, or a Boolean as it normally called in Computer Science, after George Boole, who

first studied a kind of formal logic. So we take a Boolean, which will be one of TRUE

or FALSE, and here’s what we do. We apply it to FALSE, and we apply it TRUE. And I

claim that this is a valid definition for a NOT function. But I can very

easily convince you that it’s the case, because I can do a little calculation.

So, let’s check, if we apply NOT to TRUE, that we actually get FALSE. And in just a

few steps, using the Lambda Calculus magic, we’ll find that this actually works

out. So what can we do here? Well the only thing we

can do is start to expand definitions. So, we know what the definition

of NOT is. It was lambda b, b applied to FALSE and TRUE, and then we just copy

down the TRUE. So all I’ve done in the first step here, is I’ve expanded my

definition of NOT — NOT was defined to be this Lambda Calculus expression here. Now,

I’ve got a function, which is this thing, and it’s applied to an input,

so i can just apply it. OK, and the function says if I take in a b,

I just apply that b to FALSE and TRUE. So, the thing I’m applying it to is TRUE

here, so i just do the little substitution. Rather than b, I write TRUE,

and then I copy down the FALSE, and copy down the TRUE, and I get down to here.

And at this point, you might quite rightly be thinking this

looks like complete rubbish. I’ve just written TRUE FALSE TRUE.

What does that mean? It means absolutely nothing. But it means something in the

Lambda Calculus, because we continue to expand. So, what we can do now, is expand

the definition of TRUE. We said that TRUE takes two things, and chooses the first one.

So, let’s expand it out. So, TRUE is lambda x, lambda y, x. So, it chooses the

first thing of two things, and then we just copy down the two inputs, FALSE AND TRUE.

And you can see what’s going to happen now — we’ve got a function here

which takes two things and chooses the first thing. Here the first thing is FALSE, so

when we apply the function, we just get back FALSE. So what you see

has happened here, in just a few steps, we’ve shown how using this

encoding of TRUE and FALSE, and not, we can actually get the desired behavior.

And it’s very easy to check for yourself, if you apply NOT to FALSE, you’ll get TRUE.

And I’d like to set you a little kind of puzzle at this point — think how

you could define logical AND, or logical OR, in this style as well.

And I’m interested to see what kind of definitions people come up with in

the comments. So, the very last thing I’d like to show you is this lambda

expression here, which is a very famous Lambda Calculus expression called the Y

combinator, or the Y operator. And actually, this is the key to doing recursion in

the Lambda Calculus. So, as I mentioned, Lambda Calculus has basically nothing in

it, or it’s only got three things in it: variables x, y and z, and so on; a way of

building functions; and a way of applying functions. It’s got no other control

structures, no other data types, no anything. So, if you want to do recursion,

which is the basic mechanism for defining things in terms of themselves —

again Computerphile’s had videos on this — you need to encode it. And it turns out

that this expression here is the key to encoding recursion in the Lambda

Calculus. And this expression was invented by someone called Haskell Curry,

and this is the Haskell that gives his name to the Haskell programming

language. And he was a PhD student of David Hilbert, who’s a very famous

mathematician. The last observation I’d like to leave you with here, is

something that’s interested me for many years. I think there’s a connection between

this piece of kind of abstract computer science, or abstract mathematics, and

biology. If you look at human DNA, you have this double helix structure; you

have two copies of the same thing, side-by-side, and this is the key to

allowing DNA to self-replicate. If you look at the structure of this lambda

expression here, we have two copies of the same thing side-by-side. You have

lambda x, f applied to x x, and exactly the same here. This is the key to doing

recursion, which is kind of related to self-replication in a programming

language, or in the Lambda Calculus. And for me, I don’t think this is a

coincidence — I think it’s kind of interesting philosophical observation.

The Lambda Calculus has this kind of very clever way of doing recursion, which

would take a video on its own to explain how it actually works, but you can look it

up on Wikipedia. And there’s a link here, I think, to biology. Somebody

actually found the Y Combinator so interesting, that they’ve had it

tattooed permanently on their arm, and you can find a picture of this if you do a

quick web search. What would people search for? The Y combination — the Y combinator in

mathematics or computer science. And tattoo I’m guessing. Yup!