This page is not ready to go live; just roughly copying over some material from last year.

# Rosetta Stone

Here's how it looks to say the same thing in various of these languages.

1. Function application and parentheses

In Scheme and the lambda calculus, the functions you're applying always go to the left. So you write (foo 2) and also (+ 2 3).

Mostly that's how OCaml is written too:

foo 2


But a few familiar binary operators can be written infix, so:

2 + 3


You can also write them operator-leftmost, if you put them inside parentheses to help the parser understand you:

( + ) 2 3


I'll mostly do this, for uniformity with Scheme and the lambda calculus.

In OCaml and the lambda calculus, this:

foo 2 3


means the same as:

((foo 2) 3)


These functions are "curried". foo 2 returns a 2-fooer, which waits for an argument like 3 and then foos 2 to it. ( + ) 2 returns a 2-adder, which waits for an argument like 3 and then adds 2 to it. For further reading:

2. Currying

In Scheme, on the other hand, there's a difference between ((foo 2) 3) and (foo 2 3). Scheme distinguishes between unary functions that return unary functions and binary functions. For our seminar purposes, it will be easiest if you confine yourself to unary functions in Scheme as much as possible.

Scheme is very sensitive to parentheses and whenever you want a function applied to any number of arguments, you need to wrap the function and its arguments in a parentheses. So you have to write (foo 2); if you only say foo 2, Scheme won't understand you.

Scheme uses a lot of parentheses, and they are always significant, never optional. Often the parentheses mean "apply this function to these arguments," as just described. But in a moment we'll see other constructions in Scheme where the parentheses have different roles. They do lots of different work in Scheme.

3. Binding suitable values to the variables three and two, and adding them.

In Scheme:

(let* ((three 3))
(let* ((two 2))
(+ three two)))


Most of the parentheses in this construction aren't playing the role of applying a function to some arguments---only the ones in (+ three two) are doing that.

In OCaml:

let three = 3 in
let two = 2 in
( + ) three two


In the lambda calculus:

Here we're on our own, we don't have predefined constants like + and 3 and 2 to work with. We've got to build up everything from scratch. We'll be seeing how to do that over the next weeks.

But supposing you had constructed appropriate values for + and 3 and 2, you'd place them in the ellided positions in:

(((\three (\two ((... three) two))) ...) ...)


In an ordinary imperatival language like C:

int three = 3;
int two = 2;
three + two;

4. Mutation

In C this looks almost the same as what we had before:

int x = 3;
x = 2;


Here we first initialize x to hold the value 3; then we mutate x to hold a new value.

In (the imperatival part of) Scheme, this could be done as:

(let ((x (box 3)))
(set-box! x 2))


In general, mutating operations in Scheme are named with a trailing !. There are other imperatival constructions, though, like (print ...), that don't follow that convention.

In (the imperatival part of) OCaml, this could be done as:

let x = ref 3 in
x := 2


Of course you don't need to remember any of this syntax. We're just illustrating it so that you see that in Scheme and OCaml it looks somewhat different than we had above. The difference is much more obvious than it is in C.

In the lambda calculus:

Sorry, you can't do mutation. At least, not natively. Later in the term we'll be learning how in fact, really, you can embed mutation inside the lambda calculus even though the lambda calculus has no primitive facilities for mutation.

5. Anonymous functions

Functions are "first-class values" in the lambda calculus, in Scheme, and in OCaml. What that means is that they can be arguments to, and results of, other functions. They can be stored in data structures. And so on. To read further:

We'll begin by looking at what "anonymous" functions look like. These are functions that have not been bound as values to any variables. That is, there are no variables whose value they are.

In the lambda calculus:

(\x M)


---where M is any simple or complex expression---is anonymous. It's only when you do:

((\y N) (\x M))


that (\x M) has a "name" (it's named y during the evaluation of N).

In Scheme, the same thing is written:

(lambda (x) M)


Not very different, right? For example, if M stands for (+ 3 x), then here is an anonymous function that adds 3 to whatever argument it's given:

(lambda (x) (+ 3 x))


In OCaml, we write our anonymous function like this:

fun x -> ( + ) 3 x

6. Supplying an argument to an anonymous function

Just because the functions we built aren't named doesn't mean we can't do anything with them. We can give them arguments. For example, in Scheme we can say:

((lambda (x) (+ 3 x)) 2)


The outermost parentheses here mean "apply the function (lambda (x) (+ 3 x)) to the argument 2, or equivalently, "give the value 2 as an argument to the function (lambda (x) (+ 3 x)).

In OCaml:

(fun x -> ( + ) 3 x) 2

7. Binding variables to values with "let"

Let's go back and re-consider this Scheme expression:

(let* ((three 3))
(let* ((two 2))
(+ three two)))


Scheme also has a simple let (without the *), and it permits you to group several variable bindings together in a single let- or let*-statement, like this:

(let* ((three 3) (two 2))
(+ three two))


Often you'll get the same results whether you use let* or let. However, there are cases where it makes a difference, and in those cases, let* behaves more like you'd expect. So you should just get into the habit of consistently using that. It's also good discipline for this seminar, especially while you're learning, to write things out the longer way, like this:

(let* ((three 3))
(let* ((two 2))
(+ three two)))


However, here you've got the double parentheses in (let* ((three 3)) ...). They're doubled because the syntax permits more assignments than just the assignment of the value 3 to the variable three. Myself I tend to use [ and ] for the outer of these parentheses: (let* [(three 3)] ...). Scheme can be configured to parse [...] as if they're just more (...).

It was asked in seminar if the 3 could be replaced by a more complex expression. The answer is "yes". You could also write:

(let* [(three (+ 1 2))]
(let* [(two 2)]
(+ three two)))


It was also asked whether the (+ 1 2) computation would be performed before or after it was bound to the variable three. That's a terrific question. Let's say this: both strategies could be reasonable designs for a language. We are going to discuss this carefully in coming weeks. In fact Scheme and OCaml make the same design choice. But you should think of the underlying form of the let-statement as not settling this by itself.

Repeating our starting point for reference:

(let* [(three 3)]
(let* [(two 2)]
(+ three two)))


Recall in OCaml this same computation was written:

let three = 3 in
let two = 2 in
( + ) three two

8. Binding with "let" is the same as supplying an argument to a lambda

The preceding expression in Scheme is exactly equivalent to:

(((lambda (three) (lambda (two) (+ three two))) 3) 2)


The preceding expression in OCaml is exactly equivalent to:

(fun three -> (fun two -> ( + ) three two)) 3 2


Read this several times until you understand it.

9. Functions can also be bound to variables (and hence, cease being "anonymous").

In Scheme:

(let* [(bar (lambda (x) B))] M)


then wherever bar occurs in M (and isn't rebound by a more local let or lambda), it will be interpreted as the function (lambda (x) B).

Similarly, in OCaml:

let bar = fun x -> B in
M


This in Scheme:

(let* [(bar (lambda (x) B))] (bar A))


as we've said, means the same as:

((lambda (bar) (bar A)) (lambda (x) B))


which beta-reduces to:

((lambda (x) B) A)


and that means the same as:

(let* [(x A)] B)


in other words: evaluate B with x assigned to the value A.

Similarly, this in OCaml:

let bar = fun x -> B in
bar A


is equivalent to:

(fun x -> B) A


and that means the same as:

let x = A in
B

10. Pushing a "let"-binding from now until the end

What if you want to do something like this, in Scheme?

(let* [(x A)] ... for the rest of the file or interactive session ...)


or this, in OCaml:

let x = A in
... for the rest of the file or interactive session ...


Scheme and OCaml have syntactic shorthands for doing this. In Scheme it's written like this:

(define x A)
... rest of the file or interactive session ...


In OCaml it's written like this:

let x = A;;
... rest of the file or interactive session ...


It's easy to be lulled into thinking this is a kind of imperative construction. But it's not! It's really just a shorthand for the compound let-expressions we've already been looking at, taking the maximum syntactically permissible scope. (Compare the "dot" convention in the lambda calculus, discussed above. I'm fudging a bit here, since in Scheme (define ...) is really shorthand for a letrec epression, which we'll come to in later classes.)

11. Some shorthand

OCaml permits you to abbreviate:

let bar = fun x -> B in
M


as:

let bar x = B in
M


It also permits you to abbreviate:

let bar = fun x -> B;;


as:

let bar x = B;;


Similarly, Scheme permits you to abbreviate:

(define bar (lambda (x) B))


as:

(define (bar x) B)


and this is the form you'll most often see Scheme definitions written in.

However, conceptually you should think backwards through the abbreviations and equivalences we've just presented.

(define (bar x) B)


just means:

(define bar (lambda (x) B))


which just means:

(let* [(bar (lambda (x) B))] ... rest of the file or interactive session ...)


which just means:

(lambda (bar) ... rest of the file or interactive session ...) (lambda (x) B)


or in other words, interpret the rest of the file or interactive session with bar assigned the function (lambda (x) B).

You can override a binding with a more inner binding to the same variable. For instance the following expression in OCaml:

let x = 3 in
let x = 2 in
x


will evaluate to 2, not to 3. It's easy to be lulled into thinking this is the same as what happens when we say in C:

int x = 3;
x = 2;


but it's not the same! In the latter case we have mutation, in the former case we don't. You will learn to recognize the difference as we proceed.

The OCaml expression just means:

(fun x -> ((fun x -> x) 2) 3)


and there's no more mutation going on there than there is in:

∀x. (F x or ∀x (not (F x)))


When a previously-bound variable is rebound in the way we see here, that's called shadowing: the outer binding is shadowed during the scope of the inner binding.

## Some more comparisons between Scheme and OCaml

• Simple predefined values

Numbers in Scheme: 2, 3
In OCaml: 2, 3

Booleans in Scheme: #t, #f
In OCaml: true, false

The eighth letter in the Latin alphabet, in Scheme: #\h
In OCaml: 'h'

• Compound values

These are values which are built up out of (zero or more) simple values.

Ordered pairs in Scheme: '(2 . 3) or (cons 2 3)
In OCaml: (2, 3)

Lists in Scheme: '(2 3) or (list 2 3)
In OCaml: [2; 3]
We'll be explaining the difference between pairs and lists next week.

The empty list, in Scheme: '() or (list)
In OCaml: []

The string consisting just of the eighth letter of the Latin alphabet, in Scheme: "h"
In OCaml: "h"

A longer string, in Scheme: "horse"
In OCaml: "horse"

A shorter string, in Scheme: ""
In OCaml: ""