In Scheme:
(let* ((three 3))
- (let ((two 2))
+ (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 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.
+ 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:
+ 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))) ...) ...)
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.
+ 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.
3. 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 other functions. They can be the results of the application of other functions to some arguments. They can be stored in data structures. And so on.
+ Functions are "first-class values" MORE 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.
First, we'll show 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.
(\x M)
- is always anonymous! Here `M` stands for any expression of the language, simple or complex. It's only when you do
+ ---where `M` is any simple or complex expression---is anonymous. It's only when you do:
((\y N) (\x M))
(lambda (x) M)
- Not very different, right? For example, if `M` stands for `(+ 3 x)`, then this is an anonymous function that adds 3 to whatever argument it's given:
+ 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)
-
- or:
-
- fun x -> (( + ) 3 x)
-
- In OCaml, parentheses only serve a grouping function and they often can be omitted. Or more could be added. For instance, we could equally well say:
-
fun x -> ( + ) 3 x
- or:
-
- (fun x -> (( + ) (3) (x)))
-
- As we saw above, parentheses can often be omitted in the lambda calculus too. But not in Scheme. Every parentheses has a specific role.
4. Supplying an argument to an anonymous function
((lambda (x) (+ 3 x)) 2)
- The outermost parentheses here mean "apply the function `(lambda (x) (+ 3 x))` to the argument `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:
Let's go back and re-consider this Scheme expression:
(let* ((three 3))
- (let ((two 2))
+ (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:
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))
+ (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 `(...)`.
- Someone asked in seminar if the `3` could be replaced by a more complex expression. The answer is "yes". You could also write:
+ 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)]
+ (let* [(two 2)]
(+ three two)))
- The question also came up 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.
+ 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)]
+ (let* [(two 2)]
(+ three two)))
Recall in OCaml this same computation was written:
((lambda (bar) (bar A)) (lambda (x) B))
- which, as we'll see, is equivalent to:
+ which beta-reduces to:
((lambda (x) B) A)
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.)
+ 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.)
9. Some shorthand
Some more comparisons between Scheme and OCaml
----------------------------------------------
-11. Simple predefined values
+* Simple predefined values
Numbers in Scheme: `2`, `3`
In OCaml: `2`, `3`
The eighth letter in the Latin alphabet, in Scheme: `#\h`
In OCaml: `'h'`
-12. Compound values
+* Compound values
These are values which are built up out of (zero or more) simple values.
- Ordered pairs in Scheme: `'(2 . 3)`
+ Ordered pairs in Scheme: `'(2 . 3)` or `(cons 2 3)`
In OCaml: `(2, 3)`
- Lists in Scheme: `'(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: `'()`
+ The empty list, in Scheme: `'()` or `(list)`
In OCaml: `[]`
The string consisting just of the eighth letter of the Latin alphabet, in Scheme: `"h"`