1 *This page is not ready to go live; just roughly copying over some material from last year.*
7 Here's how it looks to say the same thing in various of these languages.
9 The following site may be useful; it lets you run a Scheme interpreter inside your web browser: [Try Scheme in your web browser](http://tryscheme.sourceforge.net/). See also our links about [[learning Scheme]] and [[learning OCaml]].
13 1. Function application and parentheses
15 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)`.
17 Mostly that's how OCaml is written too:
21 But a few familiar binary operators can be written infix, so:
25 You can also write them operator-leftmost, if you put them inside parentheses to help the parser understand you:
29 I'll mostly do this, for uniformity with Scheme and the lambda calculus.
31 In OCaml and the lambda calculus, this:
39 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:
41 * [[!wikipedia Currying]]
43 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.
45 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.
47 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.
50 2. Binding suitable values to the variables `three` and `two`, and adding them.
58 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.
67 In the lambda calculus:
69 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.
71 But supposing you had constructed appropriate values for `+` and `3` and `2`, you'd place them in the ellided positions in:
73 (((\three (\two ((... three) two))) ...) ...)
75 In an ordinary imperatival language like C:
83 In C this looks almost the same as what we had before:
88 Here we first initialize `x` to hold the value 3; then we mutate `x` to hold a new value.
90 In (the imperatival part of) Scheme, this could be done as:
95 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.
97 In (the imperatival part of) OCaml, this could be done as:
102 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.
104 In the lambda calculus:
106 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.
111 3. Anonymous functions
113 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:
115 * [[!wikipedia Higher-order function]]
116 * [[!wikipedia First-class function]]
118 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.
120 In the lambda calculus:
124 ---where `M` is any simple or complex expression---is anonymous. It's only when you do:
128 that `(\x M)` has a "name" (it's named `y` during the evaluation of `N`).
130 In Scheme, the same thing is written:
134 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:
138 In OCaml, we write our anonymous function like this:
143 4. Supplying an argument to an anonymous function
145 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:
147 ((lambda (x) (+ 3 x)) 2)
149 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))`.
153 (fun x -> ( + ) 3 x) 2
156 5. Binding variables to values with "let"
158 Let's go back and re-consider this Scheme expression:
164 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:
166 (let* ((three 3) (two 2))
169 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:
175 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 `(...)`.
177 It was asked in seminar if the `3` could be replaced by a more complex expression. The answer is "yes". You could also write:
179 (let* [(three (+ 1 2))]
183 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.
185 Repeating our starting point for reference:
191 Recall in OCaml this same computation was written:
197 6. Binding with "let" is the same as supplying an argument to a lambda
199 The preceding expression in Scheme is exactly equivalent to:
201 (((lambda (three) (lambda (two) (+ three two))) 3) 2)
203 The preceding expression in OCaml is exactly equivalent to:
205 (fun three -> (fun two -> ( + ) three two)) 3 2
207 Read this several times until you understand it.
209 7. Functions can also be bound to variables (and hence, cease being "anonymous").
213 (let* [(bar (lambda (x) B))] M)
215 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)`.
219 let bar = fun x -> B in
224 (let* [(bar (lambda (x) B))] (bar A))
226 as we've said, means the same as:
228 ((lambda (bar) (bar A)) (lambda (x) B))
230 which beta-reduces to:
234 and that means the same as:
238 in other words: evaluate `B` with `x` assigned to the value `A`.
240 Similarly, this in OCaml:
242 let bar = fun x -> B in
249 and that means the same as:
254 8. Pushing a "let"-binding from now until the end
256 What if you want to do something like this, in Scheme?
258 (let* [(x A)] ... for the rest of the file or interactive session ...)
263 ... for the rest of the file or interactive session ...
265 Scheme and OCaml have syntactic shorthands for doing this. In Scheme it's written like this:
268 ... rest of the file or interactive session ...
270 In OCaml it's written like this:
273 ... rest of the file or interactive session ...
275 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.)
279 OCaml permits you to abbreviate:
281 let bar = fun x -> B in
289 It also permits you to abbreviate:
291 let bar = fun x -> B;;
297 Similarly, Scheme permits you to abbreviate:
299 (define bar (lambda (x) B))
305 and this is the form you'll most often see Scheme definitions written in.
307 However, conceptually you should think backwards through the abbreviations and equivalences we've just presented.
313 (define bar (lambda (x) B))
317 (let* [(bar (lambda (x) B))] ... rest of the file or interactive session ...)
321 (lambda (bar) ... rest of the file or interactive session ...) (lambda (x) B)
323 or in other words, interpret the rest of the file or interactive session with `bar` assigned the function `(lambda (x) B)`.
328 You can override a binding with a more inner binding to the same variable. For instance the following expression in OCaml:
334 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:
339 <em>but it's not the same!</em> In the latter case we have mutation, in the former case we don't. You will learn to recognize the difference as we proceed.
341 The OCaml expression just means:
343 (fun x -> ((fun x -> x) 2) 3)
345 and there's no more mutation going on there than there is in:
347 <pre><code>∀x. (F x or ∀x (not (F x)))
350 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.
354 * [[!wikipedia Variable shadowing]]
357 Some more comparisons between Scheme and OCaml
358 ----------------------------------------------
360 * Simple predefined values
362 Numbers in Scheme: `2`, `3`
365 Booleans in Scheme: `#t`, `#f`
366 In OCaml: `true`, `false`
368 The eighth letter in the Latin alphabet, in Scheme: `#\h`
373 These are values which are built up out of (zero or more) simple values.
375 Ordered pairs in Scheme: `'(2 . 3)` or `(cons 2 3)`
378 Lists in Scheme: `'(2 3)` or `(list 2 3)`
380 We'll be explaining the difference between pairs and lists next week.
382 The empty list, in Scheme: `'()` or `(list)`
385 The string consisting just of the eighth letter of the Latin alphabet, in Scheme: `"h"`
388 A longer string, in Scheme: `"horse"`
391 A shorter string, in Scheme: `""`