From 84c6bcbe5d95c8fe3d272a1d4ebd02dedee8b87c Mon Sep 17 00:00:00 2001 From: Jim Pryor Date: Wed, 15 Sep 2010 23:09:33 -0400 Subject: [PATCH] week1: fix markup processing? Signed-off-by: Jim Pryor --- test2.mdwn | 114 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 114 insertions(+) diff --git a/test2.mdwn b/test2.mdwn index 8ee80b44..138ee500 100644 --- a/test2.mdwn +++ b/test2.mdwn @@ -419,3 +419,117 @@ Here's how it looks to say the same thing in various of these languages. 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. + + 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. + + In the lambda calculus: + + (\x M) + + is always anonymous! Here `M` stands for any expression of the language, simple or complex. 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 this is an anonymous function that adds 3 to whatever argument it's given: + + (lambda (x) (+ 3 x)) + + Scheme uses a lot of parentheses, and they are always significant, never optional. In `(+ 3 x)` the parentheses mean "apply the function `+` to the arguments `3` and `x`. In `(lambda (x) ...)` the parentheses have a different meaning: they mark where the anonymous function you're defining begins and ends, and so on. As you'll see, parentheses have yet further roles in Scheme. I know it's confusing. + + 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 + + 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`. + + In OCaml: + + (fun x -> ( + ) 3 x) 2 + + +5. 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 `(...)`. + + Someone 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))) + + 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. + + 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 + +6. 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. + + -- 2.11.0