From f9a1fddf57a6ca3b7ea69974061e071a40f371ec Mon Sep 17 00:00:00 2001 From: Jim Pryor Date: Wed, 15 Sep 2010 23:08:05 -0400 Subject: [PATCH] week1: fix markup processing? Signed-off-by: Jim Pryor --- test2.mdwn | 432 ------------------------------------------------------------- 1 file changed, 432 deletions(-) diff --git a/test2.mdwn b/test2.mdwn index c6d64acc..58a2a03f 100644 --- a/test2.mdwn +++ b/test2.mdwn @@ -356,435 +356,3 @@ combinatorial logic   - -Rosetta Stone -============= - -Here's how it looks to say the same thing in various of these languages. - -1. Binding suitable values to the variables `three` and `two`, and adding them. - - In Scheme: - - (let* ((three 3)) - (let ((two 2)) - (+ three two))) - - In OCaml: - - let three = 3 in - let two = 2 in - three + two - - Notice OCaml lets you write the `+` in between the `three` and `two`, as you're accustomed to. However most functions need to come leftmost, even if they're binary. And you can do this with `+` too, if you enclose it in parentheses so that the OCaml parser doesn't get confused by your syntax: - - 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; - -2. 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. - - - - - -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. - -7. 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, as we'll see, is equivalent 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 - -8. 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.) - - -9. 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)`. - - -10. Shadowing - - 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 ----------------------------------------------- - -11. 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'` - -12. Compound values - - These are values which are built up out of (zero or more) simple values. - - Ordered pairs in Scheme: `'(2 . 3)` - In OCaml: `(2, 3)` - - Lists in Scheme: `'(2 3)` - In OCaml: `[2; 3]` - We'll be explaining the difference between pairs and lists next week. - - The empty list, in Scheme: `'()` - 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: `""` - -13. Function application - - Binary functions in OCaml: `foo 2 3` - - Or: `( + ) 2 3` - - These are the same as: `((foo 2) 3)`. In other words, functions in OCaml 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. - - 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. - - Additionally, as said above, 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. - - -What "sequencing" is and isn't ------------------------------- - -We mentioned before the idea that computation is a sequencing of some changes. I said we'd be discussing (fragments of, and in some cases, entire) languages that have no native notion of change. - -Neither do they have any useful notion of sequencing. But what this would be takes some care to identify. - -First off, the mere concatenation of expressions isn't what we mean by sequencing. Concatenation of expressions is how you build syntactically complex expressions out of simpler ones. The complex expressions often express a computation where a function is applied to one (or more) arguments, - -Second, the kind of rebinding we called "shadowing" doesn't involve any changes or sequencing. All the precedence facts about that kind of rebinding are just consequences of the compound syntactic structures in which it occurs. - -Third, the kinds of bindings we see in: - - (define foo A) - (foo 2) - -Or even: - - (define foo A) - (define foo B) - (foo 2) - -don't involve any changes or sequencing in the sense we're trying to identify. As we said, these programs are just syntactic variants of (single) compound syntactic structures involving "let"s and "lambda"s. - -Since Scheme and OCaml also do permit imperatival constructions, they do have syntax for genuine sequencing. In Scheme it looks like this: - - (begin A B C) - -In OCaml it looks like this: - - begin A; B; C end - -Or this: - - (A; B; C) - -In the presence of imperatival elements, sequencing order is very relevant. For example, these will behave differently: - - (begin (print "under") (print "water")) - - (begin (print "water") (print "under")) - -And so too these: - - begin x := 3; x := 2; x end - - begin x := 2; x := 3; x end - -However, if A and B are purely functional, non-imperatival expressions, then: - - begin A; B; C end - -just evaluates to C (so long as A and B evaluate to something at all). So: - - begin A; B; C end - -contributes no more to a larger context in which it's embedded than C does. This is the sense in which functional languages have no serious notion of sequencing. - -We'll discuss this more as the seminar proceeds. -- 2.11.0