X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week1.mdwn;h=b581db1b5429c6ae3559b6fa2c2cc2e86af6598b;hp=5018c711cd694565b0da85e9cb7181fb999bf2d3;hb=9f92c70ab51c8e2bcc57f5eac7c31841984142b9;hpb=59081b2acb15b435e9a486ce18bc85c1abe232c2 diff --git a/week1.mdwn b/week1.mdwn index 5018c711..b581db1b 100644 --- a/week1.mdwn +++ b/week1.mdwn @@ -1,4 +1,6 @@ -Here's what we did in seminar on Monday 9/13, (Sometimes these notes will expand on things mentioned only briefly in class, or discuss useful tangents that didn't even make it into class.) +Here's what we did in seminar on Monday 9/13, + +Sometimes these notes will expand on things mentioned only briefly in class, or discuss useful tangents that didn't even make it into class. These notes expand on *a lot*, and some of this material will be reviewed next week. Applications ============ @@ -12,7 +14,7 @@ From linguistics * (Chris: fill in other applications...) -* expressives -- at the end of the seminar we gave a demonstration of modeling [[damn]] using continuations...see the linked summary for more explanation and elaboration +* expressives -- at the end of the seminar we gave a demonstration of modeling [[damn]] using continuations...see the [summary](/damn) for more explanation and elaboration From philosophy --------------- @@ -35,7 +37,7 @@ Declarative/functional vs Imperatival/dynamic models of computation Many of you, like us, will have grown up thinking the paradigm of computation is a sequence of changes. Let go of that. It will take some care to separate the operative notion of "sequencing" here from other notions close to it, but once that's done, you'll see that languages that have no significant notions of sequencing or changes are Turing complete: they can perform any computation we know how to describe. In itself, that only puts them on equal footing with more mainstream, imperatival programming languages like C and Java and Python, which are also Turing complete. But further, the languages we want you to become familiar with can reasonably be understood to be more fundamental. They embody the elemental building blocks that computer scientists use when reasoning about and designing other languages. -Jim offered the metaphor: think of imperatival languages, which include "mutation" and "side-effects" (we'll flesh out these keywords as we proceeed), as the pate of computation. We want to teach you about the meat and potatoes, where as it turns out there is no sequencing and no changes. There's just the evaluation or simplification of complex expressions. +Jim offered the metaphor: think of imperatival languages, which include "mutation" and "side-effects" (we'll flesh out these keywords as we proceeed), as the pâté of computation. We want to teach you about the meat and potatoes, where as it turns out there is no sequencing and no changes. There's just the evaluation or simplification of complex expressions. Now, when you ask the Scheme interpreter to simplify an expression for you, that's a kind of dynamic interaction between you and the interpreter. You may wonder then why these languages should not also be understood imperatively. The difference is that in a purely declarative or functional language, there are no dynamic effects in the language itself. It's just a static semantic fact about the language that one expression reduces to another. You may have verified that fact through your dynamic interactions with the Scheme interpreter, but that's different from saying that there are dynamic effects in the language itself. @@ -47,7 +49,7 @@ For example, you'll encounter the claim that declarative languages are distingui The notion of **function** that we'll be working with will be one that, by default, sometimes counts as non-identical functions that map all their inputs to the very same outputs. For example, two functions from jumbled decks of cards to sorted decks of cards may use different algorithms and hence be different functions. -It's possible to enhance the lambda calculus so that functions do get identified when they map all the same inputs to the same outputs. This is called making the calculus **extensional**. Church called languages which didn't do this "intensional." If you try to understand this in terms of functions from worlds to extensions (an idea also associated with Church), you will hurt yourself. So too if you try to understand it in terms of mental stereotypes, another notion sometimes designated by "intension." +It's possible to enhance the lambda calculus so that functions do get identified when they map all the same inputs to the same outputs. This is called making the calculus **extensional**. Church called languages which didn't do this **intensional**. If you try to understand that kind of "intensionality" in terms of functions from worlds to extensions (an idea also associated with Church), you may hurt yourself. So too if you try to understand it in terms of mental stereotypes, another notion sometimes designated by "intension." It's often said that dynamic systems are distinguished because they are the ones in which **order matters**. However, there are many ways in which order can matter. If we have a trivalent boolean system, for example---easily had in a purely functional calculus---we might choose to give a truth-table like this for "and": @@ -407,7 +409,7 @@ Here's how it looks to say the same thing in various of these languages. which just means: - (let [(bar (lambda (x) B))] ... rest of the file or interactive session ...) + (let* [(bar (lambda (x) B))] ... rest of the file or interactive session ...) which just means: @@ -429,17 +431,17 @@ Here's how it looks to say the same thing in various of these languages. 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. + 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 change of state going on here than there is in: + and there's no more mutation going on there than there is in: -
- &exists;x. (F x and &exists;x (not (F x))) -
+ 
+	∀x. (F x or ∀x (not (F x)))
+
Some more comparisons between Scheme and OCaml @@ -447,47 +449,47 @@ Some more comparisons between Scheme and OCaml 11. Simple predefined values - Numbers in Scheme: 2, 3 - In OCaml: 2, 3 + Numbers in Scheme: `2`, `3` + In OCaml: `2`, `3` - Booleans in Scheme: #t, #f - In OCaml: true, false + Booleans in Scheme: `#t`, `#f` + In OCaml: `true`, `false` - The eighth letter in the Latin alphabet, in Scheme: #\h - In OCaml: 'h' + 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) + Ordered pairs in Scheme: `'(2 . 3)` + In OCaml: `(2, 3)` - Lists 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 empty list, in Scheme: `'()` + In OCaml: `[]` - The string consisting just of the eighth letter of the Latin alphabet, in Scheme: "h" - In OCaml: "h" + 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 longer string, in Scheme: `"horse"` + In OCaml: `"horse"` - A shorter string, in Scheme: "" - In OCaml: "" + A shorter string, in Scheme: `""` + In OCaml: `""` 13. Function application - Binary functions in OCaml: foo 2 3 + Binary functions in OCaml: `foo 2 3` - Or: ( + ) 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. + 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. + 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.