X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week7.mdwn;h=7544425c4bbcca2c3a1b72cf0cf088199a18586f;hp=8daff8a09a02adbc31a74a7a40465597d8b42e7f;hb=d3ea4212bbca7a47f8aae537023852fd9a214389;hpb=0a621c3ef165f707ce3af6ca684fe5f8f8378316 diff --git a/week7.mdwn b/week7.mdwn index 8daff8a0..7544425c 100644 --- a/week7.mdwn +++ b/week7.mdwn @@ -58,12 +58,11 @@ operations. So we have to jack up the types of the inputs:
``` let div' (u:int option) (v:int option) =
-  match v with
+  match u with
None -> None
-    | Some 0 -> None
-	| Some y -> (match u with
-					  None -> None
-                    | Some x -> Some (x / y));;
+	| Some x -> (match v with
+				  Some 0 -> None
+				| Some y -> Some (x / y));;

(*
val div' : int option -> int option -> int option =
@@ -165,7 +164,7 @@ theory of accommodation, and a theory of the situations in which
material within the sentence can satisfy presuppositions for other
material that otherwise would trigger a presupposition violation; but,
not surprisingly, these refinements will require some more
-sophisticated techniques than the super-simple option monad.]
+sophisticated techniques than the super-simple Option monad.]

@@ -228,7 +227,7 @@ that provides at least the following three elements:

So `unit` is a way to put something inside of a monadic box. It's crucial
to the usefulness of monads that there will be monadic boxes that
-	aren't the result of that operation. In the option/maybe monad, for
+	aren't the result of that operation. In the Option/Maybe monad, for
instance, there's also the empty box `None`. In another (whimsical)
example, you might have, in addition to boxes merely containing integers,
special boxes that contain integers and also sing a song when they're opened.
@@ -237,9 +236,9 @@ that provides at least the following three elements:
most straightforward way to lift an ordinary value into a monadic value
of the monadic type in question.

-*	Thirdly, an operation that's often called `bind`. This is another
+*	Thirdly, an operation that's often called `bind`. As we said before, this is another
unfortunate name: this operation is only very loosely connected to
-	what linguists usually mean by "binding." In our option/maybe monad, the
+	what linguists usually mean by "binding." In our Option/Maybe monad, the
bind operation is:

let bind u f = match u with None -> None | Some x -> f x;;
@@ -261,7 +260,7 @@ that provides at least the following three elements:

The guts of the definition of the `bind` operation amount to
specifying how to unbox the monadic value `u`.  In the `bind`
-	operator for the option monad, we unboxed the monadic value by
+	operator for the Option monad, we unboxed the monadic value by
matching it with the pattern `Some x`---whenever `u`
happened to be a box containing an integer `x`, this allowed us to
get our hands on that `x` and feed it to `f`.
@@ -282,7 +281,7 @@ that provides at least the following three elements:
For each new monadic type, this has to be worked out in an
useful way.

-So the "option/maybe monad" consists of the polymorphic `option` type, the
+So the "Option/Maybe monad" consists of the polymorphic `option` type, the
`unit`/return function, and the `bind` function.

@@ -314,12 +313,12 @@ singing box (the end result of evaluting `u`) and bind the variable
`x`".

(Note that the above "do" notation comes from Haskell. We're mentioning it here
+because you're likely to see it when reading about monads. (See our page on [[Translating between OCaml Scheme and Haskell]].) It won't work in
OCaml. In fact, the `<-` symbol already means something different in OCaml,
having to do with mutable record fields. We'll be discussing mutation someday
soon.)

-As we proceed, we'll be seeing a variety of other monad systems. For example, another monad is the list monad. Here the monadic type is:
+As we proceed, we'll be seeing a variety of other monad systems. For example, another monad is the List monad. Here the monadic type is:

# type 'a list

@@ -346,7 +345,7 @@ of `'b list`s into a single `'b list`:
# List.concat [[1]; [1;2]; [1;3]; [1;2;4]]
- : int list = [1; 1; 2; 1; 3; 1; 2; 4]

monadic system, there has to be a specification of the complex monad type,
which will be parameterized on some simpler type `'a`, and the `unit`/return
operation, and the `bind` operation. These will be different for different
@@ -366,8 +365,8 @@ Just like good robots, monads must obey three laws designed to prevent
them from hurting the people that use them or themselves.

*	**Left identity: unit is a left identity for the bind operation.**
-	That is, for all `f:'a -> 'a m`, where `'a m` is a monadic
-	type, we have `(unit x) * f == f x`.  For instance, `unit` is itself
+	That is, for all `f:'a -> 'b m`, where `'b m` is a monadic
+	type, we have `(unit x) >>= f == f x`.  For instance, `unit` is itself
a function of type `'a -> 'a m`, so we can use it for `f`:

# let unit x = Some x;;
@@ -377,14 +376,17 @@ them from hurting the people that use them or themselves.

The parentheses is the magic for telling OCaml that the
function to be defined (in this case, the name of the function
-	is `*`, pronounced "bind") is an infix operator, so we write
-	`u * f` or `( * ) u f` instead of `* u f`. Now:
+	is `>>=`, pronounced "bind") is an infix operator, so we write
+	`u >>= f` or equivalently `( >>= ) u f` instead of `>>= u
+	f`.

# unit 2;;
- : int option = Some 2
# unit 2 >>= unit;;
- : int option = Some 2

+	Now, for a less trivial instance of a function from `int`s to `int option`s:
+
# let divide x y = if 0 = y then None else Some (x/y);;
val divide : int -> int -> int option =
# divide 6 2;;
@@ -400,12 +402,19 @@ them from hurting the people that use them or themselves.

*	**Associativity: bind obeys a kind of associativity**. Like this:

-		(u >>= f) >>= g == u >>= (fun x -> f x >>= g)
+		(u >>= f) >>= g  ==  u >>= (fun x -> f x >>= g)
+
+	If you don't understand why the lambda form is necessary (the
+	"fun x -> ..." part), you need to look again at the type of `bind`.

-	If you don't understand why the lambda form is necessary (the "fun
-	x" part), you need to look again at the type of `bind`.
+	Wadler and others try to make this look nicer by phrasing it like this,
+	where U, V, and W are schematic for any expressions with the relevant monadic type:

-	Some examples of associativity in the option monad:
+		(U >>= fun x -> V) >>= fun y -> W  ==  U >>= fun x -> (V >>= fun y -> W)
+
+	Some examples of associativity in the Option monad (bear in
+	mind that in the Ocaml implementation of integer division, 2/3
+	evaluates to zero, throwing away the remainder):

# Some 3 >>= unit >>= unit;;
- : int option = Some 3
@@ -422,15 +431,15 @@ them from hurting the people that use them or themselves.
# Some 3 >>= (fun x -> divide 2 x >>= divide 6);;
- : int option = None

-Of course, associativity must hold for *arbitrary* functions of
-type `'a -> 'a m`, where `m` is the monad type.  It's easy to
-convince yourself that the `bind` operation for the option monad
-obeys associativity by dividing the inputs into cases: if `u`
-matches `None`, both computations will result in `None`; if
-`u` matches `Some x`, and `f x` evalutes to `None`, then both
-computations will again result in `None`; and if the value of
-`f x` matches `Some y`, then both computations will evaluate
-to `g y`.
+	Of course, associativity must hold for *arbitrary* functions of
+	type `'a -> 'b m`, where `m` is the monad type.  It's easy to
+	convince yourself that the `bind` operation for the Option monad
+	obeys associativity by dividing the inputs into cases: if `u`
+	matches `None`, both computations will result in `None`; if
+	`u` matches `Some x`, and `f x` evalutes to `None`, then both
+	computations will again result in `None`; and if the value of
+	`f x` matches `Some y`, then both computations will evaluate
+	to `g y`.

*	**Right identity: unit is a right identity for bind.**  That is,
`u >>= unit == u` for all monad objects `u`.  For instance,
If you studied algebra, you'll remember that a *monoid* is an
associative operation with a left and right identity.  For instance,
the natural numbers along with multiplication form a monoid with 1
-serving as the left and right identity.  That is, temporarily using
-`*` to mean arithmetic multiplication, `1 * u == u == u * 1` for all
+serving as the left and right identity.  That is, `1 * u == u == u * 1` for all
`u`, and `(u * v) * w == u * (v * w)` for all `u`, `v`, and `w`.  As
presented here, a monad is not exactly a monoid, because (unlike the
arguments of a monoid operation) the two arguments of the bind are of
different types.  But it's possible to make the connection between
monads and monoids much closer. This is discussed in [Monads in Category
+
+
+

Here are some papers that introduced monads into functional programming:

-*	[Eugenio Moggi, Notions of Computation and Monads](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf): Information and Computation 93 (1) 1991.
+*	[Eugenio Moggi, Notions of Computation and Monads](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf): Information and Computation 93 (1) 1991. Would be very difficult reading for members of this seminar. However, the following two papers should be accessible.
+
+invited talk, *19'th Symposium on Principles of Programming Languages*, ACM Press, Albuquerque, January 1992.
+

in M. Broy, editor, *Marktoberdorf Summer School on Program Design
Calculi*, Springer Verlag, NATO ASI Series F: Computer and systems
sciences, Volume 118, August 1992. Also in J. Jeuring and E. Meijer,
editors, *Advanced Functional Programming*, Springer Verlag,
-LNCS 925, 1995. Some errata fixed August 2001.  This paper has a great first
-line: **Shall I be pure, or impure?**
+LNCS 925, 1995. Some errata fixed August 2001.

-invited talk, *19'th Symposium on Principles of Programming Languages*, ACM Press, Albuquerque, January 1992.
-
-
-*	[Daniel Friedman. A Schemer's View of Monads](/schemersviewofmonads.ps): from  but the link above is to a local copy.

-There's a long list of monad tutorials on the [[Offsite Reading]] page. Skimming the titles makes me laugh.
+There's a long list of monad tutorials on the [[Offsite Reading]] page. (Skimming the titles is somewhat amusing.) If you are confused by monads, make use of these resources. Read around until you find a tutorial pitched at a level that's helpful for you.

In the presentation we gave above---which follows the functional programming conventions---we took `unit`/return and `bind` as the primitive operations. From these a number of other general monad operations can be derived. It's also possible to take some of the others as primitive. The [Monads in Category

Here are some of the other general monad operations. You don't have to master these; they're collected here for your reference.

@@ -499,12 +514,12 @@ that is:

You could also do `bind u (fun x -> v)`; we use the `_` for the function argument to be explicit that that argument is never going to be used.

-The `lift` operation we asked you to define for last week's homework is a common operation. The second argument to `bind` converts `'a` values into `'b m` values---that is, into instances of the monadic type. What if we instead had a function that merely converts `'a` values into `'b` values, and we want to use it with our monadic type. Then we "lift" that function into an operation on the monad. For example:
+The `lift` operation we asked you to define for last week's homework is a common operation. The second argument to `bind` converts `'a` values into `'b m` values---that is, into instances of the monadic type. What if we instead had a function that merely converts `'a` values into `'b` values, and we want to use it with our monadic type? Then we "lift" that function into an operation on the monad. For example:

# let even x = (x mod 2 = 0);;
val g : int -> bool =

-`even` has the type `int -> bool`. Now what if we want to convert it into an operation on the option/maybe monad?
+`even` has the type `int -> bool`. Now what if we want to convert it into an operation on the Option/Maybe monad?

# let lift g = fun u -> bind u (fun x -> Some (g x));;
val lift : ('a -> 'b) -> 'a option -> 'b option =
@@ -517,7 +532,7 @@ also define a lift operation for binary functions:

`lift2 (+)` will now be a function from `int option`s  and `int option`s to `int option`s. This should look familiar to those who did the homework.

-The `lift` operation (just `lift`, not `lift2`) is sometimes also called the `map` operation. (In Haskell, they say `fmap` or `<\$>`.) And indeed when we're working with the list monad, `lift f` is exactly `List.map f`!
+The `lift` operation (just `lift`, not `lift2`) is sometimes also called the `map` operation. (In Haskell, they say `fmap` or `<\$>`.) And indeed when we're working with the List monad, `lift f` is exactly `List.map f`!

Wherever we have a well-defined monad, we can define a lift/map operation for that monad. The examples above used `Some (g x)` and so on; in the general case we'd use `unit (g x)`, using the specific `unit` operation for the monad we're working with.

@@ -544,7 +559,7 @@ and so on. Here are the laws that any `ap` operation can be relied on to satisfy
ap (unit f) (unit x) = unit (f x)
ap u (unit x) = ap (unit (fun f -> f x)) u

-Another general monad operation is called `join`. This is the operation that takes you from an iterated monad to a single monad. Remember when we were explaining the `bind` operation for the list monad, there was a step where
+Another general monad operation is called `join`. This is the operation that takes you from an iterated monad to a single monad. Remember when we were explaining the `bind` operation for the List monad, there was a step where
we went from:

[[1]; [1;2]; [1;3]; [1;2;4]]
@@ -574,15 +589,15 @@ Monad outlook
-------------

We're going to be using monads for a number of different things in the
-weeks to come.  The first main application will be the State monad,
+weeks to come.  One major application will be the State monad,
which will enable us to model mutation: variables whose values appear
to change as the computation progresses.  Later, we will study the