X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=list_monad_as_continuation_monad.mdwn;h=e4772f88888a50876a370d8af6c53d88ad1da608;hp=a97e7e361b836303604ca382be2fb1ee080ce4bb;hb=db35cd7468a6353ab2c15a1381edbc550a2ecc61;hpb=1c1ffe645eb7ae7dd3f76e9cb0d60c7eba8657e6
diff --git a/list_monad_as_continuation_monad.mdwn b/list_monad_as_continuation_monad.mdwn
index a97e7e36..e4772f88 100644
--- a/list_monad_as_continuation_monad.mdwn
+++ b/list_monad_as_continuation_monad.mdwn
@@ -88,7 +88,7 @@ need to posit a store `s` that we can apply `u` to. Once we do so,
however, we won't have an `'a`; we'll have a pair whose first element
is an `'a`. So we have to unpack the pair:
- ... let (a, s') = u s in ... (f a) ...
+ ... let (a, s') = u s in ... f a ...
Abstracting over the `s` and adjusting the types gives the result:
@@ -112,7 +112,7 @@ will provide a connection with continuations.
Recall that `List.map` takes a function and a list and returns the
result to applying the function to the elements of the list:
- List.map (fun i -> [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]]
+ List.map (fun i -> [i; i+1]) [1; 2] ~~> [[1; 2]; [2; 3]]
and `List.concat` takes a list of lists and erases the embedded list
boundaries:
@@ -121,12 +121,12 @@ boundaries:
And sure enough,
- l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
+ l_bind [1; 2] (fun i -> [i; i+1]) ~~> [1; 2; 2; 3]
Now, why this unit, and why this bind? Well, ideally a unit should
not throw away information, so we can rule out `fun x -> []` as an
ideal unit. And units should not add more information than required,
-so there's no obvious reason to prefer `fun x -> [x;x]`. In other
+so there's no obvious reason to prefer `fun x -> [x; x]`. In other
words, `fun x -> [x]` is a reasonable choice for a unit.
As for bind, an `'a list` monadic object contains a lot of objects of
@@ -136,7 +136,7 @@ thing we know for sure we can do with an object of type `'a` is apply
the function of type `'a -> 'a list` to them. Once we've done so, we
have a collection of lists, one for each of the `'a`'s. One
possibility is that we could gather them all up in a list, so that
-`bind' [1;2] (fun i -> [i;i]) ~~> [[1;1];[2;2]]`. But that restricts
+`bind' [1; 2] (fun i -> [i; i]) ~~> [[1; 1]; [2; 2]]`. But that restricts
the object returned by the second argument of `bind` to always be of
type `'b list list`. We can eliminate that restriction by flattening
the list of lists into a single list: this is
@@ -147,7 +147,7 @@ Yet we can still desire to go deeper, and see if the appropriate bind
behavior emerges from the types, as it did for the previously
considered monads. But we can't do that if we leave the list type as
a primitive OCaml type. However, we know several ways of implementing
-lists using just functions. In what follows, we're going to use type
+lists using just functions. In what follows, we're going to use version
3 lists, the right fold implementation (though it's important and
intriguing to wonder how things would change if we used some other
strategy for implementing lists). These were the lists that made
@@ -189,16 +189,16 @@ Generalizing to lists that contain any kind of element (not just
So an `('a, 'b) list'` is a list containing elements of type `'a`,
where `'b` is the type of some part of the plumbing. This is more
general than an ordinary OCaml list, but we'll see how to map them
-into OCaml lists soon. We don't need to fully grasp the role of the `'b`'s
+into OCaml lists soon. We don't need to fully grasp the role of the `'b`s
in order to proceed to build a monad:
l'_unit (a : 'a) : ('a, 'b) list = fun a -> fun k z -> k a z
-No problem. Arriving at bind is a little more complicated, but
-exactly the same principles apply, you just have to be careful and
-systematic about it.
+Take an `'a` and return its v3-style singleton. No problem. Arriving at bind
+is a little more complicated, but exactly the same principles apply, you just
+have to be careful and systematic about it.
- l'_bind (u : ('a,'b) list') (f : 'a -> ('c, 'd) list') : ('c, 'd) list' = ...
+ l'_bind (u : ('a, 'b) list') (f : 'a -> ('c, 'd) list') : ('c, 'd) list' = ...
Unpacking the types gives:
@@ -212,22 +212,21 @@ be no more intimidated by complex types than by a linguistic tree with
deeply embedded branches: complex structure created by repeated
application of simple rules.
-[This would be a good time to try to build your own term for the types
-just given. Doing so (or attempting to do so) will make the next
+[This would be a good time to try to reason your way to your own term having the type just specified. Doing so (or attempting to do so) will make the next
paragraph much easier to follow.]
As usual, we need to unpack the `u` box. Examine the type of `u`.
This time, `u` will only deliver up its contents if we give `u` an
argument that is a function expecting an `'a` and a `'b`. `u` will
-fold that function over its type `'a` members, and that's how we'll get the `'a`s we need. Thus:
+fold that function over its type `'a` members, and that's where we can get at the `'a`s we need. Thus:
... u (fun (a : 'a) (b : 'b) -> ... f a ... ) ...
-In order for `u` to have the kind of argument it needs, the `... (f a) ...` has to evaluate to a result of type `'b`. The easiest way to do this is to collapse (or "unify") the types `'b` and `'d`, with the result that `f a` will have type `('c -> 'b -> 'b) -> 'b -> 'b`. Let's postulate an argument `k` of type `('c -> 'b -> 'b)` and supply it to `(f a)`:
+In order for `u` to have the kind of argument it needs, the `fun a b -> ... f a ...` has to have type `'a -> 'b -> 'b`; so the `... f a ...` has to evaluate to a result of type `'b`. The easiest way to do this is to collapse (or "unify") the types `'b` and `'d`, with the result that `f a` will have type `('c -> 'b -> 'b) -> 'b -> 'b`. Let's postulate an argument `k` of type `('c -> 'b -> 'b)` and supply it to `f a`:
... u (fun (a : 'a) (b : 'b) -> ... f a k ... ) ...
-Now we have an argument `b` of type `'b`, so we can supply that to `(f a) k`, getting a result of type `'b`, as we need:
+Now the function we're supplying to `u` also receives an argument `b` of type `'b`, so we can supply that to `f a k`, getting a result of type `'b`, as we need:
... u (fun (a : 'a) (b : 'b) -> f a k b) ...
@@ -252,8 +251,8 @@ Now let's think about what this does. It's a wrapper around `u`. In order to beh
Suppose we have a list' whose contents are `[1; 2; 4; 8]`---that is, our list' will be `fun f z -> f 1 (f 2 (f 4 (f 8 z)))`. We call that list' `u`. Suppose we also have a function `f` that for each `int` we give it, gives back a list of the divisors of that `int` that are greater than 1. Intuitively, then, binding `u` to `f` should give us:
- concat (map f u) =
- concat [[]; [2]; [2; 4]; [2; 4; 8]] =
+ List.concat (List.map f u) =
+ List.concat [[]; [2]; [2; 4]; [2; 4; 8]] =
[2; 2; 4; 2; 4; 8]
Or rather, it should give us a list' version of that, which takes a function `k` and value `z` as arguments, and returns the right fold of `k` and `z` over those elements. What does our formula
@@ -272,18 +271,18 @@ do? Well, for each element `a` in `u`, it applies `f` to that `a`, getting one o
So if, for example, we let `k` be `+` and `z` be `0`, then the computation would proceed:
0 ==>
- right-fold + and 0 over [2; 4; 8] = ((2+4+8+0) ==>
- right-fold + and 2+4+8+0 over [2; 4] = 2+4+(2+4+8+0) ==>
- right-fold + and 2+4+2+4+8+0 over [2] = 2+(2+4+(2+4+8+0)) ==>
- right-fold + and 2+2+4+2+4+8+0 over [] = 2+(2+4+(2+4+8+0))
+ right-fold + and 0 over [2; 4; 8] = 2+4+8+(0) ==>
+ right-fold + and 2+4+8+0 over [2; 4] = 2+4+(2+4+8+(0)) ==>
+ right-fold + and 2+4+2+4+8+0 over [2] = 2+(2+4+(2+4+8+(0))) ==>
+ right-fold + and 2+2+4+2+4+8+0 over [] = 2+(2+4+(2+4+8+(0)))
-which indeed is the result of right-folding + and 0 over `[2; 2; 4; 2; 4; 8]`. If you trace through how this works, you should be able to persuade yourself that our formula:
+which indeed is the result of right-folding `+` and `0` over `[2; 2; 4; 2; 4; 8]`. If you trace through how this works, you should be able to persuade yourself that our formula:
fun k z -> u (fun a b -> f a k b) z
-will deliver just the same folds, for arbitrary choices of `k` and `z` (with the right types), and arbitrary list's `u` and appropriately-typed `f`s, as
+will deliver just the same folds, for arbitrary choices of `k` and `z` (with the right types), and arbitrary `list'`s `u` and appropriately-typed `f`s, as
- fun k z -> List.fold_right k (concat (map f u)) z
+ fun k z -> List.fold_right k (List.concat (List.map f u)) z
would.
@@ -293,7 +292,7 @@ For future reference, we might make two eta-reductions to our formula, so that w
Let's make some more tests:
- l_bind [1;2] (fun i -> [i, i+1]) ~~> [1; 2; 2; 3]
+ l_bind [1; 2] (fun i -> [i; i+1]) ~~> [1; 2; 2; 3]
l'_bind (fun f z -> f 1 (f 2 z))
(fun i -> fun f z -> f i (f (i+1) z)) ~~>
@@ -358,6 +357,8 @@ This can be eta-reduced to:
let l'_unit a = fun k -> k a
+and:
+
let l'_bind u f =
(* we mentioned three versions of this, the fully eta-expanded: *)
fun k z -> u (fun a b -> f a k b) z