X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=list_monad_as_continuation_monad.mdwn;h=d2136dbe097ca5ff2a2213ca191f4fc33f9d23a5;hp=b4158b1638d780b89e31bbe8a5dc18d9fa045e99;hb=bd008f9ae63ba84914d12e8c3e0973382cfd9b62;hpb=5c9b178bf3c8376b621fdf5c21c80af1a66ef56b diff --git a/list_monad_as_continuation_monad.mdwn b/list_monad_as_continuation_monad.mdwn index b4158b16..d2136dbe 100644 --- a/list_monad_as_continuation_monad.mdwn +++ b/list_monad_as_continuation_monad.mdwn @@ -10,7 +10,7 @@ Rethinking the list monad To construct a monad, the key element is to settle on a type constructor, and the monad more or less naturally follows from that. We'll remind you of some examples of how monads follow from the type -constructor in a moment. This will involve some review of familair +constructor in a moment. This will involve some review of familiar material, but it's worth doing for two reasons: it will set up a pattern for the new discussion further below, and it will tie together some previously unconnected elements of the course (more specifically, @@ -76,7 +76,7 @@ The **State Monad** is similar. Once we've decided to use the following type co Then our unit is naturally: - let s_unit (a : 'a) : ('a state) = fun (s : store) -> (a, s) + let s_unit (a : 'a) : 'a state = fun (s : store) -> (a, s) And we can reason our way to the bind function in a way similar to the reasoning given above. First, we need to apply `f` to the contents of the `u` box: @@ -114,7 +114,7 @@ result to applying the function to the elements of the list: List.map (fun i -> [i;i+1]) [1;2] ~~> [[1; 2]; [2; 3]] -and List.concat takes a list of lists and erases the embdded list +and `List.concat` takes a list of lists and erases the embedded list boundaries: List.concat [[1; 2]; [2; 3]] ~~> [1; 2; 2; 3] @@ -126,7 +126,7 @@ And sure enough, 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 @@ -138,20 +138,20 @@ 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 the object returned by the second argument of `bind` to always be of -type `'b list list`. We can elimiate that restriction by flattening +type `'b list list`. We can eliminate that restriction by flattening the list of lists into a single list: this is -just List.concat applied to the output of List.map. So there is some logic to the +just `List.concat` applied to the output of `List.map`. So there is some logic to the choice of unit and bind for the list monad. 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 +a primitive OCaml type. However, we know several ways of implementing +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 implementating lists). These were the lists that made -lists look like Church numerals with extra bits embdded in them: +strategy for implementing lists). These were the lists that made +lists look like Church numerals with extra bits embedded in them: empty list: fun f z -> z list with one element: fun f z -> f 1 z @@ -174,7 +174,7 @@ types should be ourselves): We can see what the consistent, general principle types are at the end, so we can stop. These types should remind you of the simply-typed lambda calculus types for Church numerals (`(o -> o) -> o -> o`) with one extra type -thrown in, the type of the element a the head of the list +thrown in, the type of the element at the head of the list (in this case, an int). So here's our type constructor for our hand-rolled lists: @@ -182,21 +182,21 @@ So here's our type constructor for our hand-rolled lists: type 'b list' = (int -> 'b -> 'b) -> 'b -> 'b Generalizing to lists that contain any kind of element (not just -ints), we have +`int`s), we have type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b 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' = ... @@ -206,14 +206,13 @@ Unpacking the types gives: (f : 'a -> ('c -> 'd -> 'd) -> 'd -> 'd) : ('c -> 'd -> 'd) -> 'd -> 'd = ... -Perhaps a bit intimiating. +Perhaps a bit intimidating. But it's a rookie mistake to quail before complicated types. You should -be no more intimiated by complex types than by a linguistic tree with +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`.