X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=list_monad_as_continuation_monad.mdwn;h=81a5be318fafedf7a6bca8dad20bb8b6dcfb99e1;hp=8d442ae0d08ed95daff2200ae3761656d537bae7;hb=8105e467b373e9da58fa80c748c97b1d485986c3;hpb=5e8eadce75352b7c2ec2ad78c8c92b89bd6ee778 diff --git a/list_monad_as_continuation_monad.mdwn b/list_monad_as_continuation_monad.mdwn index 8d442ae0..81a5be31 100644 --- a/list_monad_as_continuation_monad.mdwn +++ b/list_monad_as_continuation_monad.mdwn @@ -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 embedded 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 @@ -140,14 +140,14 @@ possibility is that we could gather them all up in a list, so that 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 -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 +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 @@ -189,14 +189,14 @@ 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' = ...