lists-monad tweaks

[lambda.git] / list_monad_as_continuation_monad.mdwn

diff --git a/list_monad_as_continuation_monad.mdwn b/list_monad_as_continuation_monad.mdwn
index 8d442ae..d2136db 100644 (file)
--- 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:
 
 
 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:
 
 
 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]]
 
 
        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]
 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,
 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
 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
 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
 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
 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
 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
 
 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'  = ...
 


@@ -212,8 +212,7 @@ 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.
 
 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`.
 paragraph much easier to follow.]
 
 As usual, we need to unpack the `u` box.  Examine the type of `u`.