X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=zipper-lists-continuations.mdwn;h=a826eed8b22de0061fee6d8fb653b5e340406c94;hp=f05ea007bedcc6866d4afed2699b7b8a91926d94;hb=3c9d5a19d05c749c3bf4e05874bcb153db3a18c6;hpb=54ec3f0279cf6af96b0265d03d536b4a42dc0856 diff --git a/zipper-lists-continuations.mdwn b/zipper-lists-continuations.mdwn index f05ea007..a826eed8 100644 --- a/zipper-lists-continuations.mdwn +++ b/zipper-lists-continuations.mdwn @@ -22,24 +22,24 @@ and monads). For instance, take the **Reader Monad**. Once we decide that the type constructor is - type 'a reader = fun e:env -> 'a + type 'a reader = env -> 'a then we can deduce the unit and the bind: - r_unit x:'a -> 'a reader = fun (e:env) -> x + let r_unit (x : 'a) : 'a reader = fun (e : env) -> x -Since the type of an `'a reader` is `fun e:env -> 'a` (by definition), -the type of the `r_unit` function is `'a -> e:env -> 'a`, which is a +Since the type of an `'a reader` is `env -> 'a` (by definition), +the type of the `r_unit` function is `'a -> env -> 'a`, which is a specific case of the type of the *K* combinator. So it makes sense that *K* is the unit for the reader monad. Since the type of the `bind` operator is required to be - r_bind:('a reader) -> ('a -> 'b reader) -> ('b reader) + r_bind : ('a reader) -> ('a -> 'b reader) -> ('b reader) We can deduce the correct `bind` function as follows: - r_bind (u:'a reader) (f:'a -> 'b reader):('b reader) = + let r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = We have to open up the `u` box and get out the `'a` object in order to feed it to `f`. Since `u` is a function from environments to @@ -58,28 +58,28 @@ This types to `env -> 'b reader`, but we want to end up with `env -> 'b`. Once again, the easiest way to turn a `'b reader` into a `'b` is to apply it to an environment. So we end up as follows: - r_bind (u:'a reader) (f:'a -> 'b reader):('b reader) = f (u e) e + r_bind (u : 'a reader) (f : 'a -> 'b reader) : ('b reader) = f (u e) e And we're done. -[This bind is a simplified version of the careful `let a = u e in ...` -constructions we provided in earlier lectures. We use the simplified -versions here in order to emphasize similarities of structure across -monads; the official bind is still the one with the plethora of `let`'s.] +[This bind is a condensed version of the careful `let a = u e in ...` +constructions we provided in earlier lectures. We use the condensed +version here in order to emphasize similarities of structure across +monads.] The **State Monad** is similar. We somehow intuit that we want to use the following type constructor: - type 'a state = 'store -> ('a, 'store) + type 'a state = store -> ('a, store) So our unit is naturally - let s_unit (x:'a):('a state) = fun (s:'store) -> (x, s) + let s_unit (x : 'a) : ('a state) = fun (s : store) -> (x, s) And we deduce the bind in a way similar to the reasoning given above. First, we need to apply `f` to the contents of the `u` box: - let s_bind (u:'a state) (f:'a -> ('b state)):('b state) = + let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state = But unlocking the `u` box is a little more complicated. As before, we need to posit a state `s` that we can apply `u` to. Once we do so, @@ -90,8 +90,8 @@ is an `'a`. So we have to unpack the pair: Abstracting over the `s` and adjusting the types gives the result: - let s_bind (u:'a state) (f:'a -> ('b state)):('b state) = - fun (s:state) -> let (a, s') = u s in f a s' + let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state = + fun (s : store) -> let (a, s') = u s in f a s' The **Option Monad** doesn't follow the same pattern so closely, so we won't pause to explore it here, though conceptually its unit and bind @@ -101,7 +101,7 @@ Our other familiar monad is the **List Monad**, which we were told looks like this: type 'a list = ['a];; - l_unit (x:'a) = [x];; + l_unit (x : 'a) = [x];; l_bind u f = List.concat (List.map f u);; Recall that `List.map` take a function and a list and returns the @@ -135,7 +135,7 @@ extra bits embdded in them: list with two elements: fun f z -> f 2 (f 1 z) list with three elements: fun f z -> f 3 (f 2 (f 1 z)) -and so on. To save time, we'll let the Ocaml interpreter infer the +and so on. To save time, we'll let the OCaml interpreter infer the principle types of these functions (rather than deducing what the types should be): @@ -166,22 +166,22 @@ ints), we have 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 +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 in order to proceed to build a monad: - l'_unit (x:'a):(('a, 'b) list) = fun x -> fun f z -> f x z + l'_unit (x : 'a) : ('a, 'b) list = fun x -> fun f z -> f x 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. - 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' = ... Unfortunately, we'll need to spell out the types: - l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b) - (f: 'a -> ('c -> 'd -> 'd) -> 'd -> 'd) + l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b) + (f : 'a -> ('c -> 'd -> 'd) -> 'd -> 'd) : ('c -> 'd -> 'd) -> 'd -> 'd = ... It's a rookie mistake to quail before complicated types. You should @@ -194,7 +194,7 @@ This time, `u` will only deliver up its contents if we give `u` as an argument a function expecting an `'a`. Once that argument is applied to an object of type `'a`, we'll have what we need. Thus: - .... u (fun (x:'a) -> ... (f a) ... ) ... + .... u (fun (a : 'a) -> ... (f a) ... ) ... In order for `u` to have the kind of argument it needs, we have to adjust `(f a)` (which has type `('c -> 'd -> 'd) -> 'd -> 'd`) in @@ -202,21 +202,21 @@ order to deliver something of type `'b -> 'b`. The easiest way is to alias `'d` to `'b`, and provide `(f a)` with an argument of type `'c -> 'b -> 'b`. Thus: - l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b) - (f: 'a -> ('c -> 'b -> 'b) -> 'b -> 'b) + l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b) + (f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b) : ('c -> 'b -> 'b) -> 'b -> 'b = - .... u (fun (x:'a) -> f a k) ... + .... u (fun (a : 'a) -> f a k) ... -[Excercise: can you arrive at a fully general bind for this type +[Exercise: can you arrive at a fully general bind for this type constructor, one that does not collapse `'d`'s with `'b`'s?] As usual, we have to abstract over `k`, but this time, no further adjustments are needed: - l'_bind (u: ('a -> 'b -> 'b) -> 'b -> 'b) - (f: 'a -> ('c -> 'b -> 'b) -> 'b -> 'b) + l'_bind (u : ('a -> 'b -> 'b) -> 'b -> 'b) + (f : 'a -> ('c -> 'b -> 'b) -> 'b -> 'b) : ('c -> 'b -> 'b) -> 'b -> 'b = - fun (k:'c -> 'b -> 'b) -> u (fun (x:'a) -> f a k) + fun (k : 'c -> 'b -> 'b) -> u (fun (a : 'a) -> f a k) You should carefully check to make sure that this term is consistent with the typing. @@ -230,16 +230,14 @@ replicating the behavior of the standard List monad. Let's test: l'_bind (fun f z -> f 1 (f 2 z)) (fun i -> fun f z -> f i (f (i+1) z)) ~~> -Sigh. Ocaml won't show us our own list. So we have to choose an `f` -and a `z` that will turn our hand-crafted lists into standard Ocaml +Sigh. OCaml won't show us our own list. So we have to choose an `f` +and a `z` that will turn our hand-crafted lists into standard OCaml lists, so that they will print out. -
-# let cons h t = h :: t;;  (* Ocaml is stupid about :: *)
-# l'_bind (fun f z -> f 1 (f 2 z)) 
-          (fun i -> fun f z -> f i (f (i+1) z)) cons [];;
-- : int list = [1; 2; 2; 3]
-
+ # let cons h t = h :: t;; (* OCaml is stupid about :: *) + # l'_bind (fun f z -> f 1 (f 2 z)) + (fun i -> fun f z -> f i (f (i+1) z)) cons [];; + - : int list = [1; 2; 2; 3] Ta da! @@ -263,7 +261,7 @@ generalized quantifier `fun pred -> pred j` of type `(e -> t) -> t`. Let's write a general function that will map individuals into their corresponding generalized quantifier: - gqize (x:e) = fun (p:e->t) -> p x + gqize (x : e) = fun (p : e -> t) -> p x This function wraps up an individual in a fancy box. That is to say, we are in the presence of a monad. The type constructor, the unit and @@ -271,12 +269,10 @@ the bind follow naturally. We've done this enough times that we won't belabor the construction of the bind function, the derivation is similar to the List monad just given: -
-type 'a continuation = ('a -> 'b) -> 'b
-c_unit (x:'a) = fun (p:'a -> 'b) -> p x
-c_bind (u:('a -> 'b) -> 'b) (f: 'a -> ('c -> 'd) -> 'd): ('c -> 'd) -> 'd =
-  fun (k:'a -> 'b) -> u (fun (x:'a) -> f x k)
-
+ type 'a continuation = ('a -> 'b) -> 'b + c_unit (x : 'a) = fun (p : 'a -> 'b) -> p x + c_bind (u : ('a -> 'b) -> 'b) (f : 'a -> ('c -> 'd) -> 'd) : ('c -> 'd) -> 'd = + fun (k : 'a -> 'b) -> u (fun (x : 'a) -> f x k) How similar is it to the List monad? Let's examine the type constructor and the terms from the list monad derived above: @@ -291,7 +287,7 @@ The unit and the bind for the Montague continuation monad and the homemade List monad are the same terms! In other words, the behavior of the List monad and the behavior of the continuations monad are parallel in a deep sense. To emphasize the parallel, we can -instantiate the type of the list' monad using the Ocaml list type: +instantiate the type of the list' monad using the OCaml list type: type 'a c_list = ('a -> 'a list) -> 'a list