X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=zipper-lists-continuations.mdwn;h=edc3359d66015d9a0547d02ba7b35c644ff726cf;hp=a826eed8b22de0061fee6d8fb653b5e340406c94;hb=789245157230b7010000b0ce3290d45bd5a9d8fa;hpb=3c9d5a19d05c749c3bf4e05874bcb153db3a18c6 diff --git a/zipper-lists-continuations.mdwn b/zipper-lists-continuations.mdwn index a826eed8..edc3359d 100644 --- a/zipper-lists-continuations.mdwn +++ b/zipper-lists-continuations.mdwn @@ -24,76 +24,75 @@ constructor is type 'a reader = env -> 'a -then we can deduce the unit and the bind: +then the choice of unit and bind is natural: - let r_unit (x : 'a) : 'a reader = fun (e : env) -> x + let r_unit (a : 'a) : 'a reader = fun (e : env) -> 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 +specific case of the type of the *K* combinator. 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) -We can deduce the correct `bind` function as follows: +We can reason our way to the correct `bind` function as follows. We start by declaring the type: 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 +Now 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 objects of type `'a`, the way we open a box in this monad is by applying it to an environment: - .... f (u e) ... + ... f (u e) ... This subexpression types to `'b reader`, which is good. The only problem is that we invented an environment `e` that we didn't already have , so we have to abstract over that variable to balance the books: - fun e -> f (u e) ... + fun e -> f (u e) ... 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: +'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. +And we're done. This gives us a bind function of the right type. We can then check whether, in combination with the unit function we chose, it satisfies the monad laws, and behaves in the way we intend. And it does. -[This bind is a condensed version of the careful `let a = u e in ...` +[The bind we cite here 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: +The **State Monad** is similar. Once we've decided to use the following type constructor: type 'a state = store -> ('a, store) -So our unit is naturally +Then our unit is naturally: 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: +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: let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state = + ... f (...) ... 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, 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: - let s_bind (u : 'a state) (f : 'a -> 'b state) : 'b state = - fun (s : store) -> 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 +The **Option/Maybe Monad** doesn't follow the same pattern so closely, so we won't pause to explore it here, though conceptually its unit and bind follow just as naturally from its type constructor. @@ -139,25 +138,23 @@ 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): -
-# fun f z -> z;; -- : 'a -> 'b -> 'b =- -Finally, we're getting consistent principle types, 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 bit -thrown in (in this case, an int). + # fun f z -> z;; + - : 'a -> 'b -> 'b =-# fun f z -> f 1 z;; -- : (int -> 'a -> 'b) -> 'a -> 'b = -# fun f z -> f 2 (f 1 z);; -- : (int -> 'a -> 'a) -> 'a -> 'a = -# fun f z -> f 3 (f 2 (f 1 z)) -- : (int -> 'a -> 'a) -> 'a -> 'a = -