X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=from_lists_to_continuations.mdwn;h=f2e6989d14e362e2f9b10e2c5d5d07b08da49ae7;hp=0b3f4643cbcb890edd85565be664670dc38de7d6;hb=9c7ca26e6dd0c1ce1b6cd653e27a083b7379a5dd;hpb=cd3b0839319cccf085142a33c82c32df1bcf69d4 diff --git a/from_lists_to_continuations.mdwn b/from_lists_to_continuations.mdwn index 0b3f4643..f2e6989d 100644 --- a/from_lists_to_continuations.mdwn +++ b/from_lists_to_continuations.mdwn @@ -1,4 +1,3 @@ - Refunctionalizing zippers: from lists to continuations ------------------------------------------------------ @@ -148,10 +147,11 @@ The recipe for constructing the list goes like this: What is the type of each of these steps? Well, it will be a function from the result of the previous step (a list) to a new list: it will be a function of type `char list -> char list`. We'll call each step -a **continuation** of the recipe. So in this context, a continuation -is a function of type `char list -> char list`. For instance, the -continuation corresponding to the portion of the recipe below the -horizontal line is the function `fun (tail:char list) -> a::(b::tail)`. +(or group of steps) a **continuation** of the recipe. So in this +context, a continuation is a function of type `char list -> char +list`. For instance, the continuation corresponding to the portion of +the recipe below the horizontal line is the function `fun (tail:char +list) -> a::(b::tail)`. This means that we can now represent the unzipped part of our zipper--the part we've already unzipped--as a continuation: a function @@ -194,42 +194,44 @@ what the parallel would suggest. The reason is that `unzipped` is a list, but `c` is a function. That's the most crucial difference, the point of the excercise, and it should be emphasized. For instance, you can see this difference in the fact that in `tz`, we have to glue -together the two instances of `unzipped` with an explicit `List.append`. +together the two instances of `unzipped` with an explicit (and +relatively inefficient) `List.append`. In the `tc` version of the task, we simply compose `c` with itself: `c o c = fun x -> c (c x)`. Why use the identity function as the initial continuation? Well, if -you have already constructed the list "abSd", what's the next step in -the recipe to produce the desired result (which is the same list, -"abSd")? Clearly, the identity continuation. +you have already constructed the initial list `"abSd"`, what's the next +step in the recipe to produce the desired result, i.e, the very same +list, `"abSd"`? Clearly, the identity continuation. A good way to test your understanding is to figure out what the continuation function `c` must be at the point in the computation when `tc` is called with the first argument `"Sd"`. Two choices: is it -`fun x -> a::b::x`, or it is `fun x -> b::a::x`? -The way to see if you're right is to execute the following -command and see what happens: +`fun x -> a::b::x`, or it is `fun x -> b::a::x`? The way to see if +you're right is to execute the following command and see what happens: tc ['S'; 'd'] (fun x -> 'a'::'b'::x);; There are a number of interesting directions we can go with this task. -The task was chosen because the computation can be viewed as a +The reason this task was chosen is because it can be viewed as a simplified picture of a computation using continuations, where `'S'` plays the role of a control operator with some similarities to what is -often called `shift`. In the analogy, the list portrays a string of -functional applications, where `[f1; f2; f3; x]` represents `f1(f2(f3 -x))`. The limitation of the analogy is that it is only possible to -represent computations in which the applications are always -right-branching, i.e., the computation `((f1 f2) f3) x` cannot be -directly represented. +often called `shift`. In the analogy, the input list portrays a +sequence of functional applications, where `[f1; f2; f3; x]` represents +`f1(f2(f3 x))`. The limitation of the analogy is that it is only +possible to represent computations in which the applications are +always right-branching, i.e., the computation `((f1 f2) f3) x` cannot +be directly represented. One possibile development is that we could add a special symbol `'#'`, and then the task would be to copy from the target `'S'` only back to the closest `'#'`. This would allow the task to simulate delimited -continuations (for right-branching computations). +continuations with embedded prompts. + +The reason the task is well-suited to the list zipper is in part +because the list monad has an intimate connection with continuations. +The following section explores this connection. We'll return to the +list task after talking about generalized quantifiers below. + -The task is well-suited to the list zipper because the list monad has -an intimate connection with continuations. The following section -makes this connection. We'll return to the list task after talking -about generalized quantifiers below.