X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;f=from_lists_to_continuations.mdwn;h=c67d5e86c173063a1c7c455b9ada08ea09ad373c;hb=13d59901132acc7a49220d84363227d2f7d1f476;hp=0b3f4643cbcb890edd85565be664670dc38de7d6;hpb=8e50e0b21cef6e6d5aba8265bf2aa0adbf6647d2;p=lambda.git diff --git a/from_lists_to_continuations.mdwn b/from_lists_to_continuations.mdwn index 0b3f4643..c67d5e86 100644 --- a/from_lists_to_continuations.mdwn +++ b/from_lists_to_continuations.mdwn @@ -1,4 +1,3 @@ - Refunctionalizing zippers: from lists to continuations ------------------------------------------------------ @@ -7,7 +6,7 @@ to continuations is to re-functionalize a zipper. Then the concreteness and understandability of the zipper provides a way of understanding and equivalent treatment using continuations. -Let's work with lists of chars for a change. To maximize readability, we'll +Let's work with lists of `char`s for a change. To maximize readability, we'll indulge in an abbreviatory convention that "abSd" abbreviates the list `['a'; 'b'; 'S'; 'd']`. @@ -76,9 +75,9 @@ guarantees termination, and a final string without any `'S'` in it. This is a task well-suited to using a zipper. We'll define a function `tz` (for task with zippers), which accomplishes the task by mapping a -char list zipper to a char list. We'll call the two parts of the +`char list zipper` to a `char list`. We'll call the two parts of the zipper `unzipped` and `zipped`; we start with a fully zipped list, and -move elements to the zipped part by pulling the zipped down until the +move elements to the zipped part by pulling the zipper down until the entire list has been unzipped (and so the zipped half of the zipper is empty).
@@ -132,8 +131,8 @@ to get there, we'll first do the exact same thing we just did with concrete zipper using procedures. Think of a list as a procedural recipe: `['a'; 'b'; 'S'; 'd']` -is the result of the computation `a::(b::(S::(d::[])))` (or, in our old -style, `makelist a (makelist b (makelist S (makelist c empty)))`). +is the result of the computation `'a'::('b'::('S'::('d'::[])))` (or, in our old +style, `make_list 'a' (make_list 'b' (make_list 'S' (make_list 'd' empty)))`). The recipe for constructing the list goes like this:@@ -148,13 +147,14 @@ 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 +zipper---the part we've already unzipped---as a continuation: a function describing how to finish building the list. We'll write a new function, `tc` (for task with continuations), that will take an input list (not a zipper!) and a continuation and return a processed list. @@ -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.