X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=from_list_zippers_to_continuations.mdwn;h=1ef5f9c065fc670a15e571c3b36eb2ba15092d77;hp=d6eb68571d278c2ed77db97e5fc0317b13784684;hb=0feebbaaa58403d836d7ea6166cf709dd3faf1a8;hpb=49e6889d3ceb77526298a84549df44871caaf7a0 diff --git a/from_list_zippers_to_continuations.mdwn b/from_list_zippers_to_continuations.mdwn index d6eb6857..1ef5f9c0 100644 --- a/from_list_zippers_to_continuations.mdwn +++ b/from_list_zippers_to_continuations.mdwn @@ -26,7 +26,7 @@ In linguistic terms, this is a kind of anaphora resolution, where `'S'` is functioning like an anaphoric element, and the preceding string portion is the antecedent. -This deceptively simple task gives rise to some mind-bending complexity. +This task can give rise to considerable complexity. Note that it matters which 'S' you target first (the position of the * indicates the targeted 'S'): @@ -58,7 +58,7 @@ versus * ~~> ... -Aparently, this task, as simple as it is, is a form of computation, +Apparently, this task, as simple as it is, is a form of computation, and the order in which the `'S'`s get evaluated can lead to divergent behavior. @@ -75,10 +75,10 @@ entire list has been unzipped (and so the zipped half of the zipper is empty). type 'a list_zipper = ('a list) * ('a list);; let rec tz (z : char list_zipper) = - match z with - | (unzipped, []) -> List.rev(unzipped) (* Done! *) - | (unzipped, 'S'::zipped) -> tz ((List.append unzipped unzipped), zipped) - | (unzipped, target::zipped) -> tz (target::unzipped, zipped);; (* Pull zipper *) + match z with + | (unzipped, []) -> List.rev(unzipped) (* Done! *) + | (unzipped, 'S'::zipped) -> tz ((List.append unzipped unzipped), zipped) + | (unzipped, target::zipped) -> tz (target::unzipped, zipped);; (* Pull zipper *) # tz ([], ['a'; 'b'; 'S'; 'd']);; - : char list = ['a'; 'b'; 'a'; 'b'; 'd'] @@ -92,7 +92,7 @@ Task completed. One way to see exactly what is going on is to watch the zipper in action by tracing the execution of `tz`. By using the `#trace` directive in the OCaml interpreter, the system will print out the -arguments to `tz` each time it is (recurcively) called. Note that the +arguments to `tz` each time it is (recursively) called. Note that the lines with left-facing arrows (`<--`) show (recursive) calls to `tz`, giving the value of its argument (a zipper), and the lines with right-facing arrows (`-->`) show the output of each recursive call, a @@ -117,7 +117,7 @@ The nice thing about computations involving lists is that it's so easy to visualize them as a data structure. Eventually, we want to get to a place where we can talk about more abstract computations. In order to get there, we'll first do the exact same thing we just did with -concrete zipper using procedures. +concrete zipper using procedures instead. 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, @@ -134,17 +134,17 @@ 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 -(or group of steps) a **continuation** of the recipe. So in this +(or group of steps) a **continuation** of the previous steps. 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 as a continuation: a function describing how to finish building a 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. +list (not a zipper!) and a continuation `k` (it's conventional to use `k` for continuation variables) and return a processed list. The structure and the behavior will follow that of `tz` above, with some small but interesting differences. We've included the orginal `tz` to facilitate detailed comparison: @@ -155,11 +155,11 @@ some small but interesting differences. We've included the orginal | (unzipped, 'S'::zipped) -> tz ((List.append unzipped unzipped), zipped) | (unzipped, target::zipped) -> tz (target::unzipped, zipped);; (* Pull zipper *) - let rec tc (l: char list) (c: (char list) -> (char list)) = + let rec tc (l: char list) (k: (char list) -> (char list)) = match l with - | [] -> List.rev (c []) - | 'S'::zipped -> tc zipped (fun tail -> c (c tail)) - | target::zipped -> tc zipped (fun tail -> target::(c tail));; + | [] -> List.rev (k []) + | 'S'::zipped -> tc zipped (fun tail -> k (k tail)) + | target::zipped -> tc zipped (fun tail -> target::(k tail));; # tc ['a'; 'b'; 'S'; 'd'] (fun tail -> tail);; - : char list = ['a'; 'b'; 'a'; 'b'] @@ -167,30 +167,30 @@ some small but interesting differences. We've included the orginal # tc ['a'; 'S'; 'b'; 'S'] (fun tail -> tail);; - : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b'] -To emphasize the parallel, I've re-used the names `zipped` and +To emphasize the parallel, we've re-used the names `zipped` and `target`. The trace of the procedure will show that these variables take on the same values in the same series of steps as they did during the execution of `tz` above. There will once again be one initial and four recursive calls to `tc`, and `zipped` will take on the values `"bSd"`, `"Sd"`, `"d"`, and `""` (and, once again, on the final call, -the first `match` clause will fire, so the the variable `zipper` will +the first `match` clause will fire, so the the variable `zipped` will not be instantiated). -I have not called the functional argument `unzipped`, although that is +We have not called the functional argument `unzipped`, although that is 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 +list, but `k` 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 (and -relatively inefficient) `List.append`. -In the `tc` version of the task, we simply compose `c` with itself: -`c o c = fun tail -> c (c tail)`. +together the two instances of `unzipped` with an explicit (and, +computationally speaking, relatively inefficient) `List.append`. +In the `tc` version of the task, we simply compose `k` with itself: +`k o k = fun tail -> k (k tail)`. A call `tc ['a'; 'b'; 'S'; 'd']` yields a partially-applied function; it still waits for another argument, a continuation of type `char list -> char list`. We have to give it an "initial continuation" to get started. Here we supply *the identity function* as the initial continuation. Why did we choose that? Well, if you have already constructed the initial list `"abSd"`, what's the desired continuation? What's the next step in the recipe to produce the desired result, i.e, the very same list, `"abSd"`? Clearly, the identity function. 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 +continuation function `k` must be at the point in the computation when `tc` is called with the first argument `"Sd"`. Two choices: is it `fun tail -> 'a'::'b'::tail`, or it is `fun tail -> 'b'::'a'::tail`? The way to see if you're right is to execute the following command and see what happens: @@ -214,7 +214,6 @@ continuations with embedded `prompt`s (also called `reset`s). 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. +We'll explore this next.