X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;f=from_lists_to_continuations.mdwn;fp=from_lists_to_continuations.mdwn;h=0000000000000000000000000000000000000000;hb=49e6889d3ceb77526298a84549df44871caaf7a0;hp=d6eb68571d278c2ed77db97e5fc0317b13784684;hpb=87abf2f5507409788f94c7f9ac0efdec212ec11d;p=lambda.git diff --git a/from_lists_to_continuations.mdwn b/from_lists_to_continuations.mdwn deleted file mode 100644 index d6eb6857..00000000 --- a/from_lists_to_continuations.mdwn +++ /dev/null @@ -1,220 +0,0 @@ -Refunctionalizing zippers: from lists to continuations ------------------------------------------------------- - -If zippers are continuations reified (defuntionalized), then one route -to continuations is to re-functionalize a zipper. Then the -concreteness and understandability of the zipper provides a way of -understanding an equivalent treatment using continuations. - -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']`. - -We will set out to compute a deceptively simple-seeming **task: given a -string, replace each occurrence of 'S' in that string with a copy of -the string up to that point.** - -We'll define a function `t` (for "task") that maps strings to their -updated version. - -Expected behavior: - - t "abSd" ~~> "ababd" - - -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. -Note that it matters which 'S' you target first (the position of the * -indicates the targeted 'S'): - - t "aSbS" - * - ~~> t "aabS" - * - ~~> "aabaab" - -versus - - t "aSbS" - * - ~~> t "aSbaSb" - * - ~~> t "aabaSb" - * - ~~> "aabaaabab" - -versus - - t "aSbS" - * - ~~> t "aSbaSb" - * - ~~> t "aSbaaSbab" - * - ~~> t "aSbaaaSbaabab" - * - ~~> ... - -Aparently, 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. - -For now, we'll agree to always evaluate the leftmost `'S'`, which -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 -zipper `unzipped` and `zipped`; we start with a fully zipped list, and -move elements to the unzipped part by pulling the zipper down until the -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 *) - - # tz ([], ['a'; 'b'; 'S'; 'd']);; - - : char list = ['a'; 'b'; 'a'; 'b'; 'd'] - - # tz ([], ['a'; 'S'; 'b'; 'S']);; - - : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b'] - -Note that this implementation enforces the evaluate-leftmost rule. -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 -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 -simple list. - - # #trace tz;; - t1 is now traced. - # tz ([], ['a'; 'b'; 'S'; 'd']);; - tz <-- ([], ['a'; 'b'; 'S'; 'd']) - tz <-- (['a'], ['b'; 'S'; 'd']) (* Pull zipper *) - tz <-- (['b'; 'a'], ['S'; 'd']) (* Pull zipper *) - tz <-- (['b'; 'a'; 'b'; 'a'], ['d']) (* Special step *) - tz <-- (['d'; 'b'; 'a'; 'b'; 'a'], []) (* Pull zipper *) - tz --> ['a'; 'b'; 'a'; 'b'; 'd'] (* Output reversed *) - tz --> ['a'; 'b'; 'a'; 'b'; 'd'] - tz --> ['a'; 'b'; 'a'; 'b'; 'd'] - tz --> ['a'; 'b'; 'a'; 'b'; 'd'] - tz --> ['a'; 'b'; 'a'; 'b'; 'd'] - - : char list = ['a'; 'b'; 'a'; 'b'; 'd'] - -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. - -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, -`make_list 'a' (make_list 'b' (make_list 'S' (make_list 'd' empty)))`). The -recipe for constructing the list goes like this: - -> (0) Start with the empty list [] -> (1) make a new list whose first element is 'd' and whose tail is the list constructed in step (0) -> (2) make a new list whose first element is 'S' and whose tail is the list constructed in step (1) -> ----------------------------------------- -> (3) make a new list whose first element is 'b' and whose tail is the list constructed in step (2) -> (4) make a new list whose first element is 'a' and whose tail is the list constructed in step (3) - -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 -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 -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. -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: - - 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 *) - - let rec tc (l: char list) (c: (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));; - - # tc ['a'; 'b'; 'S'; 'd'] (fun tail -> tail);; - - : char list = ['a'; 'b'; 'a'; 'b'] - - # 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 -`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 -not be instantiated). - -I 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 -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)`. - -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 -`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: - - tc ['S'; 'd'] (fun tail -> 'a'::'b'::tail);; - -There are a number of interesting directions we can go with this task. -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 continuation operator. (It works like the Scheme operators `shift` or `control`; the differences between them don't manifest themselves in this example.) 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 way to extend this exercise would be to add a special symbol `'#'`, -and then the task would be to copy from the target `'S'` only back to -the closest `'#'`. This would allow our task to simulate delimited -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. - -