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 and equivalent treatment using continuations.
+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
Note that it matters which 'S' you target first (the position of the *
indicates the targeted 'S'):
- t "aSbS"
+ t "aSbS"
*
- ~~> t "aabS"
+ ~~> t "aabS"
*
~~> "aabaab"
t "aSbS"
*
- ~~> t "aSbaSb"
+ ~~> t "aSbaSb"
*
~~> t "aabaSb"
*
`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 zipped part by pulling the zipper down until the
+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) =
+ 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, 'S'::zipped) -> tz ((List.append unzipped unzipped), zipped)
| (unzipped, target::zipped) -> tz (target::unzipped, zipped);; (* Pull zipper *)
# tz ([], ['a'; 'b'; 'S'; 'd']);;
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
+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.
-
-<pre>
-# #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']
-</pre>
+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:
-
-<pre>
-(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)
-</pre>
+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
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 the list. We'll write a new
+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) =
+ 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, '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)) =
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:
+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 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 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
list task after talking about generalized quantifiers below.
-