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 and 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 zipped 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, `makelist a (makelist b (makelist S (makelist c 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 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. 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 x -> c (c x))
| target::zipped -> tc zipped (fun x -> target::(c x));;

# tc ['a'; 'b'; 'S'; 'd'] (fun x -> x);;
- : char list = ['a'; 'b'; 'a'; 'b']

# tc ['a'; 'S'; 'b'; 'S'] (fun x -> x);;
- : char list = ['a'; 'a'; 'b'; 'a'; 'a'; 'b']
```