[lambda.git] / week11.mdwn

[[!toc]]

##List Zippers##

Say you've got some moderately-complex function for searching through a list, for example:

        let find_nth (test : 'a -> bool) (n : int) (lst : 'a list) : (int * 'a) ->
                let rec helper (position : int) n lst =
                        match lst with
                        | [] -> failwith "not found"
                        | x :: xs when test x -> (if n = 1
                                then (position, x)
                                else helper (position + 1) (n - 1) xs
                        )
                        | x :: xs -> helper (position + 1) n xs
                in helper 0 n lst;;

This searches for the `n`th element of a list that satisfies the predicate `test`, and returns a pair containing the position of that element, and the element itself. Good. But now what if you wanted to retrieve a different kind of information, such as the `n`th element matching `test`, together with its preceding and succeeding elements? In a real situation, you'd want to develop some good strategy for reporting when the target element doesn't have a predecessor and successor; but we'll just simplify here and report them as having some default value:

        let find_nth' (test : 'a -> bool) (n : int) (lst : 'a list) (default : 'a) : ('a * 'a * 'a) ->
                let rec helper (predecessor : 'a) n lst =
                        match lst with
                        | [] -> failwith "not found"
                        | x :: xs when test x -> (if n = 1
                                then (predecessor, x, match xs with [] -> default | y::ys -> y)
                                else helper x (n - 1) xs
                        )
                        | x :: xs -> helper x n xs
                in helper default n lst;;

This duplicates a lot of the structure of `find_nth`; it just has enough different code to retrieve different information when the matching element is found. But now what if you wanted to retrieve yet a different kind of information...?

Ideally, there should be some way to factor out the code to find the target element---the `n`th element of the list satisfying the predicate `test`---from the code that retrieves the information you want once the target is found. We might build upon the initial `find_nth` function, since that returns the *position* of the matching element. We could hand that result off to some other function that's designed to retrieve information of a specific sort surrounding that position. But suppose our list has millions of elements, and the target element is at position 600512. The search function will already have traversed 600512 elements of the list looking for the target, then the retrieval function would have to *start again from the beginning* and traverse those same 600512 elements again. It could go a bit faster, since it doesn't have to check each element against `test` as it traverses. It already knows how far it has to travel. But still, this should seem a bit wasteful.

Here's an idea. What if we had some way of representing a list as "broken" at a specific point. For example, if our base list is:

        [10; 20; 30; 40; 50; 60; 70; 80; 90]

we might imagine the list "broken" at position 3 like this (positions are numbered starting from 0):

                    40;
                30;     50;
            20;             60;
        [10;                    70;
                                    80;
                                        90]

Then if we move one step forward in the list, it would be "broken" at position 4:

                        50;
                    40;     60;
                30;             70;
            20;                     80;
        [10;                            90]

If we had some convenient representation of these "broken" lists, then our search function could hand *that* off to the retrieval function, and the retrieval function could start right at the position where the list was broken, without having to start at the beginning and traverse many elements to get there. The retrieval function would also be able to inspect elements both forwards and backwards from the position where the list was "broken".

The kind of data structure we're looking for here is called a **list zipper**. To represent our first broken list, we'd use two lists: (1) containing the elements in the left branch, preceding the target element, *in the order reverse to their appearance in the base list*. (2) containing the target element and the rest of the list, in normal order. So:

                    40;
                30;     50;
            20;             60;
        [10;                    70;
                                    80;
                                        90]

would be represented as `([30; 20; 10], [40; 50; 60; 70; 80; 90])`. To move forward in the base list, we pop the head element `40` off of the head element of the second list in the zipper, and push it onto the first list, getting `([40; 30; 20; 10], [50; 60; 70; 80; 90])`. To move backwards again, we pop off of the first list, and push it onto the second. To reconstruct the base list, we just "move backwards" until the first list is empty. (This is supposed to evoke the image of zipping up a zipper; hence the data structure's name.)

We had some discussio in seminar of the right way to understand the "zipper" metaphor. In the case of the list zipper, I think it's best to think of the tab of the zipper being here:

                 t
                  a
                   b
                    40;
                30;     50;
            20;             60;
        [10;                    70;
                                    80;
                                        90]

And imagine that you're just seeing the left half of a real-zipper, rotated 60 degrees counter-clockwise. When the list is all "zipped up", we've "move backwards" to the state where the first element is targetted:

        ([], [10; 20; 30; 40; 50; 60; 70; 80; 90])

However you understand the "zipper" metaphor, this is a very handy datastructure, and it will become even more handy when we translate it over to more complicated base structures, like trees. To help get a good conceptual grip on how to do that, it's useful to introduce a kind of symbolism for talking about zippers. This is just a metalanguage notation, for us theorists; we don't need our programs to interpret the notation. We'll use a specification like this:

        [10; 20; 30; *; 50; 60; 70; 80; 90], * filled by 40

to represent a list zipper where the break is at position 3, and the element occupying that position is 40. For a list zipper, this is implemented using the pairs-of-lists structure described above.


##Tree Zippers##

Now how could we translate a zipper-like structure over to trees? What we're aiming for is a way to keep track of where we are in a tree, in the same way that the "broken" lists let us keep track of where we are in the base list. In a particular application, you may only need to implement this for binary trees---trees where internal nodes always have exactly two subtrees. But to get the guiding idea, it's helpful first to think about trees that permit nodes to have many subtrees. Suppose we have the following tree:

                                 1000
                            /      |  \
                         /         |    \
                      /            |      \
                   /               |        \
                /                  |          \
               500                920          950
            /   |    \          /  |  \      /  |  \
         20     50     80      91  92  93   94  95  96
        1 2 3  4 5 6  7 8 9

and we want to represent that we're at the node marked `50`. We might use the following metalanguage notation to specify this:

        {parent = ...; siblings = [node 20; *; node 80]}, * filled by node 50

This is modeled on the notation suggested above for list zippers. Here `node 20` refers not to a `int` label associated with that node, but rather to the whole subtree rooted at that node:

          20
         / | \
        1  2  3

Similarly for `node 50` and `node 80`. We haven't said yet what goes in the `parent = ...` slot. Well, the parent of a subtree targetted on `node 50` should intuitively be a tree targetted on `node 500`:

        {parent = ...; siblings = [*; node 920; node 950]}, * filled by node 500

And the parent of that targetted subtree should intuitively be a tree targetted on `node 1000`:

        {parent = None; siblings = [*]}, * filled by node 1000

This tree has no parents because it's the root of the base tree. Fully spelled out, then, our tree targetted on `node 50` would be:

        {
           parent = {
              parent = {
                 parent = None;
                 siblings = [*]
              }, * filled by node 1000;
              siblings = [*; node 920; node 950]
           }, * filled by node 500;
           siblings = [node 20; *; node 80]
        }, * filled by node 50

In fact, there's some redundancy in this structure, at the points where we have `* filled by node 1000` and `* filled by node 500`. Most of `node 1000`---with the exception of any label attached to node `1000` itself---is determined by the rest of this structure; and so too with `node 500`. So we could really work with:

        {
           parent = {
              parent = {
                 parent = None;
                 siblings = [*]
              }, label for * position (at node 1000);
              siblings = [*; node 920; node 950]
           }, label for * position (at node 500);
           siblings = [node 20; *; node 80]
        }, * filled by node 50

Or, if we only had labels on the leafs of our tree:

        {
           parent = {
              parent = {
                 parent = None;
                 siblings = [*]
              },
              siblings = [*; node 920; node 950]
           },
           siblings = [node 20; *; node 80]
        }, * filled by node 50

We're understanding the `20` here in `node 20` to just be a metalanguage marker to help us theorists keep track of which node we're referring to. We're supposing the tree structure itself doesn't associate any informative labelling information with those nodes. It only associates informative labels with the tree leafs. (We haven't represented any such labels in our diagrams.)

We still do need to keep track of what fills the outermost targetted position---`* filled by node 50`---because that contain a subtree of arbitrary complexity, that is not determined by the rest of this data structure.

For simplicity, I'll continue to use the abbreviated form:

        {parent = ...; siblings = [node 20; *; node 80]}, * filled by node 50

But that should be understood as standing for the more fully-spelled-out structure. Structures of this sort are called **tree zippers**, for a reason that will emerge. They should already seem intuitively similar to list zippers, though, at least in what we're using them to represent. I think it may initially be more helpful to call these **targetted trees**, though, and so will be switching back and forth between this different terms.

Moving left in our targetted tree that's targetted on `node 50` would be a matter of shifting the `*` leftwards:

        {parent = ...; siblings = [*; node 50; node 80]}, * filled by node 20

and similarly for moving right. If the sibling list is implemented as a list zipper, you should already know how to do that. If one were designing a tree zipper for a more restricted kind of tree, however, such as a binary tree, one would probably not represent siblings with a list zipper, but with something more special-purpose and economical.

Moving downward in the tree would be a matter of constructing a tree targetted on some child of `node 20`, with the first part of the targetted tree above as its parent:

        {
           parent = {parent = ...; siblings = [*; node 50; node 80]};
           siblings = [*; leaf 2; leaf 3]
        }, * filled by leaf 1

How would we move upward in a tree? Well, we'd build a regular, untargetted tree with a root node---let's call it `20`---and whose children are given by the outermost sibling list in the targetted tree above, after inserting the targetted subtree into the `*` position:

               node 20
            /     |    \
         /        |      \
        leaf 1  leaf 2  leaf 3

We'll call this new untargetted tree `node 20`. The result of moving upward from our previous targetted tree, targetted on `leaf 1`, would be the outermost `parent` element of that targetted tree, with `node 20` being the subtree that fills that parent's target position `*`:

        {
           parent = ...;
           siblings = [*; node 50; node 80]
        }, * filled by node 20

Or, spelling that structure out fully:

        {
           parent = {
              parent = {
                 parent = None;
                 siblings = [*]
              },
              siblings = [*; node 920; node 950]
           },
           siblings = [*; node 50; node 80]
        }, * filled by node 20

Moving upwards yet again would get us:

        {
           parent = {
              parent = None;
              siblings = [*]
           },
           siblings = [*; node 920; node 950]
        }, * filled by node 500

where `node 500` refers to a tree built from a root node whose children are given by the list `[*; node 50; node 80]`, with `node 20` inserted into the `*` position. Moving upwards yet again would get us:

        {
           parent = None;
           siblings = [*]
        }, * filled by node 1000

where the targetted element is the root of our base tree. Like the "moving backward" operation for the list zipper, this "moving upward" operation is supposed to be reminiscent of closing a zipper, and that's why these data structures are called zippers.

We haven't given you a real implementation of the tree zipper, but only a suggestive notation. We have however told you enough that you should be able to implement it yourself. Or if you're lazy, you can read:

*       [[!wikipedia Zipper (data structure)]]
*       Huet, Gerard. ["Functional Pearl: The Zipper"](http://www.st.cs.uni-sb.de/edu/seminare/2005/advanced-fp/docs/huet-zipper.pdf) Journal of Functional Programming 7 (5): 549-554, September 1997.
*       As always, [Oleg](http://okmij.org/ftp/continuations/Continuations.html#zipper) takes this a few steps deeper.

##Same-fringe using a tree zipper##

Recall back in [[Assignment4]], we asked you to enumerate the "fringe" of a leaf-labeled tree. Both of these trees:

            .                .
           / \              / \
          .   3            1   .
         / \                  / \
        1   2                2   3

have the same fringe: `[1;2;3]`. We also asked you to write a function that determined when two trees have the same fringe. The way you approached that back then was to enumerate each tree's fringe, and then compare the two lists for equality. Today, and then again in a later class, we'll encounter two new ways to approach the problem of determining when two trees have the same fringe.


Supposing you did work out an implementation of the tree zipper, then one way to determine whether two trees have the same fringe would be: go downwards (and leftwards) in each tree as far as possible. Compare the targetted leaves. If they're different, stop because the trees have different fringes. If they're the same, then for each tree, move rightward if possible; if it's not (because you're at the rightmost position in a sibling list), more upwards then try again to move rightwards. Repeat until you are able to move rightwards. Once you do move rightwards, go downwards (and leftwards) as far as possible. Then you'll be targetted on the next leaf in the tree's fringe. The operations it takes to get to "the next leaf" may be different for the two trees. For example, in these trees:

            .                .
           / \              / \
          .   3            1   .
         / \                  / \
        1   2                2   3

you won't move upwards at the same steps. Keep comparing "the next leafs" until they are different, or you exhaust the leafs of only one of the trees (then again the trees have different fringes), or you exhaust the leafs of both trees at the same time, without having found leafs with different labels. In this last case, the trees have the same fringe.

If your trees are very big---say, millions of leaves---you can imagine how this would be quicker and more memory-efficient than traversing each tree to construct a list of its fringe, and then comparing the two lists so built to see if they're equal. For one thing, the zipper method can abort early if the fringes diverge early, without needing to traverse or build a list containing the rest of each tree's fringe.

Let's sketch the implementation of this. We won't provide all the details for an implementation of the tree zipper, but we will sketch an interface for it.

First, we define a type for leaf-labeled, binary trees:

        type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)

Next, the interface for our tree zippers. We'll help ourselves to OCaml's **record types**. These are nothing more than tuples with a pretty interface. Instead of saying:

        # type blah = Blah of (int * int * (char -> bool));;

and then having to remember which element in the triple was which:

        # let b1 = Blah (1, (fun c -> c = 'M'), 2);;
        Error: This expression has type int * (char -> bool) * int
       but an expression was expected of type int * int * (char -> bool)
        # (* damnit *)
        # let b1 = Blah (1, 2, (fun c -> c = 'M'));;
        val b1 : blah = Blah (1, 2, <fun>)

records let you attach descriptive labels to the components of the tuple:

        # type blah_record = { height : int; weight : int; char_tester : char -> bool };;
        # let b2 = { height = 1; weight = 2; char_tester = fun c -> c = 'M' };;
        val b2 : blah_record = {height = 1; weight = 2; char_tester = <fun>}
        # let b3 = { height = 1; char_tester = (fun c -> c = 'K'); weight = 3 };; (* also works *)
        val b3 : blah_record = {height = 1; weight = 3; char_tester = <fun>}

These were the strategies to extract the components of an unlabeled tuple:

        let h = fst some_pair;; (* accessor functions fst and snd are only predefined for pairs *)

        let (h, w, test) = b1;; (* works for arbitrary tuples *)

        match b1 with
        | (h, w, test) -> ...;; (* same as preceding *)

Here is how you can extract the components of a labeled record:

        let h = b2.height;; (* handy! *)

        let {height = h; weight = w; char_tester = test} = b2
        in (* go on to use h, w, and test ... *)

        match test with
        | {height = h; weight = w; char_tester = test} ->
                (* go on to use h, w, and test ... *)

Anyway, using record types, we might define the tree zipper interface like so:

        type 'a starred_tree = Root | Starring_Left of 'a starred_pair | Starring_Right of 'a starred_pair
        and 'a starred_pair = { parent : 'a starred_tree; sibling: 'a tree }
        and 'a zipper = { tree : 'a starred_tree; filler: 'a tree };;

        let rec move_botleft (z : 'a zipper) : 'a zipper =
            (* returns z if the targetted node in z has no children *)
            (* else returns move_botleft (zipper which results from moving down and left in z) *)

<!--
            let {tree; filler} = z
            in match filler with
            | Leaf _ -> z
            | Node(left, right) ->
                let zdown = {tree = Starring_Left {parent = tree; sibling = right}; filler = left}
                in move_botleft zdown
            ;;
-->

        let rec move_right_or_up (z : 'a zipper) : 'a zipper option =
            (* if it's possible to move right in z, returns Some (the result of doing so) *)
            (* else if it's not possible to move any further up in z, returns None *)
            (* else returns move_right_or_up (result of moving up in z) *)

<!--
            let {tree; filler} = z
            in match tree with
            | Starring_Left {parent; sibling = right} -> Some {tree = Starring_Right {parent; sibling = filler}; filler = right}
            | Root -> None
            | Starring_Right {parent; sibling = left} ->
                let z' = {tree = parent; filler = Node(left, filler)}
                in move_right_or_up z'
            ;;
-->

The following function takes an 'a tree and returns an 'a zipper focused on its root:

        let new_zipper (t : 'a tree) : 'a zipper =
            {tree = Root; filler = t}
            ;;

Finally, we can use a mutable reference cell to define a function that enumerates a tree's fringe until it's exhausted:

        let make_fringe_enumerator (t: 'a tree) =
            (* create a zipper targetting the root of t *)
            let zstart = new_zipper t
            in let zbotleft = move_botleft zstart
            (* create a refcell initially pointing to zbotleft *)
            in let zcell = ref (Some zbotleft)
            (* construct the next_leaf function *)
            in let next_leaf () : 'a option =
                match !zcell with
                | None -> (* we've finished enumerating the fringe *)
                    None
                | Some z -> (
                    (* extract label of currently-targetted leaf *)
                    let Leaf current = z.filler
                    (* update zcell to point to next leaf, if there is one *)
                    in let () = zcell := match move_right_or_up z with
                        | None -> None
                        | Some z' -> Some (move_botleft z')
                    (* return saved label *)
                    in Some current
                )
            (* return the next_leaf function *)
            in next_leaf
            ;;

Here's an example of `make_fringe_enumerator` in action:

        # let tree1 = Leaf 1;;
        val tree1 : int tree = Leaf 1
        # let next1 = make_fringe_enumerator tree1;;
        val next1 : unit -> int option = <fun>
        # next1 ();;
        - : int option = Some 1
        # next1 ();;
        - : int option = None
        # next1 ();;
        - : int option = None
        # let tree2 = Node (Node (Leaf 1, Leaf 2), Leaf 3);;
        val tree2 : int tree = Node (Node (Leaf 1, Leaf 2), Leaf 3)
        # let next2 = make_fringe_enumerator tree2;;
        val next2 : unit -> int option = <fun>
        # next2 ();;
        - : int option = Some 1
        # next2 ();;
        - : int option = Some 2
        # next2 ();;
        - : int option = Some 3
        # next2 ();;
        - : int option = None
        # next2 ();;
        - : int option = None

You might think of it like this: `make_fringe_enumerator` returns a little subprogram that will keep returning the next leaf in a tree's fringe, in the form `Some ...`, until it gets to the end of the fringe. After that, it will keep returning `None`.

Using these fringe enumerators, we can write our `same_fringe` function like this:

        let same_fringe (t1 : 'a tree) (t2 : 'a tree) : bool =
                let next1 = make_fringe_enumerator t1
                in let next2 = make_fringe_enumerator t2
                in let rec loop () : bool =
                        match next1 (), next2 () with
                        | Some a, Some b when a = b -> loop ()
                        | None, None -> true
                        | _ -> false
                in loop ()
                ;;

The auxiliary `loop` function will keep calling itself recursively until a difference in the fringes has manifested itself---either because one fringe is exhausted before the other, or because the next leaves in the two fringes have different labels. If we get to the end of both fringes at the same time (`next1 (), next2 ()` matches the pattern `None, None`) then we've established that the trees do have the same fringe.

The technique illustrated here with our fringe enumerators is a powerful and important one. It's an example of what's sometimes called **cooperative threading**. A "thread" is a subprogram that the main computation spawns off. Threads are called "cooperative" when the code of the main computation and the thread fixes when control passes back and forth between them. (When the code doesn't control this---for example, it's determined by the operating system or the hardware in ways that the programmer can't predict---that's called "preemptive threading.") Cooperative threads are also sometimes called *coroutines* or *generators*.

With cooperative threads, one typically yields control to the thread, and then back again to the main program, multiple times. Here's the pattern in which that happens in our `same_fringe` function:

        main program            next1 thread            next2 thread
        ------------            ------------            ------------
        start next1
        (paused)                        starting
        (paused)                        calculate first leaf
        (paused)                        <--- return it
        start next2                     (paused)                        starting
        (paused)                        (paused)                        calculate first leaf
        (paused)                        (paused)                        <-- return it
        compare leaves          (paused)                        (paused)
        call loop again         (paused)                        (paused)
        call next1 again        (paused)                        (paused)
        (paused)                        calculate next leaf     (paused)
        (paused)                        <-- return it           (paused)
        ... and so on ...

If you want to read more about these kinds of threads, here are some links:

<!-- *  [[!wikipedia Computer_multitasking]]
*       [[!wikipedia Thread_(computer_science)]] -->
*       [[!wikipedia Coroutine]]
*       [[!wikipedia Iterator]]
*       [[!wikipedia Generator_(computer_science)]]
*       [[!wikipedia Fiber_(computer_science)]]
<!-- *  [[!wikipedia Green_threads]]
*       [[!wikipedia Protothreads]] -->

The way we built cooperative threads here crucially relied on two heavyweight tools. First, it relied on our having a data structure (the tree zipper) capable of being a static snapshot of where we left off in the tree whose fringe we're enumerating. Second, it relied on our using mutable reference cells so that we could update what the current snapshot (that is, tree zipper) was, so that the next invocation of the `next_leaf` function could start up again where the previous invocation left off.

In coming weeks, we'll learn about a different way to create threads, that relies on **continuations** rather than on those two tools. All of these tools are inter-related. As Oleg says, "Zipper can be viewed as a delimited continuation reified as a data structure." These different tools are also inter-related with monads. Many of these tools can be used to define the others. We'll explore some of the connections between them in the remaining weeks, but we encourage you to explore more.


##Introducing Continuations##

A continuation is "the rest of the program." Or better: an **delimited continuation** is "the rest of the program, up to a certain boundary." An **undelimited continuation** is "the rest of the program, period."

Even if you haven't read specifically about this notion (for example, even if you haven't read Chris and Ken's work on using continuations in natural language semantics), you'll have brushed shoulders with it already several times in this course.

A naive semantics for atomic sentences will say the subject term is of type `e`, and the predicate of type `e -> t`, and that the subject provides an argument to the function expressed by the predicate.

Monatague proposed we instead take subject terms to be of type `(e -> t) -> t`, and that now it'd be the predicate (still of type `e -> t`) that provides an argument to the function expressed by the subject.

If all the subject did then was supply an `e` to the `e -> t` it receives as an argument, we wouldn't have gained anything we weren't already able to do. But of course, there are other things the subject can do with the `e -> t` it receives as an argument. For instance, it can check whether anything in the domain satisfies that `e -> t`; or whether most things do; and so on.

This inversion of who is the argument and who is the function receiving the argument is paradigmatic of working with continuations. We did the same thing ourselves back in the early days of the seminar, for example in our implementation of pairs. In the untyped lambda calculus, we identified the pair `(x, y)` with a function:

        \handler. handler x y

A pair-handling function would accept the two elements of a pair as arguments, and then do something with one or both of them. The important point here is that the handler was supplied as an argument to the pair. Eventually, the handler would itself be supplied with arguments. But only after it was supplied as an argument to the pair. This inverts the order you'd expect about what is the data or argument, and what is the function that operates on it.

Consider a complex computation, such as:

        1 + 2 * (1 - g (3 + 4))

Part of this computation---`3 + 4`---leads up to supplying `g` with an argument. The rest of the computation---`1 + 2 * (1 - ___)`---waits for the result of applying `g` to that argument and will go on to do something with it (inserting the result into the `___` slot). That "rest of the computation" can be regarded as a function:

        \result. 1 + 2 * (1 - result)

This function will be applied to whatever is the result of `g (3 + 4)`. So this function can be called the *continuation* of that application of `g`. For some purposes, it's useful to be able to invert the function/argument order here, and rather than supplying the result of applying `g` to the continuation, we instead supply the continuation to `g`. Well, not to `g` itself, since `g` only wants a single `int` argument. But we might build some `g`-like function which accepts not just an `int` argument like `g` does, but also a continuation argument.

Go back and read the material on "Aborting a Search Through a List" in [[Week4]] for an example of doing this.

In very general terms, the strategy is to work with functions like this:

        let g' k (i : int) =
                ... do stuff ...
                ... if you want to abort early, supply an argument to k ...
                ... do more stuff ...
                ... normal result
        in let gcon = fun result -> 1 + 2 * (1 - result)
        in gcon (g' gcon (3 + 4))

It's a convention to use variables like `k` for continuation arguments. If the function `g'` never supplies an argument to its contination argument `k`, but instead just finishes evaluating to a normal result, that normal result will be delivered to `g'`'s continuation `gcon`, just as happens when we don't pass around any explicit continuation variables.

The above snippet of OCaml code doesn't really capture what happens when we pass explicit continuation variables. For suppose that inside `g'`, we do supply an argument to `k`. That would go into the `result` parameter in `gcon`. But then what happens once we've finished evaluating the application of `gcon` to that `result`? In the OCaml snippet above, the final value would then bubble up through the context in the body of `g'` where `k` was applied, and eventually out to the final line of the snippet, where it once again supplied an argument to `gcon`. That's not what happens with a real continuation. A real continuation works more like this:

        let g' k (i : int) =
                ... do stuff ...
                ... if you want to abort early, supply an argument to k ...
                ... do more stuff ...
                ... normal result
        in let gcon = fun result ->
                let final_value = 1 + 2 * (1 - result)
                in end_program_with final_value
        in gcon (g' gcon (3 + 4))

So once we've finished evaluating the application of `gcon` to a `result`, the program is finished. (This is how undelimited continuations behave. We'll discuss delimited continuations later.)

So now, guess what would be the result of doing the following:

        let g' k (i : int) =
                1 + k i
        in let gcon = fun result ->
                let final_value = (1, result)
                in end_program_with final_value
        in gcon (g' gcon (3 + 4))

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